1. Sketching Curves

2. Introduction This Chapter focuses on sketching Graphs
We will also be looking at using them to solve Equations
There will also be some work on Graph transformations

3. Teachings for Exercise 4A

4. A cubic equation will take one of the following shapes
Sketching Curves
Sketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
A cubic equation will take one of the following shapes
For any x3 or
For any -x3 4A

5. Sketching Curves 4A Sketching Cubics Example If y = 0 or
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
Example
Sketch the graph of the function:
If y = 0 or
So x = 2, 1 or -1
(-1,0) (1,0) and (2,0)
If x = 0
So y = 2
(0,2) 4A

6. Sketching Curves 4A Sketching Cubics Example (-1,0) (1,0) (2,0) (0,2)
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
Example
Sketch the graph of the function:
(-1,0) (1,0) (2,0) (0,2) y or 2 x -1 1 2
If we substitute in x = 3, we get a value of y = 8. The curve must be increasing after this point… 4A

7. Sketching Curves 4A Sketching Cubics Example If y = 0 or
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
Example
Sketch the graph of the function:
If y = 0 or
So x = 2, 1 or -1
(-1,0) (1,0) and (2,0)
If x = 0
So y = -2
(0,-2) 4A

8. Sketching Curves 4A Sketching Cubics Example (-1,0) (1,0) (2,0) (0,-2)
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
Example
Sketch the graph of the function:
(-1,0) (1,0) (2,0) (0,-2) y or x -1 1 2 -2
If we substitute in x = 3, we get a value of y = -8. The curve must be decreasing after this point… 4A

9. Sketching Curves 4A Sketching Cubics Example If y = 0 or
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
Example
Sketch the graph of the function:
If y = 0 or
So x = 1 or -1
(-1,0) and (1,0)
If x = 0
So y = 1
(0,1) 4A

10. Sketching Curves 4A Sketching Cubics Example (-1,0) (1,0) (0,1) y or x
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
Example
Sketch the graph of the function:
(-1,0) (1,0) (0,1) y or
‘repeated root’ 1 x -1 1
If we substitute in x = 2, we get a value of y = 3. The curve must be increasing after this point… 4A

11. Sketching Curves 4A Sketching Cubics Example or If y = 0
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
Example
Sketch the graph of the function:
Factorise
Factorise fully or
If y = 0
So x = 0, 3 or -1
(0,0) (3,0) and (-1,0)
If x = 0
So y = 0
(0,0) 4A

12. Sketching Curves 4A Sketching Cubics Example (0,0) (3,0) (-1,0) y or x
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
Example
Sketch the graph of the function:
(0,0) (3,0) (-1,0) y or x -1 3
If we substitute in x = 4, we get a value of y = 20. The curve must be increasing after this point… 4A

13. Teachings for Exercise 4B

14. Sketching Curves Sketching Cubics
You need to be able to sketch and interpret cubics that are variations of y = x3
This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it.
Example
Sketch the graph of the function: y
y = x3 x 4B

15. Sketching Curves Sketching Cubics
You need to be able to sketch and interpret cubics that are variations of y = x3
This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it.
Example
Sketch the graph of the function: y
y = x3 5 x
A cubic with a negative ‘x3’ will be reflected in the x-axis -5
y = -x3
‘Whatever you get for x3, you now have the negative of that..’ 4B

16. Sketching Curves Sketching Cubics
You need to be able to sketch and interpret cubics that are variations of y = x3
This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it.
Example
Sketch the graph of the function: y
y = (x + 1)3
y = x3 1 -1 x
When a value ‘a’ is added to a cubic, inside a bracket, it is a horizontal shift of ‘-a’
When x = 0:
‘I will now get the same values for y, but with values of x that are 1 less than before’
y-intercept 4B

17. Sketching Curves Sketching Cubics
You need to be able to sketch and interpret cubics that are variations of y = x3
This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it.
Example
Sketch the graph of the function: y
y = x3 27 x 3
y = (3 - x)3
When x = 0:
y-intercept
Reflected in the x-axis
Horizontal shift, 3 to the right 4B

18. Teachings for Exercise 4C

19. Sketching Curves y = 1/x 4C The Reciprocal Function
You need to be able to sketch the ‘reciprocal’ function. This takes the form:
Where ‘k’ is a constant.
Example
Sketch the graph of the function
These are where the graph ‘never reaches’, in this case the axes…
and its asymptotes. y
y = 1/x x -1
-0.5
-0.25
0.25
0.5 1 x y -1 -2 -4 4 2 1
You cannot divide by 0, so you get no value at this point 4C

20. The curve will be the same, but further out…
Sketching Curves
The Reciprocal Function
You need to be able to sketch the ‘reciprocal’ function. This takes the form:
Where ‘k’ is a constant.
Example
Sketch the graph of the function
and its asymptotes.
y = 3/x y
y = 1/x x
The curve will be the same, but further out… 4C

21. The curve will be the same, but reflected in the x-axis
Sketching Curves
The Reciprocal Function
You need to be able to sketch the ‘reciprocal’ function. This takes the form:
Where ‘k’ is a constant.
Example
Sketch the graph of the function
and its asymptotes. y
y = -1/x
y = 1/x x
The curve will be the same, but reflected in the x-axis 4C

22. Teachings for Exercise 4D

23. Sketching Curves and 4D Solving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
Example
On the same diagram, sketch the following curves:
and y
Quadratic ‘U’ shape
Crosses through 0 and 3 x 1 3
Cubic ‘negative’ shape
Crosses through 0 and 1. The ‘0’ is repeated so just ‘touched’ 4D

24. Sketching Curves and 4D x y Solving Equations and Sketching3 1
Solving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
Example
On the same diagram, sketch the following curves:
and
Find the co-ordinates of the points of intersection
 These will be where the graphs are equal…
Expand brackets
Group together
Factorise 4D

25. Sketching Curves and 4D Solving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
Example
On the same diagram, sketch the following curves:
and
Find the co-ordinates of the points of intersection
 These will be where the graphs are equal…
Expand brackets
x=-√3
x=0
x=√3
Group together
Factorise
(-√3 , 3+3√3)
(0,0)
(√3 , 3-3√3) 4D

26. Sketching Curves and y = 2/x 4D Solving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
Example
On the same diagram, sketch the following curves:
and y
Cubic ‘positive’ shape
Crosses through 0 and 1. The ‘0’ is repeated.
y = 2/x x
Reciprocal ‘positive’ shape 1
Does not cross any axes 4D

27. Sketching Curves and y = 2/x 4D Solving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
How does the graph show there are 2 solutions to the equation..
Example
On the same diagram, sketch the following curves:
and y
y = 2/x x 1
Set equations equal, and re-arrange
And they cross in 2 places… 4D

28. Teachings for Exercise 4E

29. Sketching Curves 4E More Transformations
You have seen that a curve with the following function:
Will be transformed horizontally ‘-a’ units.
A curve with this function:
Will be transformed vertically ‘a’ units y
f(x) + 2
f(x + 2)
f(x) x
2 units up
2 units left
f(x + 2)  The x values reduce by 2 for the same y values
f(x) + 2  The y values from the original function increase by 2 4E

30. Sketching Curves 4E y More Transformations
f(x)
More Transformations
Sketch the following functions:
f(x) = x2
 Standard curve
 Label known points
g(x) = (x + 3)2
 Moved 3 units left
 Work out new ‘key points’
h(x) = x2 + 3
 Moved 3 units up x y y
h(x)
g(x) 9 x 3 x -3 4E

31. Sketching Curves 4E y x More Transformations Given that: i) f(x) = x3
Sketch the curve where y = f(x - 1). State any locations where the graphs crosses the axes.
 f(x) = x3
 f(x – 1) = (x – 1)3
So for this curve, when x = 0, y = -1
It therefore crosses at y = -1 y
f(x) x 1 -1
f(x – 1) 4E

32. Sketching Curves 4E y x More Transformations Given that:
i) g(x) = x(x – 2)
Sketch the curve where y = g(x + 1). State any locations where the graphs crosses the axes.
 g(x) is a positive quadratic crossing at 0 and 2.
 g(x) = x(x – 2)
 g(x + 1) = (x + 1)(x + 1 – 2)
 g(x + 1) = (x + 1)(x – 1)
So for this curve, when x = 0, y = -1
It therefore crosses at y = -1 y
g(x + 1)
g(x) x -1 1 2 -1
x’s replaced with ‘x + 1’ 4E

33. Sketching Curves 4E y x h(x) + 1 More Transformations Given that:
i) h(x) = 1/x
Sketch the curve where y = h(x) + 1. State any locations where the graphs crosses the axes and the equations of any asymptotes.
 h(x) is a positive reciprocal graph
 h(x) = 1/x
 h(x) + 1 = 1/x + 1
The asymptotes are: x = 0 (the y-axis)
y = 1
It will cross the x-axis at -1 since this value will make the equation = 0 y 1
h(x) -1 x 4E

34. Teachings for Exercise 4F

35. Sketching Curves 4F Even more Transformations
You also need to be able to perform transformations of the form:
 this is a horizontal stretch of 1/a.
You also need to know:
 this is a vertical stretch by factor ‘a’
‘We will get the same y values, using half the x values’
 This is because the x values get multiplied by 2 before the y values are worked out
‘We will get y values twice as big, using the same x values’
 This is because when we work out the y values, they are doubled after 4F

36. Sketching Curves (0,9) (3,0) (-3,0) 4F y Even more Transformations
f(x)
Even more Transformations
Given that f(x) = 9 – x2, sketch the curve with equation;
a) y = f(2x)
 Sketch the original curve, working out key points.
If x = 0
If y = 0 9 x -3 3
(0,9)
(3,0)
(-3,0) 4F

37. Sketching Curves (0,9) (-1.5,0) (1.5,0) 4F y Even more Transformations
f(x)
Even more Transformations
Given that f(x) = 9 – x2, sketch the curve with equation;
a) y = f(2x)
 Substitute ‘2x’ in place of ‘x’
If x = 0
If y = 0 9 x -3 3 y
f(2x) 9
(0,9) x
-1.5
1.5
(-1.5,0)
(1.5,0) 4F

38. Sketching Curves (0,18) (3,0) (-3,0) 4F y Even more Transformations
f(x)
Even more Transformations
Given that f(x) = 9 – x2, sketch the curve with equation;
a) y = 2f(x)
 f(x), the original equation, is doubled..
If x = 0
If y = 0 9 x -3 3 18 y
2f(x)
(0,18) x -3 3
(3,0)
(-3,0) 4F

39. Summary We have learnt the shapes of several different curves
We have learnt how to apply transformations to those curves
We have also looked at how to work out the ‘key points’