Graphs
Basic Curves
Contents
- 1 Basic Curves
- 2 Sketching graphs by Addition and Subtraction of Ordinates
- 3 Reflecting Graphs in the Coordinate Axes
- 4 Sketching graphs by Multiplication of Ordinates
- 5 Reciprocal Functions and Sketching Graphs using Division of Ordinates
- 6 Graphing functions of the form or
- 7 Graphing Functions of the Form
- 8 Graphs of Composite Functions
- 9 Using Implicit Differentiation in Curve Sketching
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)
To graph these, simply plot the points and and join them.
(Note that these values can be obtained just by substituting and into the equation. We can also substitute any other or 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 |
ii)
To graph these, we can plot the points and 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
Quadratic functions are most commonly expressed in one of two forms:
(i) ( and )
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 Focus Directrix Important values: Focal length |
(ii)
Finding the vertex
1. Complete the square: has vertex
2. The x value of the vertex is given by the formula .
Determine value by substituting the x value. This yields
Finding the x-intercepts
1. Factorise: has x intercepts and
2. Use the quadratic formula to find the two x-intercepts
Finding the y-intercept
The graph has y-intercept value
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: Ordinate of axis of symmetry. |
Hints for graphing parabolas:
For both forms, if , the parabola is ‘concave up’ (or makes a happy face).
For both forms, , 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
Finding the turning points
Differentiating yields .
When , the gradient of the graph is and so there is a stationary point, which is a turning point.
Use the quadratic formula to solve for the x values when
Finding the y-intercept
The graph has y-intercept value
Finding the x-intercepts (optional)
1. Factorise: has x intercepts , and
2. Guess a suitable value that yields . Use polynomial division to factorise the expression into linear and quadratic factors and then solve the quadratic accordingly.
3. Use the cubic formula
to find the three x-intercepts
Hints:
If , the cubic points down at the left and up at the right.
If , 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 , i.e. there is a double root at , then there is a turning point at .
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
Finding the turning points
Differentiating yields .
When , the gradient of the graph is 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 , and then use the quadratic formula to solve for the other x values when
Finding the y-intercept
The graph has y-intercept value
Finding the x-intercepts (optional)
1. Factorise: has x intercepts , , and
2. Guess a suitable value that yields . Use polynomial division to factorise the expression into linear and cubic factors and then solve the cubic accordingly.
Hints:
If , the quartic points up at both ends.
If , 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 , i.e. there is a double root at , then there is a turning point at
If , i.e. there is a triple root at , then there is a horizontal point of inflexion at
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 determine a circle in the number plane. To graph the circle, complete the square for both and to obtain an equation of the form . The centre of the circle is given by and the radius is .
Rectangular Hyperbolae:
(i) ,
Then for some c.
The vertices are given by , the foci by and the asymptotes are the coordinate axes.
(ii) ,
Then for some c.
The vertices are given by , the foci by 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 , and 2 & 4 for
Check that the coordinate axes are dotted to show that they are asymptotes, and that the graph approaches the axes.
Exponential Functions
,
y-intercept value of 0, horizontal asymptote as
Students will need to plot:
Intersection with y axis Asymptotic approach to x axis At least one other point on the curve. |
Logarithmic Functions
,
These graphs are equivalent to the graph of , . They have an x-intercept value of 0, and a vertical asymptote as .
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:
(ii) Cosine graph:
(iii) Tangent graph:
(iv) Secant graph:
(v) Cosecant graph:
(vi) Cotangent graph:
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:
Students will need to plot:
Intersection with axes Endpoints |
(viii) Cosine inverse graph:
Students will need to plot:
Intersection with axes Endpoints |
(ix) Tan inverse graph:
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 can be graphed by initially graphing and then performing one translation. We can use just one transformation because the transformation applied to each point (adding 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 where is positive, translate the graph units up. This is the same as adding units to every point on the graph.
To graph where is positive, translate the graph units down.This is the same as subtracting units from every point on the graph.
2. Move the x-axis:
To graph where is positive, translate the x axis units down.
To graph where is positive, translate the x axis units up.
Note that moving the axis down increases the value of each ordinate, and moving the axis up decreases the value of each
It is interesting to note that graphing is equivalent to graphing , or decreasing the value of each point in the graph of by c. This is why we can move the axis units down to obtain our graph. Similarly, then, we can also obtain the graph of by moving the axis units to the left, or by moving the entire graph 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 |
The next example illustrates the process of graphing functions of the form . Both methods are shown.
Example 1Sketch on the same axes: a) b) c) Solution 1Method 1: Start off with the graph of . To obtain the graph of , 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 . Method 2: Start off with the graph of . To obtain the graph of , move the x axis down by 3 units so that the new x axis is at . Now relabel the axes so that the new x axis is and the old one is . 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 . 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 and 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 ordinates for turning points or x intercepts for and 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 , you can add the negative of the second curve.
Important things to consider include:
(i) Any asymptotes of result in asymptotes for . To determine the equation of the new asymptote, simply perform ordinate addition or subtraction to the asymptote and as required. Of course, the same applies for any asymptotes of .
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 .
Reflecting Graphs in the Coordinate Axes
Another simple graphical transformation is reflection. The point is the reflection of the point over the x-axis. Hence it follows that the graph is the reflection of the graph over the x-axis. Similarly, the graph is the reflection of the graph over the y-axis.
This reflective property can be used to sketch graphs of functions such as , where if and if .
Note: Be careful not to reflect over the wrong axis! One way to remember conceptually is to note that modifies the x ordinate, so everything should be reflected over the axis. Similarly, is equivalent to which modifies the 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 values have the correct sign.
Students will need to plot:
The transformed position of any important points on the original function . |
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 . On separate diagrams sketch the graphs of the following functions. a) b) Solution 3a) b) To graph , 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 and axes.
Example 4Sketch on the same set of axes: a) b) c) d) Solution 4The 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 modifies the x ordinate, the x values corresponding to each y value must be exchanged with their negative and so the graph of is reflected over the y axis. The graph of part d) can be seen in green. Because modifies the y ordinate, the graph of 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 are transformations of the graph , where every ordinate is multiplied by the same value. Thus the transformation is uniform.
The graph of where can be seen as the graph of stretched out by a factor of along the axis. This is a dilation.
The graph of where is the graph of stretched out by a factor of along the axis.
The easiest way to plot these is to note the important points or components of the original function , multiply all their ordinates by , 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 . |
Below is an example that shows the process of graphing functions of the form
Example 5Sketch on the same set of axes: a) b) c) d) Solution 5The 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 , the graph appears ‘fatter’, and when the graph appears ‘skinnier’. |
Slightly more difficult functions are of the form . When graphing , individual ordinates must be multiplied together.
Important things to consider include:
(i) If , then . The same holds for .
(ii) If , then . The same holds for .
(iii) Asymptotes: If has an asymptote , then the graph of has an asymptote at .
(iv) Critical points: If is undefined at , then is also undefined at . 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 .
Example 6 (HSC 1996 Question 4b)a) On the same set of axes, sketch and label clearly the graphs of the functions and . b) Hence, on a different set of axes, without using calculus, sketch and label clearly the graph of the function . Solution 6To plot the graph of the function , 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 and 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 leads to an asymptote in the graph of . |
Reciprocal Functions and Sketching Graphs using Division of Ordinates
Functions of the form illustrate another way of transforming a graph: by dividing ordinate. The graph of can be determined by graphing 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) and have the same sign.
(ii) If , then . The curves and will only meet when .
(iii) If , then is undefined for that x value. The graph of will have a vertical asymptote at that x value.
(iv) As , .
However, if as , the value at is undefined and an open circle should be plotted at .
(v) Minimum turning points of correspond to maximum turning points of , 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 .
Example 7a) Draw a neat sketch of the function . 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 . Clearly indicate on your sketch the equations of the vertical asymptotes and the coordinates of any stationary points. Solution 7a) Vertex: b) Where there is a vertical asymptote to . Here this occurs at and . The maximum turning point of , occurring at , corresponds to a minimum turning point of at . For , is positive and for all other , is negative. The same sign rules also apply to . As x approaches positive and negative infinity, approaches negative infinity so that approaches from below. |
Example 8Sketch on the same set of axes: a) b) Solution 8Again x intercepts of correspond to asymptotes of . As , so that . Again the sign of is the same as the sign of so that as x approaches 1 from above, approaches positive infinity and as x approaches 1 from below, approaches negative infinity. As x approaches from above, so that approaches from below. However, at the function is undefined so there is an open circle. |
More difficult functions are of the form . The graph of can be determined by using either one of two methods:
(i) Graph and and divide individual ordinates.
(ii) Graph and and multiply individual ordinates.
The features of both graphs need to be considered when graphing . Asymptotes in the graphs of and lead to asymptotes in the graph of . Similarly, turning points in both functions affect the shape of .
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 .
Example 9Without using calculus, sketch the graph of . Solution 9Method 1: Method 2: Method 1: The graph of can be seen in red, the graph of in blue and the final graph in black. The x-intercept of at corresponds to an asymptote to the graph of . Furthermore, as approaches 0, approaches negative infinity so that approaches 0. However, since it is not defined at there is an open circle at . After these points of interest- division by and infinity, stationary points- have been examined, normal division of ordinates can be used to determine the shape of the curve. The points , , and were used in this case. Method 2: The graph of can be seen in red, the graph of (determined in example 8) in blue and the final graph in black. Here the asymptote to corresponds to an asymptote to , and since as approaches both and approach 0, 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 where and are polynomials, it may be easier to first perform the division algorithm before graphing the curve.
This is particularly useful if the degree of is equal to, or one greater than, the degree of . If the degree of is equal to the degree of , there will be a vertical asymptote. If the degree of is one greater than the degree of , there will be an oblique asymptote and a vertical one.
Graphing functions of the form or
Functions of the form are part of the next big class of graphs: graphs of composite functions. In this particular case, the function is applied to , where 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 , does not exist
(ii) For , .
(iii) For , so .
(iv) For , .
(v) For , so .
(vi) As , stationary points on the graph of have the same values as stationary points on the graph of .
Note: Points (i) to (v) correspond to restricting the ranges of the ordinates for particular 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 is that of . Students often confuse the two expressions. The graph of is the graph of both and .
One way to avoid the mistake of confusing the two is to remember that while in the graph of the ordinates must be non-negative, in the graph of , 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 . |
The following example illustrates the process of graphing functions of the form and .
Graphing Functions of the Form
Functions of the form 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 .
To graph functions of the form , 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 ).
Some important points to consider when graphing these functions include:
(i) For , , meaning that x-intercepts of are also x-intercepts of .
(ii) For , .
(iii) For , for even and for odd .
(iv) For , .
(v) For , .
(vi) For , for even and for odd .
(vii) , so stationary points of have the same coordinates as x-intercepts and coordinates of stationary points of .
Note: Did you notice that the process of graphing these functions is very similar to that used when graphing functions of the forms and ? Points (i) to (vi) correspond to restricting the ranges of the ordinates for particular 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 .
Example 12 (HSC 1990 Question 3b)Consider the functions f and g, defined by and . a) Sketch the hyperbola , clearly labelling the horizontal and vertical asymptotes and the points of intersection with the x and y axes. b) Find all turning points of . c) Using the same diagram as used in (a) sketch the curve clearly labelling it. d) On a separate diagram sketch the curve given by . Solution 12a) which has asymptotes and . Note that this is a reflection, scaling and translation of the graph . 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) Possible stationary points occur at , i.e. . c) Note the minimum turning point at , the approach to the asymptotes, how for and for . 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 , where and 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 or (such as those outlined in points (i) to (iv) above) should be reflected in the graph of their composition, .
Students will need to plot:
Intersections with the x axis Intersections with the y axis The transformation of any interesting components of and |
The next example outlines the process of graphing composite functions.
Using Implicit Differentiation in Curve Sketching
Some relations between and involve powers of functions of both 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:
This shows how the gradient needs to be determined as a function of both the and 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 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 and hence on the same diagram sketch the graph of the curve . Solution 14a) 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 is not a straight-forward function of x, it makes sense to use implicit differentiation. Now since , x and y must both be non-negative so that and by substituting back into the equation, and . The critical points of the function are its endpoints, since there are no turning points. b) The function is decreasing at a decreasing rate from to . Therefore the graph looks something like this: To obtain the graph of , we note that since the domain is now the entire real line and since is even, (as a function of ) is also even. Therefore we reflect the above graph over the y axis to obtain a graph that looks something like this: |