1. No category

5.5
The x-Intercepts of a Quadratic
Relation
In the James Bond film, The Man with the Golden Gun, a complicated
spiral car jump was filmed in only one take. The exact speed of the car
had been determined by a computer, so the stunt could be performed
precisely as planned. For stunts like this and others, it is important for a
stunt coordinator to model not only where a car will begin to spiral and
its general path, but also where it will land.
Investigate
Tools
• graphing calculator
Compare the Equation of a Quadratic Relation to
Its Graph
1. a) Graph the quadratic relation y = x2 + 10x + 16. What are the
x-intercepts?
b) Factor the expression on the right side of the relation.
c) Compare the x-intercepts to the constant terms in the binomial
factors of the factored form of the relation. What do you notice?
d) Graph the factored relation in the same window. What do
you notice?
2. Copy and complete the table. Find the x-intercepts without graphing.
Relation
y=
x2
Factored Relation
x-Intercepts
+ 10x + 21
y = x2 - 8x + 15
y = x2 + 2x - 24
y = x2 - 49
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3. Graph each factored relation from question 2. Use the graph to find
intercept form
• a quadratic relation
of the form
y = a(x - r)(x - s)
• the constants, r
and s, represent the
x-intercepts of the
relation
zeros
• the x-coordinates of
the points where the
graph of a relation
crosses the x-axis
• the x-intercepts of a
relation
• the values of x for
which y = 0
Example 1
the x-intercepts. How do these x-intercepts compare to those you
found in question 2?
4. Reflect Given a quadratic relation in the form y = x2 + bx + c, how
can you find the x-intercepts without graphing?
The relation y = x2 + 5x + 6 can be expressed as y = (x + 2)(x + 3).
This is the intercept form of the quadratic relation. The x-intercepts
are -2 and -3.
In general, to find the x-intercepts of a quadratic relation
y = ax2 + bx + c, first write the relation in intercept form,
y = a(x - r)(x - s). The x-intercepts are at x = r and x = s.
Since y = 0 at the points where the graph crosses the x-axis, the
x-intercepts are also called the zeros of a quadratic relation.
Factor to Find the Zeros of a Quadratic Relation
Factor each quadratic relation. Use the factors to find the zeros. Then,
sketch the graph using the zeros and the y-intercept. Refer to the a-value
to decide if the parabola opens upward or downward.
a) y = 4x2 + 4x - 168
b) y = -3x2 + 24x - 48
c) y = x2 - 8x
d) y = x2 + 3x + 20
Solution
a) y = 4x2 + 4x - 168
= 4[x2 + x - 42]
Factor out the greatest common factor, 4.
= 4(x + 7)(x - 6)
Factor the resulting trinomial.
The zeros are at x = -7 and x = 6.
From the original relation, the y-intercept is -168. To sketch the
graph of the relation, plot the zeros and the y-intercept. Since a > 0,
the graph opens upward.
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y
20
(-7, 0)
-10 -8 -6 -4 -2 0
2
(6, 0)
4
6
8
10
x
-20
-40
-60
-80
y = 4x2 + 4x - 168
-100
-120
-140
(0, -168)
-160
b) y = -3x2 + 24x - 48
= -3[x2 - 8x + 16] Factor out the greatest common factor, -3.
= -3(x - 4)(x - 4)
= -3(x - 4)2
Since both factors are the same, there is only one zero at x = 4.
The intercept form of this relation is the same as the vertex form.
The vertex is (4, 0); this is also the x-intercept of the graph. From
the original relation, the y-intercept is -48. Since a < 0, the graph
opens downward.
y
-4 -2 0
-20
2
(4, 0)
4
6
8
10
12
14
x
y = -3x2 + 24x - 48
-40
(0, -48)
-60
-80
-100
-120
-140
-160
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c) y = x2 - 8x
= x(x - 8)
= (x + 0)(x - 8)
The greatest common factor is x.
You can also write the factors this way.
The zeros of this relation are at x = 0 and x = 8.
From the original relation, the y-intercept is 0. Since a > 0, the graph
opens upward.
y
20
16
y = x2 - 8x
12
8
4
(0, 0)
-4 -2 0
2
4
6
8
(8, 0)
10
12
14
x
-4
-8
-12
-16
d) y = x2 + 3x + 20
To factor this relation, try to find two values whose product is 20
and whose sum is 3.
There are no such values. There are two possible reasons for this:
• the relation has zeros that are not integers
• the relation has no zeros
Graph the relation to determine which reason applies to the relation.
The graph of y = x2 + 3x + 20 does not cross the x-axis, so the
relation has no zeros.
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Example 2
Different Forms of a Quadratic Relation
Consider the quadratic relation y = 3(x + 4)2 - 108.
a) What do you know about the graph of the given relation? Graph
this relation.
b) Write the relation in standard form. What do you know about the
graph of the relation from the standard form? Graph the relation.
c) Write the relation in intercept form. What do you know about the
graph of the relation given the intercept form? Graph the relation.
d) Compare the graphs of each form of the relation. Check your
graphs using technology.
Solution
a) From the given relation, the vertex is at (-4, -108).
Since a = 3, the graph opens upward.
y
-10 -8 -6 -4 -2 0
-20
2
4
6
8
10
x
y = 3(x + 4)2 - 108
-40
-60
-80
-100
(-4, -108)
b) y = 3(x + 4)2 - 108
= 3(x + 4)(x + 4) - 108
= 3[x2 + 4x + 4x + 16] - 108
= 3x2 + 12x + 12x + 48 - 108
= 3x2 + 24x - 60
From the standard form, the y-intercept is -60.
y
-10 -8 -6 -4 -2 0
-20
2
4
6
8
10
x
y = 3x2 + 24x - 60
-40
-60 (0, -60)
-80
-100
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c) y = 3x2 + 24x - 60
= 3[x2 + 8x - 20]
= 3(x - 2)(x + 10)
From the intercept form, the zeros, or x-intercepts, are at x = 2
and x = -10.
y
(-10, 0)
-10 -8 -6 -4 -2 0
2
(2, 0)
4
6
8
10
x
-20
-40
y = 3(x - 2)(x + 10)
-60
-80
-100
d) The graphs are identical for all forms of the relation. Use a graphing
calculator to check.
Example 3
Projectile Motion
A football is kicked from ground level. Its path is given by the relation
h = -4.9t2 + 22.54t, where h is the ball’s height above the ground, in
metres, and t is the time, in seconds.
a) Write the relation in intercept form.
b) Use the intercept form of the relation. Make a table of values with
times from 0.5 s to 3.5 s, in increments of 0.5 s.
c) Use the intercept form of the relation to find the zeros.
d) Plot the zeros and the points from the table of values. Draw a
smooth curve through the points.
e) When did the ball hit the ground? Explain how you found your answer.
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Solution
a) There is no constant term. Factor out -4.9t.
h = -4.9t2 + 22.54t
= -4.9t(t - 4.6)
b)
Time (s)
Height (m)
0.5
10.045
1.0
17.640
1.5
22.785
2.0
25.480
2.5
25.725
3.0
23.520
3.5
18.865
c) The intercept form is h = -4.9t(t - 4.6). The zeros are at t = 0 and
t = 4.6.
d)
Height of a Football
y
30
Height (m)
25
20
15
10
5
0
1
2
3
4
5
x
Time (s)
e) The zeros represent the times when the ball was on the ground. One
of the zeros is at t = 0, which is when the ball was kicked. The other
zero is at t = 4.6, which is when the ball landed.
The ball hit the ground at 4.6 s.
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Key Concepts
• Given a quadratic relation in intercept form, y = a(x - r)(x - s), the
zeros, or x-intercepts, are r and s.
• The vertex, standard, and intercept forms of a quadratic relation give
the same parabola when graphed.
Discuss the Concepts
D1. What do you know about the graph given each form of a quadratic
relation?
a) vertex form, y = a(x - h)2 + k
b) standard form, y = ax2 + bx + c
c) intercept form, y = a(x - r)(x - s)
D2. The x-intercepts of a quadratic relation are at x = -3 and x = 5,
and a = 5. Explain how you would find the standard form of the
quadratic relation.
Practise
A
For help with questions 1 to 5, refer to Example 1.
1. Find the x-intercepts of each quadratic relation.
a)
y
6
0
2
4
6
8
x
-6
-12
-18
-24
b)
y
-8 -6 -4 -2 0
x
-2
-4
-6
-8
-10
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2. Find the zeros of each quadratic relation.
a)
y
b)
y
24
12
20
10
16
8
12
6
8
4
4
2
-8 -4 0
4
8
-4 -2 0
x
2
4
x
3. Find the zeros of each quadratic relation.
a) y = (x - 5)(x + 3)
b) y = (x - 4)(x - 1)
c) y = 5(x - 9)(x - 9)
d) y = 3(x - 7)(x + 6)
e) y = -2(x + 8)(x + 2)
f) y = -3x(x + 5)
4. Find the zeros by factoring. Check by graphing the intercept form
and the standard form of each relation.
a) y = x2 + 10x + 16
b) y = x2 - 2x - 35
c) y = x2 - 6x - 7
d) y = 5x2 - 125
e) y = 3x2 + 39x + 108
f) y = 2x2 - 28x + 98
5. Find the zeros by factoring. Check by graphing the intercept form
and the standard form of each relation.
a) y = 4x2 - 16x
b) y = 5x2 - 125x
c) y = -5x2 + 5x + 360
d) y = -x2 - 18x - 81
e) y = -3.9x2 + 19.5x
f) y = 7.5x2 + 90x + 270
For help with questions 6 and 7, refer to Example 2.
6. Which relations have more than one zero? Explain how you know.
a) y = 3(x - 15)2 + 2
b) y = -5(x + 2)2 + 9
c) y = -(x - 8)2 - 6
d) y = 9(x + 3)2 - 10
7. Given each quadratic relation in vertex form, express the relation in
standard form and in intercept form. Then, check your answers by
graphing all three forms.
a) y = (x + 5)2 - 4
b) y = (x - 3)2 - 36
c) y = -2(x + 4)2 + 8
d) y = 6(x + 2)2 - 6
e) y = 3(x - 4)2 - 48
f) y = -4(x - 5)2 + 100
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Apply
B
8. A skateboarder jumps a gap that is 1.3 m wide. Her path can be
modelled by the relation h = -1.25d2 + 1.875d, where h is her height
above the ground and d is her horizontal distance from the edge of
the gap, both in metres.
a) Write the relation in intercept form.
b) Determine the zeros of the relation. Will the skateboarder make it
across the gap? Explain.
c) Copy and complete the table.
Horizontal Distance (m)
Height (m)
0
0.25
0.50
0.75
1.00
1.25
1.50
d) Estimate the maximum height the skateboarder reached during
her jump.
e) Graph the relation.
9. A second skateboarder
jumps off a ledge. His path
is modelled by the relation
h = -0.8d2 + 0.8d + 1.6,
where h is his height above
the ground and d is his
horizontal distance from
the ledge, both in metres.
a) What is the height of
the ledge?
b) Factor to find the zeros
of the relation.
c) At what point will the
skateboarder land on
the ground?
d) Graph the relation.
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10. In a target game at an amusement park, players launch a beanbag
toward a bucket using a mallet and a small seesaw. The path of a
beanbag that lands directly in the bucket can be modelled by the
relation h = -0.65d2 + 1.625d, where h is the beanbag’s height
above the table and d is the beanbag’s distance from the seesaw,
both in metres.
a) Find the zeros of the relation.
b) How far is the bucket from the seesaw?
c) Find the beanbag’s maximum height above the table to the nearest
tenth of a metre.
11. The path of a stunt
car can be modelled
by the relation
h = -0.1d2 + 0.5d + 3.6
where h is the car’s height
above the ground and
d is the car’s horizontal
distance from the edge of
the ramp, both in metres.
a) Find the zeros of the
relation.
b) How far from the
ramp will the car
land?
c) Suppose the stunt is
done inside a sound
studio with a ceiling
height of 10 m.
Will the car hit the
ceiling? Explain your
reasoning.
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Extend
C
12. A quadratic relation of the form y = ax2 + bx + c that cannot be
factored might still have zeros. Another method for finding the
zeros of a quadratic relation is to use the quadratic formula:
_______
-b ±√b2 - 4ac
x = __
2a
Use the quadratic formula to find the zeros of each relation.
a) y = 3x2 + 21x + 30
b) y = 16x2 - 40x - 75
c) y = 2x2 + 5x - 6
13. A quadratic relation of the form y = ax2 + bx + c has zeros
if the
_______
-b
±√b2 - 4ac
__
2a
is greater than or equal to zero. Determine if each relation has zeros.
expression b2 - 4ac in the quadratic formula x =
a) y = 5x2 + 3x + 15
b) y = 25x2 + 60x + 36
c) y = 7x2 - 10x + 5
14. A cannonball is shot from ground level with an initial velocity of
20 m/s. Ignoring air resistance, its path can be modelled by the
0.0125 d2 + (tan θ)d, where h is the cannonball’s
relation h = -_
(cos θ)2
height above the ground, in metres, d is the horizontal distance from
the cannon, in metres, and θ is the cannon’s angle of elevation, in
degrees. What angle of elevation will allow the cannonball to travel
the greatest distance?
θ
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