Introduction
Quadratic equations can be written in standard form,
factored form, and vertex form. While each form is
equivalent, certain forms easily reveal different features
of the graph of the quadratic function. In this lesson, you
will learn to use the processes of factoring and
completing the square to show key features of the graph
of a quadratic function and determine how these key
features relate to the characteristics of a real-world
situation.
1
5.6.2: Writing Equivalent Forms of Quadratic Functions
Key Concepts
• Recall that the standard form, or general form, of a
quadratic function is written as f(x) = ax2 + bx + c,
where a is the coefficient of the quadratic term, b is
the coefficient of the linear term, and c is the constant
term.
• The process of completing the square can be used to
transform a quadratic equation from standard form to
vertex form, f(x) = a(x – h)2 + k.
• Vertex form can be used to identify the key features of
a function’s graph.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Key Concepts, continued
• The vertex of a parabola is the point where the graph
changes from increasing to decreasing, or vice versa.
• In vertex form, the extremum of the graph of a
quadratic equation is easily identified using the vertex,
(h, k).
• If a < 0, the function achieves a maximum, where k is
the y-coordinate of the maximum and h is the
x-coordinate of the maximum.
• If a > 0, the function has a minimum, where k is the
y-coordinate of the minimum and h is the x-coordinate
of the minimum.
3
5.6.2: Writing Equivalent Forms of Quadratic Functions
Key Concepts, continued
• Because the axis of symmetry goes through the vertex,
the axis of symmetry is easily identified from vertex
form as x = h.
• The process of factoring can be used to transform a
quadratic equation in standard form to factored form,
f(x) = a(x – r)(x – s).
• The zeros of a function are the x-values where the
function value is 0.
• Setting the factored form equal to 0, 0 = a(x – r)(x – s),
the zeros are easily identified as r and s.
4
5.6.2: Writing Equivalent Forms of Quadratic Functions
Key Concepts, continued
• As long as the coefficients of x are 1, the x-intercepts
can be identified as (r, 0) and (s, 0).
• The axis of symmetry is easily identified from the
factored form as the axis of symmetry occurs at the
midpoint between the zeros. Therefore, the axis of
r +s
symmetry is x =
.
2
5
5.6.2: Writing Equivalent Forms of Quadratic Functions
Common Errors/Misconceptions
• confusing the attributes of different forms
• making errors in completing the square or factoring
6
5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice
Example 1
Suppose that the flight of a launched bottle rocket can
be modeled by the equation y = –x2 + 6x, where y
measures the rocket’s height above the ground in
meters and x represents the rocket’s horizontal distance
in meters from the launching spot at x = 0. How far does
the bottle rocket travel in the horizontal direction from
launch to landing? What is the maximum height the
bottle rocket reaches? How far has the bottle rocket
traveled horizontally when it reaches its maximum
height?
7
5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 1, continued
1. Identify the zeros of the function.
In the original equation, y represents the height of
the bottle rocket. At launch and landing, the height of
the bottle rocket is 0.
Write the original equation in factored form. Set it
equal to 0 to identify the zeros of the function.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 1, continued
y = –x2 + 6x
Original equation
0 = –x2 + 6x
Set the equation equal to 0.
0 = –(x2 – 6x)
Factor out –1.
0 = –x(x – 6)
Factor the binomial.
Solve for x by setting each factor equal to 0.
–x = 0 or x – 6 = 0
x = 0 or x = 6
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 1, continued
The x-intercepts are at x = 0 and x = 6. Find the
distance between the two points to determine how
far the bottle rocket travels in the horizontal direction.
6–0=6
The bottle rocket travels 6 meters in the horizontal
direction from launch to landing.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 1, continued
2. Determine the maximum height of the
bottle rocket.
The maximum height occurs at the vertex.
Write the equation in vertex form by completing the
square.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 1, continued
y = –x2 + 6x
Original equation
y = –(x2 – 6x)
Factor out the common factor,
–1, from the variable terms.
y = –(x2 – 6x + 9) + 9
Add and subtract the square of
1
of the x-term. Be sure to
2
multiply the subtracted term by
a, –1.
y = –(x – 3)2 + 9
Write the trinomial as a
binomial squared and simplify
the constant term.
5.6.2: Writing Equivalent Forms of Quadratic Functions
12
Guided Practice: Example 1, continued
The vertex form is y = –(x – 3)2 + 9. The vertex is
(3, 9). The maximum value is the y-coordinate of
the vertex, 9.
The bottle rocket reaches a maximum height of 9
meters.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 1, continued
3. Determine the horizontal distance from
the launch point to the maximum height of
the bottle rocket.
We know that the bottle rocket is launched from the
point (0, 0) and reaches a maximum height at (3, 9).
Subtract the x-values of the two points to find the
distance traveled horizontally.
3–0=3
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 1, continued
Another method is to take the total distance traveled
horizontally from launch to landing and divide it by 2
to find the same answer. This is because the
maximum value occurs halfway between the zeros of
the function.
6
=3
2
The bottle rocket has traveled 3 meters horizontally
when it reaches its maximum height.
✔
15
5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 1, continued
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice
Example 3
A football is kicked and follows a path given by
y = –0.03(x – 30)2 + 27, where y represents the height of
the ball in feet and x represents the ball’s horizontal
distance in feet. What is the maximum height the ball
reaches? What horizontal distance maximizes the
height? What are the zeros of the function? What do the
zeros represent in the context of the problem? What is
the total horizontal distance the ball travels? If the ball
reaches a height of 20.25 feet after traveling 15 feet
horizontally, will the ball make it over a 10-foot-tall goal
post that is 45 feet from the kicker?
5.6.2: Writing Equivalent Forms of Quadratic Functions
17
Guided Practice: Example 3, continued
1. Determine the maximum height of the ball.
The maximum occurs at the vertex.
The maximum value can be identified from the vertex
form of the quadratic.
The quadratic, y = –0.03(x – 30)2 + 27, is already in
vertex form, f(x) = a(x – h)2 + k, where the vertex is
(h, k).
The vertex is (30, 27) and the maximum value is 27.
The maximum height the ball reaches is 27 feet.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 3, continued
2. Determine the horizontal distance of the
ball when it reaches its maximum height.
The x-coordinate of the vertex maximizes the
quadratic.
The vertex is (30, 27).
The ball will have traveled 30 feet in the horizontal
direction when it reaches its maximum height.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 3, continued
3. Determine the zeros of the function.
The zeros of the function occur when the function
value is 0.
The factored form of the quadratic equation can be
used to identify the zeros of the function.
First write the function in standard form.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 3, continued
y = –0.03(x – 30)2 + 27
Original equation
y = –0.03(x2 – 60x + 900) + 27
Square the binomial.
y = –0.03x2 + 1.8x – 27 + 27
Distribute –0.03 over
the equation in
parentheses.
y = –0.03x2 + 1.8x
Simplify.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 3, continued
Factor the quadratic and set it equal to 0.
Standard form of the
y = –0.03x2 + 1.8x
function
Factor out the common
y = –0.03x(x – 60)
factor, –0.03x.
Set the equation in
0 = –0.03x(x – 60)
factored form equal to 0.
Set each factor equal to
–0.03x = 0 or x – 60 = 0
0 and solve for x.
x = 0 or x = 60
The zeros are 0 and 60.
5.6.2: Writing Equivalent Forms of Quadratic Functions
22
Guided Practice: Example 3, continued
4. Determine what the zeros represent in the
context of the problem.
The zeros represent where the ball was kicked from
at a horizontal distance of 0 feet, and where the ball
lands at a horizontal distance of 60 feet.
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5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 3, continued
5. Determine the total horizontal distance the
ball travels.
The distance between the two zeros gives the total
horizontal distance traveled, 60 feet.
24
5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 3, continued
6. Determine whether the ball will clear the
goal post.
The scenario asked if the ball would make it over a
10-foot-tall goal post that is 45 feet from the kicker if
the ball reached a height of 20.25 feet after traveling
15 feet horizontally.
The distance between the two zeros is 60 feet.
The axis of symmetry is half the distance between the
zeros at x = 30.
25
5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 3, continued
Fifteen feet is 15 feet to the left of the axis of
symmetry.
The height of the ball 15 units to the right of the axis
of symmetry will be the same as the height 15 units
to the left of the axis of symmetry, or 20.25 feet.
This is above the goal-post height of 10 feet.
The ball will go over the 10-foot-tall goal post.
✔
26
5.6.2: Writing Equivalent Forms of Quadratic Functions
Guided Practice: Example 3, continued
27
5.6.2: Writing Equivalent Forms of Quadratic Functions