Download Graphs of quadratic functions

SCHOLAR Study Guide
National 5 Mathematics
Course Materials
Topic 18: Graphs of quadratic
functions
Authored by:
Margaret Ferguson
Reviewed by:
Jillian Hornby
Previously authored by:
Eddie Mullan
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
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SCHOLAR Study Guide Course Materials Topic 18: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
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1
Topic 1
Graphs of quadratic functions
Contents
18.1 Identifying features of a quadratic function . . . . . . . . . . . . . . . . . . . .
3
18.2 Recognising and determining equations of quadratic functions from graphs . .
18.3 Sketching quadratic functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
22
18.4 Using function notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.5 Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
37
18.6 End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Learning objectives
By the end of this topic, you should be able to:
•
identify the shape, zeros and y-intercept of a quadratic function;
•
determine the turning point and equation of the axis of symmetry of a quadratic
function;
•
recognise a quadratic function from its graph;
•
determine the equation of a quadratic function from its graph;
•
sketch a quadratic function;
•
recognise and use function notation.
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
1.1
Identifying features of a quadratic function
A quadratic function has a graph called a parabola. There are two basic shapes of
parabola:
A good way to describe the shape of
this function is a smiley face.
1.
The function has a minimum turning
point because if you run your finger
over the shape of the graph it
decreases to a low point then turns and
increases. This is the nature of the
function.
A parabola is symmetrical and has a
line of symmetry. The equation of the
line of symmetry takes the form x = a.
The line of symmetry passes through
the turning point so the coordinates of
the turning point are (a , . . .). The line
of symmetry is normally called the axis
of symmetry.
A good way to describe the shape of
this function is a sad face.
2.
The nature of this function is a
maximum because if you run your
finger over the shape of the graph it
increases to a high point then turns and
decreases.
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
The y-intercept is the place where the
graph crosses the y axis. The
coordinates of the y-intercept are
(0 , c).
The graph may cross the x axis in two
places, one place or not at all. If the
graph crosses the x axis these point(s)
are called the zeros, roots or
x-intercepts. The coordinates of the
zeros or roots are (p , 0) and (q , 0).
Key point
•
The shape of the graph of a quadratic will be smiley if the x 2 term is positive
e.g. y = 5x2 looks like,
•
The shape of the graph of a quadratic will be sad if the x 2 term is negative
e.g. y = − x2 looks like,
Examples
1.
Problem:
A quadratic function has the equation y = − 2x 2 + 5x − 1.
State the shape of the graph and its nature.
Solution:
The x2 term is negative because its coefficient is -2 so it is a sad face and looks like
this,
The nature of the function is a maximum because it increases to a high point, turns and
decreases.
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
2.
Problem:
Identify the coordinates of the zeros and the y-intercept of the quadratic y = x 2 + 6x +
8.
Solution:
The zeros or roots are the points where the graph crosses the x axis so we get (2,0)
and (4,0).
The y-intercept is the point where the graph crosses the y axis so we get (0,8).
Notice the +8 on the end of the equation y = x 2 + 6x + 8, this also helps to identify
the y-intercept.
..........................................
3.
Problem:
Identify the equation of the axis of symmetry and the coordinates of the turning point of
the quadratic y = − x2 + 6x − 8
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Solution:
The axis of symmetry goes vertically half way between the zeros. The zeros are at 2
and 4, 3 lies half way between 2 and 4.
An easy way to calculate it is to find the average of the roots
2+4
2
=
6
2
= 3.
So the equation of the axis of symmetry is x = 3.
We now know that the maximum turning point is (3, . . .).
If we substitute x = 3 into y = − x2 + 6x − 8 we get,
y =
=
- (3)2 + 6 × 3 − 8
- 9 + 18 − 8
= 1
The coordinates of the maximum turning point are (3 , 1).
..........................................
Identifying the features of a quadratic function exercise
Identifying the shape of a quadratic function
Go online
Q1:
Identify the shape of the quadratic y = 3x 2
a)
b)
..........................................
Q2:
Identify the shape of the quadratic y = − 2x 2
a)
b)
..........................................
Identifying the nature of a quadratic function
Q3:
Identify the nature of the quadratic y = − 3x 2
a) Maximum
b) Minimum
..........................................
Q4:
Identify the nature of the quadratic y = 2x 2
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
a) Maximum
b) Minimum
..........................................
Identifying the equation of the line of symmetry of a quadratic
Q5: Identify the equation of the axis of symmetry of the quadratic y = x 2 − 6x + 8.
..........................................
Identifying the coordinates of the turning point of a quadratic function
Q6: Identify the coordinates of the turning point of y = x 2 + 8x + 15.
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Identifying the zeros of a quadratic function
Q7:
Identify the coordinates of the zeros or roots of y = x 2 − x − 6.
..........................................
Identifying the y-intercept of a quadratic function
Q8:
Identify the coordinates of the y-intercept of y = x 2 − 2x − 3.
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Identifying features of a quadratic function
A quadratic has the equation y = x 2 + 8x + 12.
Q9: Identify the shape of the quadratic function.
a)
b)
..........................................
Q10: Identify the nature of the quadratic function.
a) Maximum
b) Minimum
..........................................
Q11: Identify the coordinates of the y-intercept.
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
A quadratic has the equation y = x 2 + 8x + 12.
Q12: Identify the coordinates of the Zeros.
..........................................
Q13: Identify the equation of the line of symmetry.
..........................................
Q14: Identify the coordinates of the turning point.
..........................................
..........................................
1.2
Recognising and determining equations of quadratic
functions from graphs
Quadratic functions of the form y = kx2
Go online
This is a graph of the quadratic function y = kx 2 . The curve is called a parabola. If we
change the value of k notice how the shape of the quadratic changes.
If we set k = 1 we will achieve the following quadratic. Notice that the point (1,k) lies
on this curve, in this case the point (1,1).
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
If we set k = − 2 we will achieve this quadratic. Notice that the point (1,k) lies on this
curve, in this case the point (1,-2).
..........................................
Key point
If the equation of a quadratic function is of the form y = kx 2 then:
•
the turning point will be at the origin, (0,0);
•
the graph will pass through the point (1,k).
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Example
Problem:
Write down the equation of the curve in the form y = kx 2 .
Solution:
The curve passes through (1, 5) substitute x = 1 and y = 5 into y = kx 2 .
⇒ 5 = k × 12
⇒5 = k
⇒k = 5
The equation is y = 5x 2 .
Note we can pick the coordinates of any point where the curve passes through the
corner of a grid square. Here we could have chosen (-1,5). The value for k would still
be the same.
The only point we cannot use is (0,0).
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
13
Quadratic functions of the form y = kx2 practice
Q15:
Write down the equation of the curve in the form y = kx 2 .
..........................................
Q16:
Write down the equation of the curve in the form y = kx 2 .
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Quadratic functions of the form y = (x − a)2 + b
Go online
This is a graph of the quadratic function y = (x − a) 2 + b. You may recognise this as
completed square form. The curve is called a parabola. If we change the values of a
and b notice what happens to the turning point.
If we set a = 1 and b = 1 we will achieve the following quadratic. Notice what happens
to the turning point.
If we set a = − 3 and b = − 2 we will achieve the following quadratic. Notice what
happens to the turning point.
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
15
Quadratic functions of the form y = − (x − a)2 + b
This is a graph of the quadratic function y = − (x − a) 2 + b. The curve is called a
parabola. If we change the values of a and b notice what happens to the turning point.
If we set a = 1 and b = 1 we will achieve the following quadratic. Notice what happens
to the turning point.
If we set a = − 3 and b = 3 we will achieve the following quadratic. Notice what
happens to the turning point.
..........................................
Key point
If the equation of a quadratic function is of the form
y = (x − a)2 + b or y = − (x − a)2 + b
then the coordinates of the turning point will be (a, b).
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16
TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Example
Problem:
This is the graph of a quadratic function whose equation is of the form y = (x − a) 2 + b.
Find the values of a and b and write down the equation of the function.
Solution:
By inspection the turning point is (-5,-1).
Thus a = − 5 and b = − 1.
The equation is y = (x − (−5))2 + (−1)
or simply y = (x + 5)2 − 1.
..........................................
Quadratic functions of the form y = ± (x − a)2 + b practice
Q17:
Go online
This is the graph of a quadratic function whose equation is of the form y = (x − a) 2 + b.
Find the values of a and b and write down the equation of the function.
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
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..........................................
Q18:
This is the graph of a quadratic function whose equation is of the form y = − (x −
a)2 + b.
Find the values of a and b and write down the equation of the function.
..........................................
Determining equations of quadratic functions from graphs exercise
Q19: The diagram shows a graph of the quadratic function y = kx 2 . What is the value
of k?
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Q20: The diagram shows a graph of the quadratic function y = kx 2 . What is the value
of k?
..........................................
Q21: The diagram shows a graph of the quadratic function y = kx 2 . What is the value
of k?
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Q22: The diagram shows a graph of the quadratic function y = kx 2 . What is the value
of k?
..........................................
Q23: The diagram shows a graph of the quadratic function y = kx 2 . What is the value
of k?
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Q24: The diagram shows the graph of a quadratic function of the form y = (x − a) 2 +
b.
Give the equation of the function.
..........................................
Q25: The diagram shows the graph of a quadratic function of the form y = − (x −
a)2 + b.
Give the equation of the function.
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Q26: The diagram shows the graph of a quadratic function of the form y = (x − a) 2 +
b.
Give the equation of the function.
..........................................
Q27: The diagram shows the graph of a quadratic function of the form y = − (x −
a)2 + b.
Give the equation of the function.
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
1.3
Sketching quadratic functions
Sketching a quadratic graph from its equation (1)
Go online
This is a graph of the quadratic function y = k(x − a) 2 + b. When k = 1, a = 0 and
b = 0 the parabola has y-intercept (0,0).
Notice that the point (a, b) is a minimum turning point, in this case, (0,0) and the
parabola has an axis of symmetry x = 0
If we set k = 1, a = 3 and b = − 5 the parabola has y-intercept (0,4).
Notice that the point (a, b) is a minimum turning point, in this case, (3,-5) and the
parabola has an axis of symmetry x = 3
If we set k = − 1, a = − 4 and b = 2 the parabola has y-intercept (0,-14).
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Notice that the point (a, b) is a maximum turning point, in this case, (-4,2) and the
parabola has an axis of symmetry x = − 4
..........................................
Key point
If the equation of a quadratic function is of the form y = (x − a) 2 + b then:
•
the turning point at (a, b) will be a minimum;
•
the axis of symmetry will have equation x = a;
•
the y-intercept occurs when x = 0. Substitute x = 0 into y = (x − a)2 +
b to find the y coordinate of the y-intercept.
Key point
If the equation of a quadratic function is of the form y = − (x − a) 2 + b then:
•
the turning point at (a, b) will be a maximum;
•
the axis of symmetry will have equation x = a;
•
the y-intercept occurs when x = 0. Substitute x = 0 into y = − (x −
a)2 + b to find the y coordinate of the y-intercept.
Examples
1.
Problem:
A quadratic function has the equation y = (x − 3) 2 + 1.
a) Write down the coordinates of the turning point.
b) Say whether it is a maximum or minimum.
c) Give the axis of symmetry of the parabola.
d) Write down the coordinates of the y-intercept.
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
e) Sketch the graph
Solution:
a) The turning point is (3, 1).
x = 3 is the value of x which makes the bracket zero. (i.e. x − 3 = 0 gives
x = 3)
y = 1 is the value of y when the bracket is worth zero.
b) It is a minimum turning point.
The x2 term is positive so we have a smiley face.
c) The axis of symmetry is x = 3.
The equation of the vertical line passing through the turning point.
d) The y-intercept is (0,10).
Any point on the y axis has an x coordinate of zero.
If we substitute x = 0 we get:
y = (0 − 3)2 + 1
= ( - 3)2 + 1
= 9 + 1
= 10
e)
..........................................
2.
Problem:
A quadratic function has the equation y = − (x + 1) 2 − 2.
a) Write down the coordinates of the turning point.
b) Say whether it is a maximum or minimum.
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
c) Give the axis of symmetry of the parabola.
d) Write down the coordinates of the y-intercept.
e) Sketch the graph
Solution:
a) The turning point is (-1, -2).
x = − 1 is the value of x which makes the bracket zero. (i.e. x + 1 = 0 gives
x = − 1)
y = − 2 is the value of y when the bracket is worth zero.
b) It is a maximum turning point.
The x2 term is negative so we have a sad face.
c) The axis of symmetry is x = − 1.
The equation of the vertical line passing through the turning point.
d) The y-intercept is (0,-3).
Any point on the y axis has an x coordinate of zero.
If we substitute x = 0 we get:
y = - (0 + 1)2 − 2
=
- (1)2 − 2
=
-1 − 2
=
-3
e)
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Top tip
On your sketch write the coordinates of the turning point and the y-intercept
beside the points.
Write the equation beside the curve e.g. y = (x − 3) 2 + 1
Sketching a quadratic graph from its equation practice (1)
Q28:
Go online
A quadratic function has the equation y = − (x + 2) 2 + 3.
a) Write down the coordinates of the turning point.
b) Say whether it is a maximum or minimum.
c) What is the equation of the axis of symmetry?
d) Write down the coordinates of the y-intercept.
e) Sketch the graph
..........................................
Q29:
A quadratic function has the equation y = (x − 1) 2 + 2.
a) Write down the coordinates of the turning point.
b) Say whether it is a maximum or minimum.
c) What is the equation of the axis of symmetry?
d) Write down the coordinates of the y-intercept.
e) Sketch the graph.
..........................................
Sketching a quadratic graph from its equation (2)
This is a graph of the quadratic function y = (x − m)(x − n).
Go online
The x-coordinate of the turning point is x = (m + n)/2.
The y-coordinate of the turning point can be found by substituting x = (m + n)/2 into
the equation of the function y = (x − m)(x − n).
The axis of symmetry x = (m + n) ÷ 2, exactly in the middle of m and n.
When m = 0 and n = 0 notice that the equation of the function is y = x 2 . The root,
or x-intercept, the minimum turning point and y-intercept are all at the same point and
have coordinates (0,0).
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
When m = 4 and n = − 2 notice that the roots, or x-intercept, are (0,4) and (0,-2),
the parabola has y-intercept (0,-8) and the axis of symmetry is x = 1.
When m = − 2 and n = 2 notice that the roots, or x-intercept, are (0,-2) and (0,2),
the parabola has y-intercept (0,-4) and the axis of symmetry is x = 0.
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
..........................................
Key point
If the equation of a quadratic function is of the form y = (x − m)(x − n) then:
•
the roots have coordinates (m,0) and (n,0);
•
the y-intercept has x coordinate = 0. We find the y-coordinate by
substituting x = 0 into y = (x − m)(x − n);
•
the axis of symmetry is a vertical line half way between the roots;
•
the equation of the axis of symmetry is x = (m + n)/2;
•
the x-coordinate of the turning point is (m + n)/2. Substitute x = (m + n)/2
into y = (x − m)(x − n) to find the y-coordinate of the turning point.
Top tip
If the equation of the quadratic function is of the form y = ax 2 + bx + c then:
•
factorise to help find the roots.
•
the coordinates of the y-intercept are (0,c).
Examples
1.
Problem:
A quadratic function has the equation y = (x − 2)(x − 6).
a) Write down the coordinates of the roots.
b) Write down the coordinates of the y-intercept.
c) Write down the equation of the axis of symmetry.
d) State the nature of the turning point.
e) Write down the coordinates of the turning point.
f) Sketch the graph
Solution:
a) The roots are (2,0) and (6,0).
The value(s) of x which make each bracket zero. (i.e. x − 2 = 0 gives x = 2
and x − 6 = 0 gives x = 6)
b) The coordinates of y-intercept is (0,12).
The value of y when x = 0.
If we substitute x = 0 we get:
y = (x − 2) (x − 6)
= (0 − 2) (0 − 6)
= ( - 2 ) × ( - 6)
= 12
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
c) The axis of symmetry is x = 4.
The exact middle of 2 and 6 is (2 +2 6) =
8
2
= 4.
d) It is a minimum turning point.
(x − 2)(x − 6) = x2 − 8x + 12 so since x2 is positive the shape is a smiley
face and hence a minimum turning point.
e) The minimum turning point is (4,-4).
Substitute x = 4 into y = (x − 2)(x − 6) we get:
y = (4 − 2) (4 − 6)
= 2 × ( - 2)
=
-4
f)
..........................................
2.
Problem:
A quadratic function has the equation y = x 2 − 2x − 15.
a) Write down the coordinates of the roots.
b) Write down the coordinates of the y-intercept.
c) Write down the equation of the axis of symmetry.
d) State the nature of the turning point.
e) Write down the coordinates of the turning point.
f) Sketch the graph
Solution:
a) The roots are (-3,0) and (5,0).
If we factorise x2 − 2x − 15 we get (x − 5)(x + 3). (i.e. x − 5 = 0 gives x = 5
and x + 3 = 0 gives x = − 3)
The value(s) of x which make each bracket zero.
b) The coordinates of y-intercept is (0,-15).
The value of y when x = 0.
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
If we substitute x = 0 we get:
y = (0 − 5) (0 + 3)
= ( - 5) × 3
=
- 15
c) The axis of symmetry is x = 1.
The exact middle of -3 and 5 is ((−3)2 + 5) =
2
2
= 1.
d) It is a minimum turning point.
Since x2 is positive in the original equation the shape is a smiley face and hence a
minimum turning point.
e) The minimum turning point is (1,-16).
Substitute x = 1 into y = (x − 5)(x + 3) we get:
y = (1 − 5) (1 + 3)
= ( - 4) × 4
=
- 16
f)
..........................................
Sketching a quadratic graph from its equation practice (2)
Q30:
Go online
A quadratic function has the equation y = (x + 3)(x − 1).
a) Write down the coordinates of the roots.
b) Write down the coordinates of the y-intercept.
c) Give the axis of symmetry.
d) State the nature of the turning point.
e) Write down the coordinates of the turning point.
f) Sketch the graph
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
31
Q31:
A quadratic function has the equation y = x 2 + 5x + 6.
a) Write down the coordinates of the roots.
b) Write down the coordinates of the y-intercept.
c) Give the axis of symmetry.
d) State the nature of the turning point.
e) Write down the coordinates of the turning point.
f) Sketch the graph
..........................................
Sketching quadratic functions exercise
Q32:
A quadratic function has the equation y = − (x − 3) 2 − 5.
a) Write down the coordinates of the turning point.
b) State the nature of the turning point.
c) Identify the equation of the axis of symmetry.
d) Write down the coordinates of the y-intercept.
..........................................
Q33:
A quadratic function has the equation y = − (x + 5) 2 + 4.
a) Write down the coordinates of the turning point.
b) State the nature of the turning point.
c) Identify the equation of the axis of symmetry.
d) Write down the coordinates of the y-intercept.
..........................................
Q34:
A quadratic function has the equation y = (x − 2) 2 + 4.
a) Write down the coordinates of the turning point.
b) State the nature of the turning point.
c) Identify the equation of the axis of symmetry.
d) Write down the coordinates of the y-intercept.
..........................................
Q35:
A quadratic function has the equation y = (x + 1) 2 − 4.
a) Write down the coordinates of the turning point.
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
b) State the nature of the turning point.
c) Identify the equation of the axis of symmetry.
d) Write down the coordinates of the y-intercept.
..........................................
Q36:
A quadratic function has the equation y = (x + 6)(x − 4).
a) Write down the coordinates of the roots.
b) Write down the coordinates of the y-intercept.
c) Identify the equation of the axis of symmetry.
d) State the nature of the turning point.
e) Write down the coordinates of the turning point.
..........................................
Q37:
A quadratic function has the equation y = (x − 2)(x + 3).
a) Write down the coordinates of the roots.
b) Write down the coordinates of the y-intercept.
c) Identify the equation of the axis of symmetry.
d) State the nature of the turning point.
e) Write down the coordinates of the turning point.
..........................................
Q38:
A quadratic function has the equation y = x(x − 6).
a) Write down the coordinates of the roots.
b) Write down the coordinates of the y-intercept.
c) Identify the equation of the axis of symmetry.
d) State the nature of the turning point.
e) Write down the coordinates of the turning point.
..........................................
Q39:
A quadratic function has the equation y = x 2 + 4x − 21.
a) Write down the coordinates of the roots.
b) Write down the coordinates of the y-intercept.
c) Identify the equation of the axis of symmetry.
d) State the nature of the turning point.
e) Write down the coordinates of the turning point.
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
1.4
Using function notation
Some function definitions are needed and this requires us to start to have an
understanding of sets of numbers.
The standard number sets are:
•
N = {1, 2, 3, 4, 5, . . .} the set of natural numbers.
•
W = {0, 1, 2, 3, 4, 5, . . .} the set of whole numbers.
•
Z = {. . ., -3, -2, -1, 0, 1, 2, 3, . . .} the set of integers.
•
Q = the set of all numbers which can be written as fractions, called the set of
rational numbers.
√
R = the set of rational and irrational numbers (such as 2 ), called the set of real
numbers.
√
I = the set of all real numbers which are not rational numbers e.g. 2, π
•
•
A function f from set A to set B is a rule which assigns each element in A to exactly
one element in B. This is often written as f : A → B.
The set A contains all the input values and this is called the domain.
The set B contains all the output values and is called the range.
Example
Problem:
You may have already met function machines.
The function machine multiplies the input value by 3 to get the output value.
Solution:
This could be expressed as x → 3x and is equivalent to f (x) = 3x.
This function could also have been expressed as the equation y = 3x.
Since all the input values are integers we could define the function as f (x) = 3x for
x ∈ Z.
..........................................
This symbol, ∈, means "is a member of".
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Function mapping
You may already have met mappings.
Go online
..........................................
This could be expressed as x ⇒ x2 and is equivalent to f (x) = x2 .
This function is quadratic and could also have been expressed as the equation y = x 2 .
The domain for this function is the set of real numbers giving f (x) = x 2 , x ∈ R.
Example
Problem:
A function is defined in function notation as f (x) = x 2 + 1 and is described as "f of x
equals x squared plus 1".
Solution:
Then,
f (1) = 12 + 1 = 2
f (2) = 22 + 1 = 5
f (3) = 32 + 1 = 10
..........................................
Q40: f (x) = 3x − 4. Find the value of f (5).
..........................................
Q41: f (x) = x2 + 2x + 1. Find the value of f (−2).
..........................................
Q42: g(x) = x3 + x + 7. Find the value of g(0).
..........................................
Sometimes we are given the output value of a function and need to find the
corresponding value of x.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
35
Examples
1.
Problem:
h(x) = 5x − 1. Find the value of x when h(x) = 19.
Solution:
In this case we know the answer to the function so if we replace h(x) with 5x − 1 in
h(x) = 19 we get:
5x − 1 = 19
5x = 20
x = 4
solving the equation gives
and
..........................................
2.
Problem:
k(x) = x3 . Find the value of x when k(x) = 8.
Solution:
x3 = 8
√
3
8
x =
x = 2
..........................................
Function notation exercise
Number sets
Go online
Q43:
a) Does
b) Does
c) Does
√
√
√
3 belong to the set of integers Z?
3 belong to the set of natural numbers N?
3 belong to the set of irrational numbers I?
d) Does 5·4 belong to the set of real numbers R?
..........................................
Function notation
Q44:
a) Given that f (x) = 6x − 1 evaluate f (−1).
b) p(x) = x2 + 3. Find the value of p(8).
c) Given that v(x) = x3 + x evaluate v(2).
d) e(x) =
1/ .
x
Find the value of e(4).
e) Given that g(x) = x2 + 4x + 3 evaluate g(−2).
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
..........................................
Q45:
a) h(x) = 2x + 5. Find the value of x when h(x) = 17.
b) Given that m(x) = x3 + 1 evaluate the value of x when m(x) = 28.
c) a(x) = 24 /x . Find the value of x when a(x) = 6.
..........................................
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
1.5
Learning points
Features of quadratic functions
•
The shape of the graph of a quadratic will be smiley if the x 2 term is positive
e.g. y = 5x2 looks like
•
The shape of the graph of a quadratic will be sad if the x 2 term is negative
e.g. y = − x2 looks like
•
If the shape is smiley the nature will be a minimum.
•
If the shape is sad the shape will be a maximum.
•
The graph is symmetrical and the equation of the axis of symmetry will take the
form x = a.
•
The zeros or roots of a quadratic are the point(s) where the graph crosses the
x-axis and will take the form (p,0).
•
The y-intercept is the point where the graph crosses the y-axis and will take the
form (0,c) where c can be identified from y = ax 2 + bx + c.
Determining the equation of a quadratic function from its graph
•
•
If the equation of the quadratic takes the form y = kx 2 :
◦
find the coordinates of a point on the graph (Note: you cannot use the origin);
◦
substitute the values of x and y into y = kx 2 ;
◦
calculate the value of k;
◦
state the equation of the function.
If the equation of the quadratic takes the form y = k(x − a) 2 + b:
◦
find the coordinates of the turning point from the graph;
◦
replace a with the x-coordinate;
◦
replace b with the y-coordinate;
◦
state the equation of the function
Sketching the graph of a quadratic function
•
Find the coordinates of the y-intercept
◦
If the equation takes the form ax 2 + bx + c = 0 then (0,c).
◦
If the equation takes the form y = (x − m)(x − n) then substitute x = 0
into the function to calculate the y-coordinate (0, m × n)
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
◦
•
•
•
If the equation takes the form y = (x − a)2 + b then substitute x = 0 into
the function to calculate the y-coordinate.
Find the coordinates of the zeros or roots
◦
If the equation takes the form y = (x − m)(x − n) then the roots are (m,0)
and (n,0).
◦
If the equation takes the form y = ax 2 + bx + c you must factorise the
expression first.
Find the equation of the axis of symmetry
◦
Find the value in the middle of the zeros by inspection or calculating the
n
average of the roots m +
2 .
◦
State the equation in the form x =
◦
If the equation takes the form y = (x − a)2 + b then the equation of the
axis of symmetry is x = a.
m+n
2 .
Find the coordinates of the turning point
◦
The x-coordinate is the value in the middle of the roots.
◦
Substitute for x into the equation of the function to determine the y-coordinate.
◦
If the function is in the form y = (x − a)2 + b then the turning point is (a, b).
Sketching the graph
•
Identify the shape of the function.
•
Identify the nature of the turning point.
•
Draw a set of axes.
•
Plot the points for the roots, turning point and y-intercept. (You may not always
know the coordinates of the roots.)
•
Bearing in mind the shape, sketch the graph.
•
Write the coordinates beside the roots, turning point and y-intercept on your graph.
•
Label your graph with its equation e.g. y = x 2 + 2x − 3
Function Notation
•
A function can be expressed as an equation e.g. y = x 2 + 6x − 16 or in function
notation e.g. f (x) = x2 + 6x − 16.
•
Functions have a domain the set of input values.
•
Functions have a range the set of output values.
•
A function is a rule which maps each input value to exactly one output value.
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
1.6
39
End of topic test
End of topic 18 test
Identifying features of a quadratic function
A quadratic has the equation y = x 2 − 2x + 1.
Q46: Identify the shape of the quadratic function.
a)
b)
..........................................
Q47: Identify the nature of the quadratic function.
a) Maximum
b) Minimum
..........................................
Q48: Identify the coordinates of the y-intercept.
..........................................
A quadratic has the equation y = (x + 5)(x − 1).
Q49: Identify the coordinates of the Zeros.
..........................................
Q50: Identify the equation of the axis of symmetry.
..........................................
© H ERIOT-WATT U NIVERSITY
Go online
40
TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Q51: Identify the coordinates of the turning point.
..........................................
A quadratic has the equation y = x 2 + 8x + 12.
Q52: Identify the coordinates of the Zeros.
..........................................
Q53: Identify the equation of the axis of symmetry.
..........................................
Q54: Identify the nature of the turning point.
..........................................
Q55: Identify the coordinates of the turning point.
..........................................
A quadratic has the equation y = − (x − 5) 2 + 4.
Q56: Identify the shape of the quadratic function.
a)
b)
..........................................
Q57: Identify the nature of the turning point.
..........................................
Q58: Identify the coordinates of the turning point.
..........................................
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Q59: Identify the equation of the axis of symmetry.
..........................................
Equation of a quadratic function
Q60: The graph shows a quadratic function of the form y = kx 2 . What is the value of
k?
..........................................
Q61: The graph shows a quadratic function of the form y = kx 2 . What is the value of
k?
..........................................
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Q62: The graph shows a quadratic function of the form y = (x − a) 2 + b.
a) What is the value of a?
b) What is the value of b?
..........................................
Q63: The graph shows a quadratic function of the form y = − (x − a) 2 + b.
a) What is the value of a?
b) What is the value of b?
..........................................
A quadratic function has the equation y = − (x + 3) 2 + 6
Q64: What is the nature of the turning point?
..........................................
Q65: The axis of symmetry is x = ?
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Q66: What are the coordinates of the turning point?
..........................................
Sketching the graph of a quadratic function
A quadratic function has equation y = (x − 2)(x + 3).
Q67: What are the coordinates of the roots?
..........................................
Q68: What are the coordinates of the y-intercept?
..........................................
Q69: What is the axis of symmetry?
..........................................
Q70: State the nature of the turning point.
..........................................
Q71: What are the coordinates of the turning point?
..........................................
Q72: State the shape of the sketch.
..........................................
A quadratic function has equation y = (x − 2)(3 − x).
Q73: What are the coordinates of the roots?
..........................................
Q74: What are the coordinates of the y-intercept?
..........................................
Q75: What is the axis of symmetry?
..........................................
Q76: State the nature of the turning point.
..........................................
Q77: What are the coordinates of the turning point?
..........................................
Q78: State the shape of the sketch.
..........................................
A quadratic function has equation y = x 2 + 6x + 8
Q79: What are the coordinates of the y-intercept?
..........................................
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS
Q80: Express the equation of the graph in completed square form y = (x + a) 2 + b.
..........................................
Q81: What are the coordinates of the turning point ?
..........................................
Q82: What is the shape of the function?
..........................................
The profit made by a company selling mobile phone covers can be calculated from
P (x) = 5x(600 − x) where P (x) is the profit, in pounds, when x is the price, in pence,
of the phone cover.
Q83: At what price should the company sell the phone cover to make maximum profit?
..........................................
Q84: What is the maximum profit the company can make from the sale of phone covers?
..........................................
Function notation
Q85:
a) f (x) = 6x − 1. Find the value of f (−1).
b) g(x) = x2 + 4x + 3. Find the value of g(2).
c) Given that n(x) = x2 + 3x − 5 evaluate n(−4)
d) h(x) =
8/ .
x
Find the value of h(10).
..........................................
Q86:
a) p(x) = 9x − 2. Find x when p(x) = 43.
b) q(x) = x3 + 5. Find x when q(x) = 6.
..........................................
..........................................
© H ERIOT-WATT U NIVERSITY
GLOSSARY
Glossary
coefficient
a number used to multiply a variable is called a coefficient, e.g. In 3x the coefficient
is 3; sometimes a letter will replace the number e.g. In px the coefficient is p
domain
the set A contains all the input values and this is called the domain
function f
a function f from set A to set B is a rule which assigns each element in A exactly
one element in B; this is often written as f : A → B
quadratic function
in a quadratic function the greatest power of the variable, normally x, is 2; y =
3x2 + 2x − 8 and y = − 4x2 − x + 6 are examples of quadratic functions
range
the set B contains all the output values and is called the range
standard number sets
the standard number sets are:
•
N = {1, 2, 3, 4, 5, . . .} the set of natural numbers.
•
W = {0, 1, 2, 3, 4, 5, . . .} the set of whole numbers.
•
Z = {. . ., -3, -2, -1, 0, 1, 2, 3, . . .} the set of integers.
•
Q = the set of all numbers which can be written as fractions, called the set of
rational numbers.
√
R = the set of rational and irrational numbers (such as 2 ), called the set of
real numbers.
√
I = the set of all real numbers which are not rational numbers e.g. 2, π
•
•
© H ERIOT-WATT U NIVERSITY
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46
ANSWERS: TOPIC 18
Answers to questions and activities
18 Graphs of quadratic functions
Identifying the features of a quadratic function exercise (page 6)
Q1: a.
This is because the 3 is positive.
Q2: b.
This is because the -2 is negative.
Q3: a. Maximum
This is because the -3 is negative.
Q4: b. Minimum
This is because the 2 is positive.
Q5:
x =
Q6:
(-4,-1)
2+4
2
= 3
x = - 5 +2 ( - 3) = - 4
y = ( - 4)2 + 8 × - 4 + 15
= 16 − 32 + 15
= -1
Q7:
(-2,0) and (3,0)
Q8:
(0, -3)
Q9:
a)
Q10: b) Minimum
Q11: (0,12)
Q12: (-2,0) and (-6,0)
Q13: x =
( - 6 + ( - 2))
2
=
-4
Q14: (-4,-4)
x = −4
y = ( - 4)2 + 8 × - 4 + 12
= 16 − 32 + 12
=
-4
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 18
Quadratic functions of the form y = kx 2 practice (page 13)
Q15:
The curve passes through (1, 10) substitute x = 1 and y = 10 into y = kx 2 .
⇒ 10 = k × 12
⇒ 10 = k
⇒ k = 10
The equation is y = 10x 2 .
Q16:
The curve passes through (2, -12) substitute x = 2 and y = − 12 into y = kx 2 .
⇒ −12 = k × 22
⇒ −12 = k × 4
⇒ −12 = 4k
⇒k = −3
The equation is y = − 3x 2 .
© H ERIOT-WATT U NIVERSITY
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48
ANSWERS: TOPIC 18
Quadratic functions of the form y = ± (x − a) 2 + b practice (page 16)
Q17:
By inspection the turning point is (1,-2).
Thus a = 1 and b = − 2.
The equation is y = (x − 1))2 + (−2)
or simply y = (x − 1)2 − 2.
Q18:
By inspection the turning point is (3,2).
Thus a = 3 and b = 2.
The equation is y = − (x − 3)2 + 2.
Determining equations of quadratic functions from graphs exercise (page 17)
Q19: k = 5
Q20: k = − 2
Q21: k = 1/4
Q22: k = 2
Q23: k = − 5
Q24: y = (x − 4)2 + 2
Q25: y = − (x + 1)2 − 3
Q26: y = (x − 3)2 − 4
Q27: y = − (x + 3)2 + 5
Sketching a quadratic graph from its equation practice (1) (page 26)
Q28:
a) The turning point is at (-2, 3).
b) It is a maximum turning point.
c) The axis of symmetry is x = − 2.
d) The y-intercept is (0, -1).
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 18
e)
Q29:
a) The turning point is at (1, 2).
b) It is a minimum turning point.
c) The axis of symmetry is x = 1.
d) The y-intercept is (0, 3).
e)
Sketching a quadratic graph from its equation practice (2) (page 30)
Q30:
a) The roots are (-3,0) and (1,0).
b) The coordinates of y-intercept is (0,-3).
c) The axis of symmetry is x = − 1.
d) It is a minimum turning point.
e) The minimum turning point is (-1,-4).
© H ERIOT-WATT U NIVERSITY
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50
ANSWERS: TOPIC 18
f)
Q31:
a) When the equation is factorised we get y = (x + 2)(x + 3). The roots are (-2,0)
and (-3,0).
b) The coordinates of y-intercept is (0,6).
c) The axis of symmetry is x = − 2 · 5.
d) It is a minimum turning point.
e) The minimum turning point is (-2·5,-0·25).
f)
Sketching quadratic functions exercise (page 31)
Q32:
a) The turning point is (3,-5).
b) It is a maximum turning point.
c) The axis of symmetry is x = 3.
d) The coordinates of y-intercept is (0,-14).
Q33:
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 18
a)
b)
c)
d)
The turning point is (-5,4).
It is a maximum turning point.
The axis of symmetry is x = −5.
The coordinates of y-intercept is (0,-21).
Q34:
a)
b)
c)
d)
The turning point is (2,4).
It is a minimum turning point.
The axis of symmetry is x = 2.
The coordinates of y-intercept is (0,8).
Q35:
a)
b)
c)
d)
The turning point is (-1,-4).
It is a minimum turning point.
The axis of symmetry is x = − 1.
The coordinates of y-intercept is (0,-3).
Q36:
a)
b)
c)
d)
e)
The roots are (-6,0) and (4,0).
The coordinates of y-intercept is (0,-24).
The axis of symmetry is x = − 1.
It is a minimum turning point.
The minimum turning point is (-1,-25).
Q37:
a)
b)
c)
d)
e)
The roots are (-3,0) and (2,0).
The coordinates of y-intercept is (0,-6).
The axis of symmetry is x = − 0 · 5.
It is a minimum turning point.
The minimum turning point is (-0·5,-6·25).
Q38:
a)
b)
c)
d)
e)
The roots are (0,0) and (6,0).
The coordinates of y-intercept is (0,0).
The axis of symmetry is x = 3.
It is a minimum turning point.
The minimum turning point is (3,-9).
Q39:
a)
b)
c)
d)
e)
The roots are (-7,0) and (3,0).
The coordinates of y-intercept is (0,-21).
The axis of symmetry is x = − 2.
It is a minimum turning point.
The minimum turning point is (-2,-25).
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52
ANSWERS: TOPIC 18
Answers from page 34.
Q40:
f (x) = 3x − 4
f (5) = 3 × 5 − 4 = 11
f (x) = x2 + 2x + 1
Q41: f ( - 2) = ( - 2)2 + 2 × ( - 2) + 1
= 4 + ( - 4) + 1 = 1
Q42: Any letter can be used to define a function but most commonly f is used.
g (x) = x3 + x + 7
g (0) = (0)3 + 0 + 7
= 0 + 0 + 7 = 7
Function notation exercise (page 35)
Q43:
a) No
b) No
c) Yes
d) Yes
Q44:
a) -7
b) 67
c) 10
d) 1/4
e) -1
Q45:
a) 6
b) 3
c) 4
End of topic 18 test (page 39)
Q46: a)
Q47: b) Minimum
Q48: (0,1)
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 18
53
Q49: (-5,0) and (1,0)
Q50: x =
( - 5 + 1)
2
=
-2
Q51: (-2,-12)
Q52: (-6,0) and (-2,0)
Q53: x =
( - 6 + ( - 2))
2
=
Q54: Minimum
Q55: (-4,-4)
Q56: b)
Q57: Maximum
Q58: (5,4)
Q59: x = 5
Q60: k = 7
Q61: k = − 2
Q62:
a) -1
b) 3
Q63:
a) 3
b) 5
Q64: Maximum
Q65: -3
Q66: (-3,6)
Q67: (-3,0) and (2,0)
Q68: (0,-6)
Q69: x = -0·5
Q70: Minimum
Q71: (-0·5, -6·25)
© H ERIOT-WATT U NIVERSITY
-4
54
ANSWERS: TOPIC 18
Q72: "Smiley" face
Q73: (2,0) and (3,0)
Q74: (0,-6)
Q75: x = 2·5
Q76: Maximum
Q77: (2·5, 0·25)
Q78: "Sad" face
Q79: (0,8)
Q80: y = (x + 3)2 - 1
Q81: (-3,-1)
Q82: "Smiley" face
Q83:
Steps:
•
To work out the price required to make maximum profit, you must find the value of
x which gives the maximum value of P (x) = 5x(600 − x).
•
Identify the equation of the axis of symmetry.
Answer: £300
pence
Q84:
Steps:
•
To find the maximum value of P (x) = 5x(600 − x), substitute the value of x into
the equation.
Answer: £450000
Q85:
a) -7
b) 15
c) -1
d) 0·8 or 4 /5
Q86:
a) 5
b) 1
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