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Lesson 4.5 Exercises, pages 295–299
A
Students should verify that two forms of an equation represent the same
quadratic function, where necessary.
3. Determine the number that would be added to each binomial to get a
perfect square trinomial. Add the number, then factor the trinomial.
a) x2 + 12x
b) x2 - 8x
2
12
a b ⴝ 36
2
a
x ⴙ 12x ⴙ 36 ⴝ (x ⴙ 6)
2
2
x ⴚ 8x ⴙ 16 ⴝ (x ⴚ 4)2
2
d) x2 + 2x
2
49
7
a b ⴝ
2
4
22
2
2
b ⴝ 16
3
c) x2 + 7x
x2 ⴙ 7x ⴙ
ⴚ8
49
7
ⴝ ax ⴙ b
4
2
2
3
2
3 2
2
2
9
3
P Q ⴝ a4b , or 16
2
x2 ⴙ x ⴙ
9
3
ⴝ ax ⴙ b
16
4
2
4.5 Equivalent Forms of the Equation of a Quadratic Function—Solutions
DO NOT COPY.
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4. Do the equations in each pair represent the same quadratic function?
a) y = x2 + 4x - 1; y = (x + 2)2 - 5
Expand: y ⴝ (x ⴙ 2)2 ⴚ 5
y ⴝ x2 ⴙ 4x ⴙ 4 ⴚ 5
y ⴝ x2 ⴙ 4x ⴚ 1
This matches the other equation. So, the equations represent the
same quadratic function.
b) y = - 2x2 - 6x + 1; y = -2(x + 3)2 + 1
Expand: y ⴝ ⴚ2(x ⴙ 3)2 ⴙ 1
y ⴝ ⴚ2(x2 ⴙ 6x ⴙ 9) ⴙ 1
y ⴝ ⴚ2x2 ⴚ 12x ⴚ 18 ⴙ 1
y ⴝ ⴚ2x2 ⴚ 12x ⴚ 17
This does not match the other equation. So, the equations do not
represent the same quadratic function.
B
5. Write each equation in standard form. Verify algebraically that the
two forms of the equation represent the same quadratic function.
a) y = x2 + 6x + 1
b) y = x2 - 2x - 4
2
Add and subtract: a b ⴝ 9
Add and subtract: a
y ⴝ (x2 ⴙ 6x) ⴙ 1
y ⴝ (x2 ⴚ 2x) ⴚ 4
6
2
2
ⴚ2
b ⴝ1
2
ⴝ (x2 ⴙ 6x ⴙ 9 ⴚ 9) ⴙ 1
ⴝ (x2 ⴚ 2x ⴙ 1 ⴚ 1) ⴚ 4
ⴝ (x2 ⴙ 6x ⴙ 9) ⴚ 9 ⴙ 1
ⴝ (x2 ⴚ 2x ⴙ 1) ⴚ 1 ⴚ 4
ⴝ (x ⴙ 3) ⴚ 8
ⴝ (x ⴚ 1)2 ⴚ 5
2
6. Write each equation in standard form. Use a graphing calculator to
verify that the two forms of the equation represent the same
quadratic function.
a) y = 2x2 + 8x - 4
b) y = 3x2 - 12x - 1
y ⴝ 2(x2 ⴙ 4x) ⴚ 4
2
2
ⴚ4
b ⴝ4
2
Add and subtract: a b ⴝ 4
Add and subtract: a
y ⴝ 2(x2 ⴙ 4x ⴙ 4 ⴚ 4) ⴚ 4
y ⴝ 3(x2 ⴚ 4x ⴙ 4 ⴚ 4) ⴚ 1
4
2
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y ⴝ 3(x2 ⴚ 4x) ⴚ 1
ⴝ 2(x2 ⴙ 4x ⴙ 4) ⴚ 2(4) ⴚ 4
ⴝ 3(x2 ⴚ 4x ⴙ 4) ⴚ 3(4) ⴚ 1
ⴝ 2(x ⴙ 2)2 ⴚ 12
ⴝ 3(x ⴚ 2)2 ⴚ 13
DO NOT COPY.
4.5 Equivalent Forms of the Equation of a Quadratic Function—Solutions
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7. Write each equation in standard form.
3
a) y = 4x2 - 6x + 2
3
4
y ⴝ (x2 ⴚ 8x) ⴙ 2
Add and subtract: a
2
ⴚ8
b ⴝ 16
2
3
4
3
3
ⴝ (x2 ⴚ 8x ⴙ 16) ⴚ (16) ⴙ 2
4
4
y ⴝ (x2 ⴚ 8x ⴙ 16 ⴚ 16) ⴙ 2
ⴝ 0.75(x ⴚ 4)2 ⴚ 12 ⴙ 2
ⴝ 0.75(x ⴚ 4)2 ⴚ 10
1
b) y = - 2x2 + 5x + 1
1
2
y ⴝ ⴚ (x2 ⴚ 10x) ⴙ 1
Add and subtract: a
2
ⴚ10
b ⴝ 25
2
1
2
1
1
ⴝ ⴚ (x2 ⴚ 10x ⴙ 25) ⴚ (ⴚ25) ⴙ 1
2
2
y ⴝ ⴚ (x2 ⴚ 10x ⴙ 25 ⴚ 25) ⴙ 1
ⴝ ⴚ0.5( x ⴚ 5)2 ⴙ 12.5 ⴙ 1
ⴝ ⴚ0.5(x ⴚ 5)2 ⴙ 13.5
8. Write each equation in standard form, then identify the given
characteristic of the graph of the function.
a) y = 2x2 + 5x - 3; the coordinates of the vertex
y ⴝ 2 ax 2 ⴙ xb ⴚ 3
5
2
2
5
2
2
25
5
Add and subtract: P Q ⴝ a b , or
4
16
2
y ⴝ 2 ax 2 ⴙ x ⴙ
5
2
ⴝ 2 ax 2 ⴙ x ⴙ
5
2
2
ⴝ 2 ax ⴙ b ⴚ
5
4
25
25
ⴚ b ⴚ3
16
16
25
25
b ⴚ 2a b ⴚ 3
16
16
25
ⴚ3
8
ⴝ 2(x ⴙ 1.25)2 ⴚ 6.125
Compare this with y ⴝ a(x ⴚ p)2 ⴙ q.
The vertex of the parabola has coordinates (ⴚ1.25, ⴚ6.125).
24
4.5 Equivalent Forms of the Equation of a Quadratic Function—Solutions
DO NOT COPY.
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b) y = -4x2 + 11x + 12; the y-coordinate of the vertex
y ⴝ ⴚ4 ax 2 ⴚ
11
xb ⴙ 12
4
11 2
2
4
11
121
Add and subtract: P
Q ⴝ aⴚ 8 b , or 64
2
ⴚ
y ⴝ ⴚ4 ax 2 ⴚ
ⴝ ⴚ4 ax 2 ⴚ
11
121
121
xⴙ
ⴚ
b ⴙ 12
4
64
64
11
121
121
xⴙ
b ⴚ 4 aⴚ
b ⴙ 12
4
64
64
2
ⴝ ⴚ4 ax ⴚ
11
121
ⴙ 12
b ⴙ
8
16
ⴝ ⴚ4 ax ⴚ
11
313
b ⴙ
8
16
2
Compare this with y ⴝ a(x ⴚ p)2 ⴙ q.
The vertex of the parabola has coordinates a ,
So, the y-coordinate of the vertex is
313
.
16
11 313
b.
8 16
9. Compare the two solutions for completing the square. Identify the
error, then explain each step for the correct solution.
Solution A
Solution B
2
3
2
- (x 2 + 6x) - 10
3
2
- (x 2 + 6x + 9 - 9) - 10
3
2
- (x 2 + 6x + 9) + 6 - 10
3
2
- (x + 3)2 - 4
3
2
3
2 2
- (x + 6x) - 10
3
2
- (x 2 + 6x + 9 - 9) - 10
3
2
- (x 2 + 6x + 9) - 9 - 10
3
2
- (x + 3)2 - 19
3
y = - x 2 - 4x - 10
y = - x 2 - 4x - 10
Step 1
y =
y =
Step 2
y =
Step 3
y =
Step 4
y =
y =
y =
y =
Solution A is correct.
Solution B has an error in the 4th line. When ⴚ9 was taken out of the
2
brackets, it should have been multiplied by ⴚ .
2
3
3
Step 1: Remove ⴚ as a common factor from the first 2 terms.
Step 2: Add and subtract the square of one-half of 6, the coefficient of x.
2
3
Step 3: Take ⴚ9 outside of the brackets by multiplying it by ⴚ .
Step 4: Write the terms in the brackets as a perfect square. Simplify the
terms outside of the brackets.
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DO NOT COPY.
4.5 Equivalent Forms of the Equation of a Quadratic Function—Solutions
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10. Identify the errors in this solution of completing the square.
Write the correct solution.
y
y
y
y
=
=
=
=
-3x2 - 6x + 4
- 3(x2 - 2x) + 4
-3(x2 - 2x + 1) + 4 + 3
- 3(x - 1)2 + 7
There is an error in line 2 of the solution: when a factor of ⴚ3 was
removed from ⴚ6x, the result should have been 2x, not ⴚ2x.
A correct solution is:
y ⴝ ⴚ3x2 ⴚ 6x ⴙ 4
y ⴝ ⴚ3(x2 ⴙ 2x) ⴙ 4
y ⴝ ⴚ3(x2 ⴙ 2x ⴙ 1 ⴚ 1) ⴙ 4
y ⴝ ⴚ3(x2 ⴙ 2x ⴙ 1) ⴚ 3(ⴚ1) ⴙ 4
y ⴝ ⴚ3(x ⴙ 1)2 ⴙ 7
C
11. What are the coordinates of the vertex of the graph of
y = ax2 + bx + c?
Complete the square.
y ⴝ ax2 ⴙ bx ⴙ c
y ⴝ a ax 2 ⴙ a xb ⴙ c
b
y ⴝ a ax 2 ⴙ a x ⴙ
b
2
Add and subtract: a b ⴝ
b
2a
b2
4a2
b2
b2
ⴚ 2b ⴙ c
2
4a
4a
b2
b2
y ⴝ a ax 2 ⴙ a x ⴙ 2b ⴚ a a 2b ⴙ c
4a
4a
b
2
y ⴝ a ax ⴙ
b
b2
ⴙc
b ⴚ
2a
4a
y ⴝ a ax ⴙ
b
ⴚ b2 ⴙ 4ac
b ⴙ
2a
4a
2
Compare this with y ⴝ a(x ⴚ p)2 ⴙ q.
The vertex of the parabola has coordinates aⴚ
26
b ⴚ b2 ⴙ 4ac
,
b.
2a
4a
4.5 Equivalent Forms of the Equation of a Quadratic Function—Solutions
DO NOT COPY.
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