Mathematics Extension 2 – Graphs

Graphs

Basic Curves

Before graphing more difficult functions, it is useful to revise the graphs of commonly graphed functions. In the extension 2 course, students will often be asked to graph functions that are compositions of these equations.

Linear Functions

Linear equations are most commonly expressed in one of two forms:

i) $Latex formula$

To graph these, simply plot the points $Latex formula$ and $Latex formula$ and join them.
(Note that these values can be obtained just by substituting $Latex formula$ and $Latex formula$ into the equation. We can also substitute any other $Latex formula$ or $Latex formula$ values to find two points on the line and join them. There is no need to memorise the formulas for these points.)

 Students will need to plot: Intersection with x axis Intersection with y axis Important values: Gradient $Latex formula$

ii) $Latex formula$

To graph these, we can plot the points $Latex formula$ and $Latex formula$ and join them. Again, any two points on the line can be used.

 Students will need to plot: Intersection with x axis Intersection with y axis

Quadratic functions are most commonly expressed in one of two forms:

(i) $Latex formula$( and $Latex formula$)

To plot these functions, plot the vertex and at least one other point to give the general shape of the parabola, and then join appropriately. Always mark in at least one, but preferably two points apart from the vertex to show the value of a.
(Any three points define a unique parabola)

 Students will need to plot: Vertex $Latex formula$ Focus $Latex formula$ Directrix $Latex formula$ Important values: Focal length $Latex formula$

(ii) $Latex formula$

Finding the vertex

1. Complete the square:$Latex formula$ has vertex $Latex formula$
2. The x value of the vertex is given by the formula $Latex formula$.
Determine $Latex formula$ value by substituting the x value. This yields

$Latex formula$

Finding the x-intercepts

1. Factorise: $Latex formula$ has x intercepts $Latex formula$ and $Latex formula$
2. Use the quadratic formula $Latex formula$ to find the two x-intercepts

Finding the y-intercept

The graph has y-intercept value $Latex formula$

 Students will need to plot: VertexIntersections with x axis Intersection with y axis Note: As a general rule of thumb, plot at least 2 or 3 points for any polynomial function. Intercepts are included. Important values: $Latex formula$Ordinate of axis of symmetry.

Hints for graphing parabolas:

For both forms, if $Latex formula$, the parabola is ‘concave up’ (or makes a happy face).
For both forms, $Latex formula$, the parabola is ‘concave down’.
Doing a quick check that theparabola curves in the right direction and that the y-intercept will end up at the right place can prevent silly mistakes.

Cubic Functions

$Latex formula$

Finding the turning points

Differentiating yields $Latex formula$.
When $Latex formula$, the gradient of the graph is $Latex formula$ and so there is a stationary point, which is a turning point.
Use the quadratic formula to solve for the x values when $Latex formula$

Finding the y-intercept

The graph has y-intercept value $Latex formula$

Finding the x-intercepts (optional)

1. Factorise: $Latex formula$ has x intercepts $Latex formula$, $Latex formula$ and $Latex formula$
2. Guess a suitable value $Latex formula$ that yields $Latex formula$. Use polynomial division to factorise the expression into linear and quadratic factors $Latex formula$ and then solve the quadratic accordingly.
3. Use the cubic formula

to find the three x-intercepts

Hints:

If $Latex formula$, the cubic points down at the left and up at the right.

If $Latex formula$, the cubic points up at the left and down at the right.

Doing a quick check that the y-intercept will end up at the right place can prevent silly mistakes.

If $Latex formula$, i.e. there is a double root at $Latex formula$, then there is a turning point at $Latex formula$.

 Students will need to plot: Turning points Intersection with y axis Intersections with x axis (optional if the x-intercept values cannot be determined using the first 2 methods)

Quartic Functions

$Latex formula$

Finding the turning points

Differentiating yields $Latex formula$.
When $Latex formula$, the gradient of the graph is $Latex formula$ and so there is a stationary point, which is a turning point.
Use the ‘guess and check’ method to find one value of x for which $Latex formula$, and then use the quadratic formula to solve for the other x values when $Latex formula$

Finding the y-intercept

The graph has y-intercept value $Latex formula$

Finding the x-intercepts (optional)

1. Factorise: $Latex formula$ has x intercepts $Latex formula$, $Latex formula$, $Latex formula$ and $Latex formula$
2. Guess a suitable value $Latex formula$ that yields $Latex formula$. Use polynomial division to factorise the expression into linear and cubic factors $Latex formula$ and then solve the cubic accordingly.

Hints:

If $Latex formula$, the quartic points up at both ends.

If $Latex formula$, the quartic points down at both ends.

Doing a quick check that the y-intercept will end up at the right place can prevent silly mistakes.

If $Latex formula$, i.e. there is a double root at $Latex formula$, then there is a turning point at $Latex formula$

If $Latex formula$, i.e. there is a triple root at $Latex formula$, then there is a horizontal point of inflexion at $Latex formula$

 Students will need to plot: Turning points Intersection with y axis Intersections with x axis (optional if the x-intercept values cannot easily be determined)

Circles

Equations of the form $Latex formula$ determine a circle in the number plane. To graph the circle, complete the square for both $Latex formula$ and $Latex formula$ to obtain an equation of the form $Latex formula$. The centre of the circle is given by $Latex formula$ and the radius is $Latex formula$.

 Students will need to plot: Center of circle Intersections with x and y axes Note: If there are no intersections with the axes, plot another point on the circle to show the radius. Important values: Radius $Latex formula$

Rectangular Hyperbolae: $Latex formula$

(i) $Latex formula$$Latex formula$

Then $Latex formula$ for some c.

The vertices are given by $Latex formula$, the foci by $Latex formula$ and the asymptotes are the coordinate axes.

(ii) $Latex formula$$Latex formula$

Then $Latex formula$ for some c.

The vertices are given by $Latex formula$, the foci by $Latex formula$ and the asymptotes are the coordinate axes.

 Students will need to plot: Vertices Foci At least one other point on the curve

Hints:
Check that the graph is in the right quadrants: 1 & 3 for $Latex formula$, and 2 & 4 for $Latex formula$
Check that the coordinate axes are dotted to show that they are asymptotes, and that the graph approaches the axes.

Exponential Functions

$Latex formula$$Latex formula$

y-intercept value of 0, horizontal asymptote $Latex formula$ as $Latex formula$

 Students will need to plot: Intersection with y axis Asymptotic approach to x axis At least one other point on the curve.

Logarithmic Functions

$Latex formula$$Latex formula$

These graphs are equivalent to the graph of $Latex formula$, $Latex formula$. They have an x-intercept value of 0, and a vertical asymptote $Latex formula$ as $Latex formula$.

 Students will need to plot: Intersection with x axis Asymptotic approach to y axis At least one other point on the curve

Trigonometric Functions

(i) Sine graph: $Latex formula$

(ii) Cosine graph: $Latex formula$

(iii) Tangent graph: $Latex formula$

(iv) Secant graph: $Latex formula$

(v) Cosecant graph: $Latex formula$

(vi) Cotangent graph: $Latex formula$

 Students will need to plot: Intersections with x axis Intersection with y axis All turning points All asymptotes

Inverse Trigonometric Functions

(vii) Sine inverse graph: $Latex formula$

$Latex formula$
 Students will need to plot: Intersection with axes Endpoints

(viii) Cosine inverse graph: $Latex formula$

$Latex formula$
 Students will need to plot: Intersection with axes Endpoints

(ix) Tan inverse graph: $Latex formula$

$Latex formula$
 Students will need to plot: Intersection with axes Both asymptotes Note: It is also advisable to plot the two points of inflexion, but any other point is also sufficient.

Sketching graphs by Addition and Subtraction of Ordinates

There are three transformations regularly used in curve sketching. These are translation, dilation and reflection.

Functions of the form $Latex formula$ can be graphed by initially graphing $Latex formula$ and then performing one translation. We can use just one transformation because the transformation applied to each point (adding $Latex formula$ units to the y ordinate) is the same.

Such functions can be graphed using either one of two methods:

1. Translate the graph:

To graph $Latex formula$ where $Latex formula$ is positive, translate the graph $Latex formula$ units up. This is the same as adding $Latex formula$ units to every point on the graph.

To graph $Latex formula$ where $Latex formula$ is positive, translate the graph $Latex formula$ units down.This is the same as subtracting $Latex formula$ units from every point on the graph.

2. Move the x-axis:

To graph $Latex formula$ where $Latex formula$ is positive, translate the x axis $Latex formula$ units down.

To graph $Latex formula$ where $Latex formula$ is positive, translate the x axis $Latex formula$ units up.
Note that moving the $Latex formula$ axis down increases the value of each $Latex formula$ ordinate, and moving the $Latex formula$ axis up decreases the value of each $Latex formula$

It is interesting to note that graphing $Latex formula$ is equivalent to graphing $Latex formula$, or decreasing the $Latex formula$ value of each point in the graph of $Latex formula$ by c. This is why we can move the $Latex formula$ axis $Latex formula$ units down to obtain our graph. Similarly, then, we can also obtain the graph of $Latex formula$ by moving the $Latex formula$ axis $Latex formula$ units to the left, or by moving the entire graph $Latex formula$ units to the right.

 Students will need to plot: Intersections with both axes The translation of any important points or asymptotes in the plot of$Latex formula$

The next example illustrates the process of graphing functions of the form $Latex formula$. Both methods are shown.

Example 1

Sketch on the same axes:

a) $Latex formula$

b) $Latex formula$

c) $Latex formula$

Solution 1

Method 1:

Start off with the graph of $Latex formula$. To obtain the graph of $Latex formula$, take a few points and add 3 to the y ordinate of each. Plot the new points to obtain the general shape of the curve and join as appropriate. Do the same to plot $Latex formula$.

Method 2:

Start off with the graph of $Latex formula$. To obtain the graph of $Latex formula$, move the x axis down by 3 units so that the new x axis is at $Latex formula$. Now relabel the axes so that the new x axis is $Latex formula$ and the old one is $Latex formula$.

This question shows obvious problem with this method. It is difficult to obtain the graphs on the same set of axes, something that is commonly requested by HSC questions. However, it is a quick and useful method for graphing single functions. If you find this method easier to follow, use it where applicable.

Slightly more difficult questions will ask students to graph functions of the form $Latex formula$. These also require the addition and subtraction of ordinates. However, points at different x values will need to be treated differently.
To graph these, initially graph both $Latex formula$ and $Latex formula$ and then add or subtract the ordinates on the two graphs to sketch the new curve. The easiest way to determine the shape of the new curve is to calculate the new $Latex formula$ordinates for turning points or x intercepts for $Latex formula$ and $Latex formula$ and plot the points. The turning points oftentranslate to turning points or points of inflexion in the new function and the intercepts can show the shape of the curve between turning points. Note that the calculation does not need to be done numerically. It can be done graphically by adding or subtracting the distances from the x axis.
Note: Instead of using subtraction of ordinates for plotting $Latex formula$, you can add the negative of the second curve.

Important things to consider include:

(i) Any asymptotes of $Latex formula$ result in asymptotes for $Latex formula$. To determine the equation of the new asymptote, simply perform ordinate addition or subtraction to the asymptote and $Latex formula$ as required. Of course, the same applies for any asymptotes of $Latex formula$.

 Students will need to plot: Intersections with both axes Any asymptotes Turning points or points of inflexion (depending on ease of calculation)

Below is an example that shows how to graph functions of the form $Latex formula$.

Example 2

Sketch the graphs of $Latex formula$ and $Latex formula$ for $Latex formula$ on the same axes. Hence sketch the graph of $Latex formula$ for $Latex formula$.

Solution 2

First, any asymptotes must be identified. In this case, as $Latex formula$, $Latex formula$ so that $Latex formula$ also approaches negative infinity.

A few important points must then be plotted. These included places where $Latex formula$ and $Latex formula$ are zero or equal, as these result in well defined points. In this case, $Latex formula$ at $Latex formula$, $Latex formula$ at $Latex formula$, $Latex formula$ and $Latex formula$ and $Latex formula$ at some unknown x ordinate. The resulting points can be determined by adding or subtracting lengths, as shown by the dotted lines.

Finally, a few more points can be obtained by simply adding or subtracting lengths at regular intervals along the x axis. At $Latex formula$ and $Latex formula$ the addition of $Latex formula$ to $Latex formula$ can be used as shown.

Reflecting Graphs in the Coordinate Axes

Another simple graphical transformation is reflection. The point $Latex formula$ is the reflection of the point $Latex formula$ over the x-axis. Hence it follows that the graph $Latex formula$ is the reflection of the graph $Latex formula$ over the x-axis. Similarly, the graph $Latex formula$ is the reflection of the graph $Latex formula$ over the y-axis.

This reflective property can be used to sketch graphs of functions such as $Latex formula$, where $Latex formula$ if $Latex formula$ and $Latex formula$ if $Latex formula$.

Note: Be careful not to reflect over the wrong axis! One way to remember conceptually is to note that $Latex formula$ modifies the x ordinate, so everything should be reflected over the axis. Similarly, $Latex formula$ is equivalent to $Latex formula$ which modifies the $Latex formula$ ordinate, so everything should be reflected over the x-axis. If you are still unsure, numerically test a few x values and see if the $Latex formula$ values have the correct sign.

 Students will need to plot: The transformed position of any important points on the original function $Latex formula$.

Below is an example that illustrates the process of graphing a function that is the reflection of another function.

Example 3 (HSC 1993 Question 4a)

Let $Latex formula$. On separate diagrams sketch the graphs of the following functions.

a) $Latex formula$

b) $Latex formula$

Solution 3

a)

b) To graph $Latex formula$, simply take your graph from part a) and reflect anything below the x axis above the x axis. The final graph is shown below in black, and can be compared to the graph in part a), which is shown in blue:

The next example illustrates the difference between the equations that result in reflecting over the $Latex formula$ and $Latex formula$ axes.

Example 4

Sketch on the same set of axes:

a) $Latex formula$

b) $Latex formula$

c) $Latex formula$

d) $Latex formula$

Solution 4

The graph of part a) can be seen in red. This is a typical sine curve.

The graph of part b) can be seen in green. Notice how the shape of the curve is preserved. The curve is simply shifted up one unit.

The graph of part c) can be seen in blue. Because $Latex formula$ modifies the x ordinate, the x values corresponding to each y value must be exchanged with their negative and so the graph of $Latex formula$ is reflected over the y axis.

The graph of part d) can be seen in green. Because $Latex formula$ modifies the y ordinate, the graph of $Latex formula$ is reflected over the x axis.

Note: Make sure you understand why c) and d) look the way that they do! If you are unsure, try plotting a few points yourself.

Sketching graphs by Multiplication of Ordinates

Functions of the form $Latex formula$ are transformations of the graph $Latex formula$, where every ordinate is multiplied by the same value. Thus the transformation is uniform.

The graph of $Latex formula$ where $Latex formula$ can be seen as the graph of $Latex formula$ stretched out by a factor of $Latex formula$ along the $Latex formula$ axis. This is a dilation.

The graph of $Latex formula$ where $Latex formula$ is the graph of $Latex formula$ stretched out by a factor of $Latex formula$ along the $Latex formula$ axis.

The easiest way to plot these is to note the important points or components of the original function $Latex formula$, multiply all their $Latex formula$ ordinates by $Latex formula$, plot the resulting points and join, following a curve similar to the original.

 Students will need to plot: The transformed position of any important points on the original function $Latex formula$.

Below is an example that shows the process of graphing functions of the form $Latex formula$

Example 5

Sketch on the same set of axes:

a) $Latex formula$

b) $Latex formula$

c) $Latex formula$

d) $Latex formula$

Solution 5

The vertices, x intercepts and y intercepts should always be noted first. In this case, they total to three points, which defines a unique parabola. Therefore they can be joined in a parabolic shape with the given vertex. This yields the following plot:

Note how when $Latex formula$, the graph appears ‘fatter’, and when $Latex formula$ the graph appears ‘skinnier’.

Slightly more difficult functions are of the form $Latex formula$. When graphing $Latex formula$, individual ordinates must be multiplied together.

Important things to consider include:

(i) $Latex formula$ If $Latex formula$, then $Latex formula$. The same holds for $Latex formula$.

(ii) $Latex formula$ If $Latex formula$, then $Latex formula$. The same holds for $Latex formula$.

(iii) Asymptotes: If $Latex formula$ has an asymptote $Latex formula$, then the graph of $Latex formula$ has an asymptote at $Latex formula$.

(iv) Critical points: If $Latex formula$ is undefined at $Latex formula$, then $Latex formula$ is also undefined at $Latex formula$These will need to be indicated by something such as an open circle.

 Students will need to plot:Intersections with the x axis Intersections with the y axis Asymptotes Any easily determined turning points

The next example demonstrates the process of graphing functions of the form $Latex formula$.

Example 6 (HSC 1996 Question 4b)

a) On the same set of axes, sketch and label clearly the graphs of the functions $Latex formula$ and $Latex formula$.

b) Hence, on a different set of axes, without using calculus, sketch and label clearly the graph of the function $Latex formula$.

Solution 6

To plot the graph of the function $Latex formula$, a few points with the same x ordinates should be taken from both graphs, the resulting y ordinate calculated and the new point plotted. To make calculations simpler, use points on $Latex formula$ and $Latex formula$ with y ordinates of 0 and 1. Ideally, around 5 to 10 points with a wide range of x ordinates (a few in the middle and a few near the endpoints) should be plotted to estimate the shape of the curve before being joined to create a new curve.

Notice how the asymptote in $Latex formula$ leads to an asymptote in the graph of $Latex formula$.

Reciprocal Functions and Sketching Graphs using Division of Ordinates

Functions of the form $Latex formula$ illustrate another way of transforming a graph: by dividing ordinate. The graph of $Latex formula$ can be determined by graphing $Latex formula$ and then taking the reciprocal of every ordinate and plotting the new point. By doing this many times, the general shape of the new curve can be determined.

Important things to consider:

(i) $Latex formula$and $Latex formula$ have the same sign.

(ii) $Latex formula$If $Latex formula$, then $Latex formula$. The curves $Latex formula$ and $Latex formula$ will only meet when $Latex formula$.

(iii) $Latex formula$If $Latex formula$, then $Latex formula$ is undefined for that x value. The graph of $Latex formula$ will have a vertical asymptote at that x value.

(iv) As $Latex formula$, $Latex formula$.
However, if $Latex formula$ as $Latex formula$, the $Latex formula$ value at $Latex formula$ is undefined and an open circle should be plotted at $Latex formula$.

(v) Minimum turning points of $Latex formula$ correspond to maximum turning points of $Latex formula$, and vice versa.

 Students will need to plot: Intersections with the x axis Intersections with the y axis AsymptotesTurning points Critical points

Below are some examples that illustrate the process of graphing functions of the form $Latex formula$.

Example 7

a) Draw a neat sketch of the function $Latex formula$. State the coordinates of its vertex and of its points of intersection with both coordinate axes.

b) Hence draw a neat sketch of the function $Latex formula$. Clearly indicate on your sketch the equations of the vertical asymptotes and the coordinates of any stationary points.

Solution 7

a)

Vertex: $Latex formula$
Points of intersection with axes:
Substituting $Latex formula$ gives $Latex formula$: intersection with y axis $Latex formula$
Substituting $Latex formula$ gives $Latex formula$ and $Latex formula$: intersections with x axis $Latex formula$ and $Latex formula$

b) Where $Latex formula$ there is a vertical asymptote to $Latex formula$. Here this occurs at $Latex formula$ and $Latex formula$.

The maximum turning point of $Latex formula$, $Latex formula$ occurring at $Latex formula$, corresponds to a minimum turning point of $Latex formula$ at $Latex formula$.
Since this is the only stationary point of , only has one stationary point and the shape of the graph can be determined by approaching the asymptotes accordingly:

For $Latex formula$, $Latex formula$ is positive and for all other $Latex formula$, $Latex formula$ is negative. The same sign rules also apply to $Latex formula$. As x approaches positive and negative infinity, $Latex formula$ approaches negative infinity so that $Latex formula$ approaches $Latex formula$ from below.
As x approaches 2 and 6, $Latex formula$ approaches $Latex formula$ so that $Latex formula$ diverges to infinity and approaches the asymptotes.

Example 8

Sketch on the same set of axes:

a) $Latex formula$

b) $Latex formula$

Solution 8

Again x intercepts of $Latex formula$ correspond to asymptotes of $Latex formula$. As $Latex formula$, $Latex formula$ so that $Latex formula$. Again the sign of $Latex formula$ is the same as the sign of $Latex formula$ so that as x approaches 1 from above, $Latex formula$approaches positive infinity and as x approaches 1 from below, $Latex formula$ approaches negative infinity.

As x approaches from above, $Latex formula$ so that $Latex formula$approaches $Latex formula$ from below. However, at $Latex formula$ the function is undefined so there is an open circle.

More difficult functions are of the form $Latex formula$. The graph of $Latex formula$ can be determined by using either one of two methods:

(i) Graph $Latex formula$ and $Latex formula$ and divide individual ordinates.

(ii) Graph $Latex formula$ and $Latex formula$ and multiply individual ordinates.

The features of both graphs need to be considered when graphing $Latex formula$. Asymptotes in the graphs of $Latex formula$ and $Latex formula$ lead to asymptotes in the graph of $Latex formula$. Similarly, turning points in both functions affect the shape of $Latex formula$.

 Students will need to plot: Intersections with the x axis Intersections with the y axis Asymptotes Easily determined turning points Critical points

The next example demonstrates how to use both methods to graph functions of the form $Latex formula$.

Example 9

Without using calculus, sketch the graph of $Latex formula$.

Solution 9

Method 1:

Method 2:

Method 1:

The graph of $Latex formula$ can be seen in red, the graph of $Latex formula$ in blue and the final graph in black. The x-intercept of $Latex formula$ at $Latex formula$ corresponds to an asymptote to the graph of $Latex formula$. Furthermore, as $Latex formula$ approaches 0, $Latex formula$ approaches negative infinity so that $Latex formula$ approaches 0. However, since it is not defined at $Latex formula$ there is an open circle at $Latex formula$.

After these points of interest- division by $Latex formula$ and infinity, stationary points- have been examined, normal division of ordinates can be used to determine the shape of the curve. The points $Latex formula$, $Latex formula$, $Latex formula$ and $Latex formula$ were used in this case.

Method 2:

The graph of $Latex formula$ can be seen in red, the graph of $Latex formula$ (determined in example 8) in blue and the final graph in black. Here the asymptote to $Latex formula$ corresponds to an asymptote to $Latex formula$, and since as $Latex formula$ approaches $Latex formula$ both $Latex formula$ and $Latex formula$ approach 0, $Latex formula$ also approaches 0. However, there is an open circle since the function is undefined at that point. Multiplication of individual ordinates can be used to show the general shape of the rest of the curve.

If $Latex formula$ where $Latex formula$ and $Latex formula$ are polynomials, it may be easier to first perform the division algorithm before graphing the curve.

This is particularly useful if the degree of $Latex formula$ is equal to, or one greater than, the degree of $Latex formula$. If the degree of $Latex formula$ is equal to the degree of $Latex formula$, there will be a vertical asymptote. If the degree of $Latex formula$ is one greater than the degree of $Latex formula$, there will be an oblique asymptote and a vertical one.

Example 10 (EBHS Trial 1990 Question 4(i) b-c)

a) Determine all the asymptotes of the curve $Latex formula$

b) Draw a careful sketch of the curve $Latex formula$

Solution 10

a) Using the division algorithm yields $Latex formula$.
Therefore $Latex formula$ and as $Latex formula$, $Latex formula$ so that $Latex formula$.

The asymptotes are $Latex formula$ and $Latex formula$.
Notice that since the degree of the numerator is greater than the degree of the denominator, there is both an oblique asymptote $Latex formula$ and a vertical asymptote $Latex formula$.

b) Using the division algorithm reveals that the function doesn’t require division of ordinates at all. It is simply the sum of two common functions: $Latex formula$ and $Latex formula$. Using the calculated asymptotes and plotting the graphs of $Latex formula$ and $Latex formula$ simplifies the question even more. There remains only to determine intercepts, which can be done easily by substituting $Latex formula$ and $Latex formula$. This yields $Latex formula$ and $Latex formula$.

Note the inclusion of the asymptotic approaches to both asymptotes. These must be indicated on the graph.

Graphing functions of the form $Latex formula$ or $Latex formula$

Functions of the form $Latex formula$ are part of the next big class of graphs: graphs of composite functions. In this particular case, the function $Latex formula$ is applied to $Latex formula$, where $Latex formula$ is a function of x. It is important to understand conceptually how to graph these functions, because they are one of the families of composite functions that occur the most in this course.

There are a few important points to remember when graphing these functions (here we consider only non-negative values of y):

(i) For $Latex formula$, $Latex formula$ does not exist

(ii) For $Latex formula$, $Latex formula$.

(iii) For $Latex formula$, $Latex formula$so $Latex formula$.

(iv) For $Latex formula$, $Latex formula$.

(v) For $Latex formula$, $Latex formula$ so $Latex formula$.

(vi) As $Latex formula$, stationary points on the graph of $Latex formula$ have the same $Latex formula$ values as stationary points on the graph of $Latex formula$.

Note: Points (i) to (v) correspond to restricting the ranges of the $Latex formula$ ordinates for particular $Latex formula$ values, point (ii) also corresponds to determining the relationship between intercepts and point (vi) corresponds to finding the relationship between stationary points. These concepts become important when graphing other composite functions.

Another graph closely related to that of $Latex formula$ is that of $Latex formula$. Students often confuse the two expressions. The graph of $Latex formula$ is the graph of both $Latex formula$ and $Latex formula$.

One way to avoid the mistake of confusing the two is to remember that while in the graph of $Latex formula$ the ordinates must be non-negative, in the graph of $Latex formula$, the ordinates can be both positive and negative.

 Students will need to plot: Intersections with the x axis Intersections with the y axis The transformations of any important components of the original graph of $Latex formula$.

The following example illustrates the process of graphing functions of the form $Latex formula$ and $Latex formula$.

Example 11

Let $Latex formula$.

a) Sketch the graph of $Latex formula$ showing clearly the coordinates and nature of any stationary points and the equations of any asymptotes.

b) Hence sketch on the same axis the graph of $Latex formula$.

Solution 11

a)

$Latex formula$
Possible stationary points occur at $Latex formula$ i.e. $Latex formula$.
When $Latex formula$, $Latex formula$ and when $Latex formula$ $Latex formula$.
For $Latex formula$, $Latex formula$ and for $Latex formula$ $Latex formula$ so that there is a local minimum turning point at $Latex formula$ and a local maximum turning point at $Latex formula$.
As $Latex formula$ $Latex formula$ so that the x axis is an asymptote.
Putting all this information together yields the following graph:

b) Plotting $Latex formula$ is equivalent to plotting $Latex formula$ and $Latex formula$. Since zeros, turning points and asymptotes are preserved, these should be graphed first. After that, the general shape of the curve can be estimated by consider the shape of $Latex formula$. For $Latex formula$, the graph lies above the original curve but below $Latex formula$ and for $Latex formula$, the graph lies below the original curve but above $Latex formula$.

In this case, $Latex formula$ lies entirely between $Latex formula$ and $Latex formula$. $Latex formula$ is simply the reflection of $Latex formula$ over the x axis.

Graphing Functions of the Form $Latex formula$

Functions of the form $Latex formula$ are again composite functions, this time applying a polynomial function to a given function. Because polynomials are usually determined by turning points and intercepts, these become very important in plotting functions of the form $Latex formula$.

To graph functions of the form $Latex formula$, use calculus to determine the position and nature of stationary points. The general shape of the graph can then be determined by finding the sign of the gradient of the slope between stationary points (sign of $Latex formula$).

Some important points to consider when graphing these functions include:

(i) For $Latex formula$, $Latex formula$, meaning that x-intercepts of $Latex formula$ are also x-intercepts of $Latex formula$.

(ii) For $Latex formula$, $Latex formula$.

(iii) For $Latex formula$, $Latex formula$ for even $Latex formula$ and $Latex formula$ for odd $Latex formula$.

(iv) For $Latex formula$, $Latex formula$.

(v) For $Latex formula$, $Latex formula$.

(vi) For $Latex formula$, $Latex formula$ for even $Latex formula$ and $Latex formula$ for odd $Latex formula$.

(vii) $Latex formula$, so stationary points of $Latex formula$ have the same $Latex formula$coordinates as x-intercepts and $Latex formula$ coordinates of stationary points of $Latex formula$.

Note: Did you notice that the process of graphing these functions is very similar to that used when graphing functions of the forms $Latex formula$ and $Latex formula$? Points (i) to (vi) correspond to restricting the ranges of the $Latex formula$ ordinates for particular $Latex formula$ values, point (i) also corresponds to determining the relationship between intercepts and point (vii) corresponds to finding the relationship between stationary points.

 Students will need to plot: Intersections with the x axis Intersections with the y axis Turning points Asymptotes

Below is an example that illustrates the process of graphing functions of the form $Latex formula$.

Example 12 (HSC 1990 Question 3b)

Consider the functions f and g, defined by $Latex formula$ and $Latex formula$.

a) Sketch the hyperbola $Latex formula$, clearly labelling the horizontal and vertical asymptotes and the points of intersection with the x and y axes.

b) Find all turning points of $Latex formula$.

c) Using the same diagram as used in (a) sketch the curve $Latex formula$ clearly labelling it.

d) On a separate diagram sketch the curve given by $Latex formula$.

Solution 12

a) $Latex formula$ which has asymptotes $Latex formula$ and $Latex formula$.

Note that this is a reflection, scaling and translation of the graph $Latex formula$.

Since the sign in front of the fractional part of the function is negative, the graph must lie in the second and fourth ‘quadrants’ created by the asymptotes. It looks something like this:

b) $Latex formula$
To find possible turning points, we must differentiate.

$Latex formula$

Possible stationary points occur at $Latex formula$, i.e. $Latex formula$.
Since for $Latex formula$ $Latex formula$ and for $Latex formula$ $Latex formula$, there is a local minimum turning point at $Latex formula$. At this point, $Latex formula$.
The only stationary point is a local minimum turning point at $Latex formula$.

c) Note the minimum turning point at $Latex formula$, the approach to the asymptotes, how $Latex formula$ for $Latex formula$ and $Latex formula$ for $Latex formula$. These can all be used to determine the approximate shape of the graph.

d) The graph for d) is simply the reflection over the y axis of the graph for x.

Graphs of Composite Functions

Composite functions are functions of the form $Latex formula$, where $Latex formula$ and $Latex formula$ are both functions. A lot of graphing questions that are worth a lot of marks simply ask students to graph the composition of common functions, such as those outlined in the revision section.

Graphs of composite functions may be constructed by considering the properties of the basic functions involved.

These may include:

(i) domain and range

(ii) x-intercepts and y-intercepts

(iii) asymptotes

(iv) turning points and other stationary points

(v) whether the function is increasing or decreasing

In general, any interesting components of either $Latex formula$ or $Latex formula$ (such as those outlined in points (i) to (iv) above) should be reflected in the graph of their composition, $Latex formula$.

 Students will need to plot: Intersections with the x axis Intersections with the y axis The transformation of any interesting components of $Latex formula$ and $Latex formula$

The next example outlines the process of graphing composite functions.

Example 13 (STHS Trial 2001 Question 5a (ii))

Sketch the graph of $Latex formula$, showing all its important features. Do not use calculus.

Solution 13

Method 1:

Consider first the properties of $Latex formula$.

Its domain and range are both all real $Latex formula$, which means that it has no intercepts with the coordinate axes.

As $Latex formula$, $Latex formula$, as $Latex formula$, $Latex formula$, as $Latex formula$, $Latex formula$ and as $Latex formula$, $Latex formula$. This means that for $Latex formula$, the function is strictly decreasing, and for $Latex formula$, the function is strictly increasing. It has no turning points or stationary points.

Consider next the properties of $Latex formula$.

Its domain is all positive $Latex formula$ and its range is all real $Latex formula$. This means that is has no y-intercepts, but it has one x-intercept at $Latex formula$.
It has a vertical asymptote at $Latex formula$ with $Latex formula$ as $Latex formula$ and is strictly increasing over its domain.
It has no turning points or stationary points.

Together, these let us determine the properties of $Latex formula$:
Its domain is the set of all x such that $Latex formula$ is in the domain of $Latex formula$. Since only positive reals are in the domain of $Latex formula$, and $Latex formula$ is positive only for positive $Latex formula$, the domain of $Latex formula$ is all positive $Latex formula$. Its range is therefore all real .
It has a vertical asymptote at $Latex formula$.
Since $Latex formula$ is strictly increasing for positive $Latex formula$, $Latex formula$ is also strictly increasing.
Since both $Latex formula$ and $Latex formula$ have no turning points or stationary points, $Latex formula$ also does not have any.
The x-intercept of $Latex formula$ occurs at $Latex formula$, or $Latex formula$. It does not have any y-intercepts.

Now by substituting a few values of x into $Latex formula$ and comparing with the shapes of $Latex formula$ and $Latex formula$, the shape of the curve can be determined.

Using $Latex formula$, $Latex formula$, and $Latex formula$ yields:

(Note: The coordinates of the plotted points should only be used for personal reference. They should not be included in solutions to examination questions.)

The above method is a little tedious, but helps in building a greater grasp of the general shapes and behaviours of these graphs. It is therefore the recommended method. However, students running short on time could try the following method:

Method 2:

This method again requires the student to determine asymptotes and intercepts as in method 1. However, instead of considering the nature of the functions or numerically calculating values, students can simply graphically determine the positions of a few points by using lengths and then join the dots. This can be seen in the black dotted lines in the following figure:

Consider first the thick, vertical black dotted line. This has x-intercept $Latex formula$. The aim is to find the position of $Latex formula$.

First, determine the length of the thick vertical black line, $Latex formula$.

Then find the value of $Latex formula$ by making $Latex formula$ the x ordinate and reading up to $Latex formula$ using the thin vertical dotted black line, $Latex formula$.
Finally, find the position of $Latex formula$ by finding the point of intersection of the thick black vertical line, $Latex formula$ and the thin black horizontal line, $Latex formula$.

The same method can be used to determine the position of a number of points, as shown in green.

The obvious drawback with this method is that it is difficult to be precise. As a result, a lot of the finer details in the composite function would be lost. However, it is a method that can be used even in conjunction with method 1, to determine extra points when you aren’t sure about the shape of the graph or when the composite function is messy to enter into your calculator.

Using Implicit Differentiation in Curve Sketching

Some relations between $Latex formula$ and $Latex formula$ involve powers of functions of both $Latex formula$ and y. To graph these, it is useful to know the gradient at different points on the curve, as this can show the general shape of the curve. However, to find the gradients for these relations, implicit differentiation must be used.

The concept of implicit differentiation makes use of the chain rule as follows:

$Latex formula$
$Latex formula$
$Latex formula$

This shows how the gradient needs to be determined as a function of both the $Latex formula$ and $Latex formula$ values.

The following example shows how implicit differentiation can be used to determine the shape of a curve.

Example 14 (CSSA 1996 Trial Question 5a (i)-(ii))

Let k be a positive constant.

a) Show that $Latex formula$ defines y to be a monotonic decreasing function of x throughout its domain. Comment on the gradient of the tangent at any critical point on the curve.

b) Sketch the graph of the curve $Latex formula$ and hence on the same diagram sketch the graph of the curve $Latex formula$.

Solution 14

a) To determine the gradient of the function (and therefore the nature of its increasing and decreasing), we must somehow differentiate it with respect to x. However, since $Latex formula$ is not a straight-forward function of x, it makes sense to use implicit differentiation.
Differentiating implicitly with respect to x yields

$Latex formula$$Latex formula$$Latex formula$

Now since $Latex formula$, x and y must both be non-negative so that $Latex formula$ and by substituting back into the equation, $Latex formula$ and $Latex formula$.
Therefore $Latex formula$ and since $Latex formula$ for $Latex formula$, y is monotonically decreasing throughout its domain.

The critical points of the function are its endpoints, since there are no turning points.
At $Latex formula$ the gradient is undefined and there is a vertical tangent.
At $Latex formula$, $Latex formula$ so the gradient is 0 and there is a horizontal tangent.

b) The function is decreasing at a decreasing rate from $Latex formula$ to $Latex formula$. Therefore the graph looks something like this:

To obtain the graph of $Latex formula$, we note that since $Latex formula$ the domain is now the entire real line and since $Latex formula$ is even, $Latex formula$ (as a function of $Latex formula$) is also even. Therefore we reflect the above graph over the y axis to obtain a graph that looks something like this: