1. Quadratic Functions Functions Quadratic Functions y = ax2
Int 2
Functions
Quadratic Functions y = ax2
Quadratics y = ax2 +c
Quadratics y = a(x-b)2
Quadratics y = a(x-b)2 + c
Factorised form y = (x-a)(x-b)

2. Starter
Int 2

3. Functions www.mathsrevision.com Int 2 Learning Intention
Success Criteria
To explain the term function.
Understand the term function.
Work out values for a given function.

4. Functions www.mathsrevision.com
Int 2
A roll of carpet is 5m wide. It is solid in strips by the area.
If the length of a strip is x m then the area. A square metres,
is given by A = 5x.
The value of A depends on the value of x.
We say A is a function of x. We write :
A(x) =5x
Example
A(1) = 5 x 1 =5
A(2) = 5 x 2 =10
A(t) = 5 x t = 5t

5. Functions x 1 2 3 4 5 A 10 15 20 25 www.mathsrevision.com
Int 2
Using the formula for the function we can make a table and
draw a graph using A as the y coordinate. x 1 2 3 4 5 A 10 15 20 25
In the case
The graph is a
straight line
We can this a
Linear function.

6. Functions www.mathsrevision.com
Int 2
For the following functions write down the gradient
and were the function crosses the y-axis
f(x) = 2x - 1
f(x) = 0.5x + 7
f(x) = -3x
Sketch the following functions.
f(x) = x
f(x) = 2x + 7
f(x) = x +1

7. Functions
Int 2
Now try MIA Ex 1
Ch14 (page 216)

8. Starter
Int 2

9. Quadratic Functions www.mathsrevision.com Int 2 Learning Intention
Success Criteria
To explain the main properties of the basic quadratic function y = ax2
using graphical methods.
To know the properties of a quadratic function.
Understand the links between graphs of the form y = x2 and y = ax2

10. Quadratic Functions www.mathsrevision.com A function of the form
Int 2
A function of the form
f(x) = a x2 + b x + c
is called a quadratic function
The simplest quadratics have the form
f(x) = a x2
Lets investigate

11. Quadratic Functions Now try MIA Ex 2 Q2 P 219 www.mathsrevision.com
Int 2
Now try MIA Ex 2
Q2 P 219

12. Quadratic of the form f(x) = ax2
Key Features
Symmetry about x =0
Vertex at (0,0)
The bigger the value of a the steeper the curve.
-x2 flips the curve about x - axis

13. Quadratic Functions www.mathsrevision.com Example
Int 2
Example
The parabola has the form y = ax2 graph opposite.
The point (3,36) lies on the graph.
Find the equation of the function.
(3,36)
Solution
f(3) = 36
36 = a x 9
a = 36 ÷ 9
a = 4
f(x) = 4x2

14. Quadratic Functions Now try MIA Ex 2 Q3 (page 219)
Int 2
Now try MIA Ex 2
Q3 (page 219)

15. Starter www.mathsrevision.com
Int 2
Q1. Write down the equation of the quadratic.
f(x) = ax2
(2,100)
Solution
f(2) = 100
100 = a x 4
a = 100 ÷ 4
a = 25
f(x) = 25x2
(x-4)(x-3)

16. Quadratic Functions www.mathsrevision.com y = ax2+ c Int 2
Learning Intention
Success Criteria
To explain the main properties of the basic quadratic function
y = ax2+ c
using graphical methods.
To know the properties of a quadratic function.
y = ax2+ c
Understand the links between graphs of the form y = x2 and y = ax2 + c

17. Quadratic Functions Now try MIA Ex 2 Q5 (page 220)
Int 2
Now try MIA Ex 2
Q5 (page 220)
Quadratic of the form f(x) = ax2 + c

18. Quadratic of the form f(x) = ax2 + c
Key Features
Symmetry about x = 0
Vertex at (0,C)
a > 0 the vertex (0,C) is a minimum turning point.
a < 0 the vertex (0,C) is a maximum turning point.

19. Quadratic Functions www.mathsrevision.com Example
Int 2
Example
The parabola has the form y = ax2 + c graph opposite.
The vertex is the point (0,2) so c = 2. The point (3,38)
lies on the graph. Find the equation of the function.
(3,38)
Solution
f(x) = a x2 + c
(0,2)
f(3) = a x
38 = a x 9 +2
a = (38 -2) ÷ 9
a = 4
f(x) = 4x2 + 2

20. Quadratic Functions Now try MIA Ex 2 Q7 (page 221)
Int 2
Now try MIA Ex 2
Q7 (page 221)

21. Starter www.mathsrevision.com
Int 2
Q1. Write down the equation of the quadratic.
(9,81)
Solution
f(9) = 81
81 = a x 9
a = 81 ÷ 9
a = 9
f(x) = 9x2
(x-5)(x-6)

22. Quadratic Functions www.mathsrevision.com y = a(x – b)2 Int 2
Learning Intention
Success Criteria
To explain the main properties of the basic quadratic function
y = a(x - b)2
using graphical methods.
To know the properties of a quadratic function.
y = a(x – b)2
Understand the links between graphs of the form
y = x2 and y = a(x – b)2

23. Quadratic Functions Now try MIA Ex 3 Q2 (page 222)
Int 2
Now try MIA Ex 3
Q2 (page 222)
Quadratic of the form f(x) = a(x - b)2

24. Quadratic of the form f(x) = a(x - b)2
Key Features
Symmetry about x = b
Vertex at (b,0)
Cuts y - axis at x = 0
a > 0 the vertex (b,0) is a minimum turning point.
a < 0 the vertex (b,0) is a maximum turning point.

25. Quadratic Functions www.mathsrevision.com Example
Int 2
Example
The parabola has the form f(x) = a(x – b)2.
The vertex is the point (2,0) so b = 2. The point (5,36)
lies on the graph. Find the equation of the function.
Solution
f(x) = a (x - b)2
(5,36)
f(5) = a ( 5 - 2)2
(2,0)
36 = a x 9
a = 36 ÷ 9
a = 4
f(x) = 4(x-2)2

26. Quadratic Functions Now try MIA Ex 3 Q4 and Q5 (page 222)
Int 2
Now try MIA Ex 3
Q4 and Q5 (page 222)

27. Quadratic Functions Homework MIA Ex 4 (page 222) www.mathsrevision.com
Int 2
Homework
MIA Ex 4 (page 222)

28. Starter
Int 2
f(x)
(5,25) x

29. Quadratic Functions www.mathsrevision.com Int 2 Learning Intention
Success Criteria
To explain the main properties of the basic quadratic function
y = a(x-b)2 + c
using graphical methods.
To know the properties of a quadratic function.
Understand the links between the graph of the form
y = x2
and
y = a(x-b)2 + c

30. Quadratic Functions y = a(x - b)2+c www.mathsrevision.com
Int 2
Every quadratic function can be written in the form
y = a(x - b)2+c
The curve y= f(x) is a parabola
axis of symmetry at x = b
Y - intercept
Vertex or turning point at (b,c)
(b,c)
Cuts y-axis when
x = 0 y = a(x – b)2 + c
x = b
a > 0 minimum turning point
a < 0 maximum turning point

31. Example 1 Sketch the graph y = (x - 3)2 + 2
Quadratic Functions
y = a(x-b)2+c
Int 2
Example 1 Sketch the graph y = (x - 3)2 + 2
a = 1
b = 3
c = 2
Vertex / turning point is (b,c)
= (3,2) y
Axis of symmetry at b = 3
(0,11)
y = (0 - 3)2 + 2
= 11
(3,2) x

32. Example2 Sketch the graph y = -(x + 2)2 + 1
Quadratic Functions
y = a(x-b)2+c
Int 2
Example2 Sketch the graph y = -(x + 2)2 + 1
a = -1
b = -2
c = 1
Vertex / turning point is (b,c)
= (-2,1) y
Axis of symmetry at b = -2
(-2,1)
y = -(0 + 2)2 + 1
= -3 x
(0,-3)

33. Example Write down equation of the curve
Quadratic Functions
y = a(x-b)2+c
Int 2
Example Write down equation of the curve
Given a = 1 or a = -1
a < 0 maximum turning point
a = -1
(-3,5)
Vertex / turning point is (-3,5)
b = -3
(0,-4)
c = 5
y = -(x + 3)2 + 5

34. Quadratic Functions Now try MIA Ex 5 Q1 and Q2 (page 225)
Int 2
Now try MIA Ex 5
Q1 and Q2 (page 225)

35. Quadratic of the form f(x) = a(x - b)2 + c
Cuts y - axis when x=0
Symmetry about x =b
Vertex / turning point at (b,c)
a > 0 the vertex is a minimum.
a < 0 the vertex is a maximum.

36. Quadratic Functions Now try MIA Ex6 (page 226) www.mathsrevision.com
Int 2
Now try MIA Ex6
(page 226)

37. Starter
Int 2
f(x) x
(3,-6)

38. Quadratic Functions www.mathsrevision.com Int 2 Learning Intention
Success Criteria
To show factorised form of a quadratic function.
To interpret the keyPoints of the factorised form of a quadratic function.

39. Quadratic Functions y = (x - a)(x - b) www.mathsrevision.com
Int 2
Some quadratic functions can be written in the
factorised form
y = (x - a)(x - b)
The zeros / roots of this function occur when
y = (x - a)(x - b) = x = a and x = b
Note: The a,b in this form are NOT the a,b in the form
f(x) ax2 + bx + c

40. Q. Find the zeros, axis of symmetry and turning point
for f(x) = (x - 2)(x - 4)
Zero’s at
x = 2 and x = 4
Axis of symmetry
ALWAYS halfway
between
x = 2 and x = 4
x =3
Y – coordinate - turning point y = (3 - 2)(3 - 4) = -1
(3,-1)

41. Quadratic Functions Now try MIA Ex7 (page 227) www.mathsrevision.com
Int 2
Now try MIA Ex7
(page 227)