Relating The Standard And Factored Forms Worksheet - Chapter 3.2 Page 8
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113.2
In Summary
Key Ideas
• Some quadratic functions in standard form, f(x) 5 ax
1 bx 1 c, can be
2
expressed in factored form as f(x) 5 a(x 2 r)(x 2 s) by factoring. The two
forms of the quadratic function are equivalent.
• All quadratic functions in factored form can be expressed in standard form
by expanding. The two forms of the quadratic function are equivalent.
Need to Know
y
y
• Both the standard and factored forms
a
x
s)
0
(r
2
c
provide useful information for graphing
the parabola.
When a quadratic function is expressed
•
in factored form, f(x) 5 a(x 2 r)(x 2 s),
r
s
x
r
s
r and s are the x-intercepts, or
x
zeros, of the function. The axis of
symmetry is the vertical line that runs
c
through the midpoint of the zeros and
a
x
s)
0
(r
2
is defined by x 5 (r 1 s) 4 2. This value
is also the x-coordinate of the vertex.
When a quadratic function is expressed in standard form,
•
f(x) 5 ax
1 bx 1 c, c is the y-intercept of the function.
2
The value of a in both factored and standard forms determines
•
the direction that the parabola opens. If a . 0, the parabola
open up; if a , 0, the parabola opens down.
CHECK Your Understanding
y
4
x
y 5 x
2
.
The parabola shown is congruent to
1.
2 0
6
4
2
4
What are the zeros of the function?
a)
4
Write the equation in factored form.
b)
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Express each quadratic function in factored form. Then determine the
2.
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zeros, the equation of the axis of symmetry, and the coordinates of the
vertex.
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f (x) 5 2x
1 12x
f (x) 5 2x
1 100
2
2
a)
c)
f (x) 5 x
2 7x 1 12
f (x) 5 2x
1 5x 2 3
2
2
b)
d)
Tip
Communication
Express each quadratic function in standard form. Determine the
3.
Two parabolas that are
y-intercept.
congruent have exactly the
f (x) 5 3x(x 2 4)
f (x) 5 2(x 2 4) (3x 1 2)
same size and shape.
a)
c)
f (x) 5 (x 2 5) (x 1 7)
f (x) 5 (3x 2 4) (2x 1 5)
b)
d)
139
Working with Quadratic Functions: Standard and Factored Forms
NEL
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