H Algebra 2/Trig · Unit 2
Read each graph. Use the labeled roots or the labeled vertex to find a, then write the equation in standard form.
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y = ax² + bx + c
find a, then write standard form
The x-intercepts are the roots.
x = −2, so (x + 2) is a factor.
x = 4, so (x − 4) is a factor.
The labeled point is (0, −8).
y = a(x − r)(x − s)
y = a(x + 2)(x − 4)
Substitute (0, −8).
−8 = a(0 + 2)(0 − 4)
−8 = a(2)(−4)
−8 = −8a
Divide both sides by −8.
a = −8−8
a = 1
y = 1(x + 2)(x − 4)
Now write that equation in standard form.
Expand the parenthesis.
y = a(x − r)(x − s)
y = (x + 2)(x − 4)
Multiply the binomials.
(x + 2)(x − 4)
x² − 4x + 2x − 8
x² − 2x − 8
Distribute the a-value.
a = 1
y = 1(x² − 2x − 8)
The roots are labeled, and one other point is labeled. Write the equation in root form, solve for a, then write the equation in standard form.
The x-intercepts are the roots.
x = −1, so (x + 1) is a factor.
x = 3, so (x − 3) is a factor.
The labeled point is (0, −3).
y = a(x − r)(x − s)
y = a(x + 1)(x − 3)
Substitute (0, −3).
−3 = a(0 + 1)(0 − 3)
−3 = a(1)(−3)
−3 = −3a
Divide both sides by −3.
a = −3−3
a = 1
y = 1(x + 1)(x − 3)
Now write that equation in standard form.
Expand the parenthesis.
y = a(x − r)(x − s)
y = (x + 1)(x − 3)
Multiply the binomials.
(x + 1)(x − 3)
x² − 3x + x − 3
x² − 2x − 3
Distribute the a-value.
a = 1
y = 1(x² − 2x − 3)
y = x² − 2x − 3
Answer: y = x² − 2x − 3
The x-intercepts are the roots.
x = 1, so (x − 1) is a factor.
x = 3, so (x − 3) is a factor.
The labeled point is (0, 6).
y = a(x − r)(x − s)
y = a(x − 1)(x − 3)
Substitute (0, 6).
6 = a(0 − 1)(0 − 3)
6 = a(−1)(−3)
6 = 3a
Divide both sides by 3.
a = 63
a = 2
y = 2(x − 1)(x − 3)
Now write that equation in standard form.
Expand the parenthesis.
y = a(x − r)(x − s)
y = 2(x − 1)(x − 3)
Multiply the binomials.
(x − 1)(x − 3)
x² − 3x − x + 3
x² − 4x + 3
Distribute the a-value.
a = 2
y = 2(x² − 4x + 3)
y = 2x² − 8x + 6
Answer: y = 2x² − 8x + 6
The x-intercepts are the roots.
x = −2, so (x + 2) is a factor.
x = 2, so (x − 2) is a factor.
The labeled point is (1, 3).
y = a(x − r)(x − s)
y = a(x + 2)(x − 2)
Substitute (1, 3).
3 = a(1 + 2)(1 − 2)
3 = a(3)(−1)
3 = −3a
Divide both sides by −3.
a = 3−3
a = −1
y = −1(x + 2)(x − 2)
Now write that equation in standard form.
Expand the parenthesis.
y = a(x − r)(x − s)
y = −(x + 2)(x − 2)
Multiply the binomials.
(x + 2)(x − 2)
x² − 2x + 2x − 4
x² − 4
Distribute the a-value.
a = −1
y = −1(x² − 4)
y = −x² + 4
Answer: y = −x² + 4
The x-intercepts are the roots.
x = −3, so (x + 3) is a factor.
x = −1, so (x + 1) is a factor.
The labeled point is (0, −6).
y = a(x − r)(x − s)
y = a(x + 3)(x + 1)
Substitute (0, −6).
−6 = a(0 + 3)(0 + 1)
−6 = a(3)(1)
−6 = 3a
Divide both sides by 3.
a = −63
a = −2
y = −2(x + 3)(x + 1)
Now write that equation in standard form.
Expand the parenthesis.
y = a(x − r)(x − s)
y = −2(x + 3)(x + 1)
Multiply the binomials.
(x + 3)(x + 1)
x² + x + 3x + 3
x² + 4x + 3
Distribute the a-value.
a = −2
y = −2(x² + 4x + 3)
y = −2x² − 8x − 6
Answer: y = −2x² − 8x − 6
The x-intercepts are the roots.
x = 0, so x is a factor.
x = 4, so (x − 4) is a factor.
The labeled point is (3, 3).
y = a(x − r)(x − s)
y = a(x)(x − 4)
Substitute (3, 3).
3 = a(3)(3 − 4)
3 = a(3)(−1)
3 = −3a
Divide both sides by −3.
a = 3−3
a = −1
y = −1(x)(x − 4)
Now write that equation in standard form.
Expand the parenthesis.
y = a(x − r)(x − s)
y = −x(x − 4)
Multiply the binomials.
x(x − 4)
x² − 4x
Distribute the a-value.
a = −1
y = −1(x² − 4x)
y = −x² + 4x
Answer: y = −x² + 4x
The vertex and one other point are labeled. Write the equation in vertex form, solve for a, then write the equation in standard form.
The formula is
y = a(x − h)² + k
Vertex (3, −2).
Inside the parentheses, move left and right, opposite of the inside.
Right 3 is (x − 3)².
Outside the parentheses, move up and down. It is what it is.
Down 2 is −2.
y = a(x − 3)² − 2
Substitute the point (5, −6).
−6 = a(5 − 3)² − 2
−6 = a(2)² − 2
−6 = 4a − 2
−6 + 2 = 4a
−4 = 4a
a = −44
a = −1
y = −(x − 3)² − 2
Now write that equation in standard form.
Expand the parenthesis.
(x − h)² = (x − h)(x − h)
(x − 3)²
(x − 3)(x − 3)
y = −(x − 3)(x − 3) − 2
Multiply the binomials.
(x − 3)(x − 3)
x² − 3x − 3x + 9
x² − 6x + 9
Distribute the a-value.
a = −1
y = −1(x² − 6x + 9) − 2
y = −x² + 6x − 9 − 2
Combine like terms.
−9 − 2 = −11
The formula is
y = a(x − h)² + k
Vertex (2, 3).
Inside the parentheses, move left and right, opposite of the inside.
Right 2 is (x − 2)².
Outside the parentheses, move up and down. It is what it is.
Up 3 is +3.
y = a(x − 2)² + 3
Substitute the point (4, 7).
7 = a(4 − 2)² + 3
7 = a(2)² + 3
7 = 4a + 3
7 − 3 = 4a
4 = 4a
a = 44
a = 1
y = (x − 2)² + 3
Now write that equation in standard form.
Expand the parenthesis.
(x − h)² = (x − h)(x − h)
(x − 2)²
(x − 2)(x − 2)
y = (x − 2)(x − 2) + 3
Multiply the binomials.
(x − 2)(x − 2)
x² − 2x − 2x + 4
x² − 4x + 4
Distribute the a-value.
a = 1
y = 1(x² − 4x + 4) + 3
y = x² − 4x + 4 + 3
Combine like terms.
4 + 3 = 7
y = x² − 4x + 7
Answer: y = x² − 4x + 7
The formula is
y = a(x − h)² + k
Vertex (1, 4).
Inside the parentheses, move left and right, opposite of the inside.
Right 1 is (x − 1)².
Outside the parentheses, move up and down. It is what it is.
Up 4 is +4.
y = a(x − 1)² + 4
Substitute the point (3, 12).
12 = a(3 − 1)² + 4
12 = a(2)² + 4
12 = 4a + 4
12 − 4 = 4a
8 = 4a
a = 84
a = 2
y = 2(x − 1)² + 4
Now write that equation in standard form.
Expand the parenthesis.
(x − h)² = (x − h)(x − h)
(x − 1)²
(x − 1)(x − 1)
y = 2(x − 1)(x − 1) + 4
Multiply the binomials.
(x − 1)(x − 1)
x² − x − x + 1
x² − 2x + 1
Distribute the a-value.
a = 2
y = 2(x² − 2x + 1) + 4
y = 2x² − 4x + 2 + 4
Combine like terms.
2 + 4 = 6
y = 2x² − 4x + 6
Answer: y = 2x² − 4x + 6
The formula is
y = a(x − h)² + k
Vertex (1, −2).
Inside the parentheses, move left and right, opposite of the inside.
Right 1 is (x − 1)².
Outside the parentheses, move up and down. It is what it is.
Down 2 is −2.
y = a(x − 1)² − 2
Substitute the point (−1, −6).
−6 = a(−1 − 1)² − 2
−6 = a(−2)² − 2
−6 = 4a − 2
−6 + 2 = 4a
−4 = 4a
a = −44
a = −1
y = −(x − 1)² − 2
Now write that equation in standard form.
Expand the parenthesis.
(x − h)² = (x − h)(x − h)
(x − 1)²
(x − 1)(x − 1)
y = −(x − 1)(x − 1) − 2
Multiply the binomials.
(x − 1)(x − 1)
x² − x − x + 1
x² − 2x + 1
Distribute the a-value.
a = −1
y = −1(x² − 2x + 1) − 2
y = −x² + 2x − 1 − 2
Combine like terms.
−1 − 2 = −3
y = −x² + 2x − 3
Answer: y = −x² + 2x − 3
The formula is
y = a(x − h)² + k
Vertex (−1, −2).
Inside the parentheses, move left and right, opposite of the inside.
Left 1 is (x + 1)².
Outside the parentheses, move up and down. It is what it is.
Down 2 is −2.
y = a(x + 1)² − 2
Substitute the point (1, −10).
−10 = a(1 + 1)² − 2
−10 = a(2)² − 2
−10 = 4a − 2
−10 + 2 = 4a
−8 = 4a
a = −84
a = −2
y = −2(x + 1)² − 2
Now write that equation in standard form.
Expand the parenthesis.
(x − h)² = (x − h)(x − h)
(x + 1)²
(x + 1)(x + 1)
y = −2(x + 1)(x + 1) − 2
Multiply the binomials.
(x + 1)(x + 1)
x² + x + x + 1
x² + 2x + 1
Distribute the a-value.
a = −2
y = −2(x² + 2x + 1) − 2
y = −2x² − 4x − 2 − 2
Combine like terms.
−2 − 2 = −4
y = −2x² − 4x − 4
Answer: y = −2x² − 4x − 4
The formula is
y = a(x − h)² + k
Vertex (2, −1).
Inside the parentheses, move left and right, opposite of the inside.
Right 2 is (x − 2)².
Outside the parentheses, move up and down. It is what it is.
Down 1 is −1.
y = a(x − 2)² − 1
Substitute the point (4, −5).
−5 = a(4 − 2)² − 1
−5 = a(2)² − 1
−5 = 4a − 1
−5 + 1 = 4a
−4 = 4a
a = −44
a = −1
y = −(x − 2)² − 1
Now write that equation in standard form.
Expand the parenthesis.
(x − h)² = (x − h)(x − h)
(x − 2)²
(x − 2)(x − 2)
y = −(x − 2)(x − 2) − 1
Multiply the binomials.
(x − 2)(x − 2)
x² − 2x − 2x + 4
x² − 4x + 4
Distribute the a-value.
a = −1
y = −1(x² − 4x + 4) − 1
y = −x² + 4x − 4 − 1
Combine like terms.
−4 − 1 = −5
y = −x² + 4x − 5
Answer: y = −x² + 4x − 5
Each graph gives either the roots and a point, or the vertex and a point. The parabola may open up or down. Write the equation in standard form.
The x-intercepts are the roots.
x = −2, so (x + 2) is a factor.
x = 0, so x is a factor.
The labeled point is (1, 3).
y = a(x − r)(x − s)
y = a(x + 2)(x)
Substitute (1, 3).
3 = a(1 + 2)(1)
3 = a(3)(1)
3 = 3a
Divide both sides by 3.
a = 33
a = 1
y = 1(x + 2)(x)
Now write that equation in standard form.
Expand the parenthesis.
y = a(x − r)(x − s)
y = (x + 2)x
Multiply the binomials.
(x + 2)x
x² + 2x
Distribute the a-value.
a = 1
y = 1(x² + 2x)
y = x² + 2x
Answer: y = x² + 2x
The formula is
y = a(x − h)² + k
Vertex (−2, 3).
Inside the parentheses, move left and right, opposite of the inside.
Left 2 is (x + 2)².
Outside the parentheses, move up and down. It is what it is.
Up 3 is +3.
y = a(x + 2)² + 3
Substitute the point (0, 11).
11 = a(0 + 2)² + 3
11 = a(2)² + 3
11 = 4a + 3
11 − 3 = 4a
8 = 4a
a = 84
a = 2
y = 2(x + 2)² + 3
Now write that equation in standard form.
Expand the parenthesis.
(x − h)² = (x − h)(x − h)
(x + 2)²
(x + 2)(x + 2)
y = 2(x + 2)(x + 2) + 3
Multiply the binomials.
(x + 2)(x + 2)
x² + 2x + 2x + 4
x² + 4x + 4
Distribute the a-value.
a = 2
y = 2(x² + 4x + 4) + 3
y = 2x² + 8x + 8 + 3
Combine like terms.
8 + 3 = 11
y = 2x² + 8x + 11
Answer: y = 2x² + 8x + 11
The formula is
y = a(x − h)² + k
Vertex (0, −2).
Inside the parentheses, move left and right, opposite of the inside.
The vertex is on the y-axis, so the squared part is x².
Outside the parentheses, move up and down. It is what it is.
Down 2 is −2.
y = ax² − 2
Substitute the point (2, −6).
−6 = a(2)² − 2
−6 = 4a − 2
−6 + 2 = 4a
−4 = 4a
a = −44
a = −1
y = −x² − 2
Now write that equation in standard form.
Expand the parenthesis.
(x − h)² = (x − h)(x − h)
x²
xx
y = −xx − 2
Multiply the binomials.
xx
x²
Distribute the a-value.
a = −1
y = −1(x²) − 2
y = −x² − 2
Answer: y = −x² − 2
The x-intercepts are the roots.
x = 2, so (x − 2) is a factor.
x = 6, so (x − 6) is a factor.
The labeled point is (3, 3).
y = a(x − r)(x − s)
y = a(x − 2)(x − 6)
Substitute (3, 3).
3 = a(3 − 2)(3 − 6)
3 = a(1)(−3)
3 = −3a
Divide both sides by −3.
a = 3−3
a = −1
y = −1(x − 2)(x − 6)
Now write that equation in standard form.
Expand the parenthesis.
y = a(x − r)(x − s)
y = −(x − 2)(x − 6)
Multiply the binomials.
(x − 2)(x − 6)
x² − 6x − 2x + 12
x² − 8x + 12
Distribute the a-value.
a = −1
y = −1(x² − 8x + 12)
y = −x² + 8x − 12
Answer: y = −x² + 8x − 12
The formula is
y = a(x − h)² + k
Vertex (4, −3).
Inside the parentheses, move left and right, opposite of the inside.
Right 4 is (x − 4)².
Outside the parentheses, move up and down. It is what it is.
Down 3 is −3.
y = a(x − 4)² − 3
Substitute the point (2, −7).
−7 = a(2 − 4)² − 3
−7 = a(−2)² − 3
−7 = 4a − 3
−7 + 3 = 4a
−4 = 4a
a = −44
a = −1
y = −(x − 4)² − 3
Now write that equation in standard form.
Expand the parenthesis.
(x − h)² = (x − h)(x − h)
(x − 4)²
(x − 4)(x − 4)
y = −(x − 4)(x − 4) − 3
Multiply the binomials.
(x − 4)(x − 4)
x² − 4x − 4x + 16
x² − 8x + 16
Distribute the a-value.
a = −1
y = −1(x² − 8x + 16) − 3
y = −x² + 8x − 16 − 3
Combine like terms.
−16 − 3 = −19
y = −x² + 8x − 19
Answer: y = −x² + 8x − 19
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