Solved Practice Worksheet: Graphing Quadratic Functions in | Chegg.com - Free Printable

Let’s solve each problem one by one. We’ll find a, b, c; whether the parabola opens up or down; if the vertex is a max or min; x-intercepts; y-intercept; axis of symmetry; and the vertex coordinates.

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Problem 3: y = x² - 4x + 8

Step 1: Identify a, b, c
This is in standard form: y = ax² + bx + c
→ a = 1, b = -4, c = 8

Step 2: Opens up or down?
Since a = 1 (positive), it opens up.

Step 3: Is vertex a max or min?
Opens up → vertex is a min

Step 4: Find x-intercepts
Set y = 0:
0 = x² - 4x + 8
Use quadratic formula: x = [-b ± √(b² - 4ac)] / (2a)
Discriminant = (-4)² - 4(1)(8) = 16 - 32 = -16 → negative
→ No real x-intercepts (graph doesn’t cross x-axis)

Step 5: y-intercept
Set x = 0: y = 0 - 0 + 8 = 8 → (0, 8)

Step 6: Axis of Symmetry
x = -b/(2a) = -(-4)/(2*1) = 4/2 = 2

Step 7: Vertex
Plug x = 2 into equation:
y = (2)² - 4(2) + 8 = 4 - 8 + 8 = 4
→ Vertex: (2, 4)

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Problem 4: y = 2x² + 8x

Step 1: Identify a, b, c
a = 2, b = 8, c = 0 (no constant term)

Step 2: Opens up or down?
a = 2 > 0 → opens up

Step 3: Vertex is max or min?
Opens up → min

Step 4: x-intercepts
Set y = 0:
0 = 2x² + 8x
Factor: 2x(x + 4) = 0
→ x = 0 or x = -4
→ x-intercepts: (0, 0) and (-4, 0)

Step 5: y-intercept
Set x = 0: y = 0 → (0, 0)

Step 6: Axis of Symmetry
x = -b/(2a) = -8/(2*2) = -8/4 = -2

Step 7: Vertex
Plug x = -2 into equation:
y = 2(-2)² + 8(-2) = 2(4) - 16 = 8 - 16 = -8
→ Vertex: (-2, -8)

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Problem 5: y = -3x² -12x + 1

Step 1: Identify a, b, c
a = -3, b = -12, c = 1

Step 2: Opens up or down?
a = -3 < 0 → opens down

Step 3: Vertex is max or min?
Opens down → max

Step 4: x-intercepts
Set y = 0:
0 = -3x² -12x + 1
Use quadratic formula:
x = [-(-12) ± √((-12)² - 4*(-3)*1)] / (2*(-3))
= [12 ± √(144 + 12)] / (-6)
= [12 ± √156] / (-6)
√156 = √(4*39) = 2√39 ≈ 12.49
So:
x₁ = (12 + 12.49)/(-6) ≈ 24.49/-6 ≈ -4.08
x₂ = (12 - 12.49)/(-6) ≈ (-0.49)/(-6) ≈ 0.08
We can leave exact form:
x = [12 ± 2√39]/(-6) = [6 ± √39]/(-3)
But for school level, approximate decimals are okay unless asked otherwise.
→ x-intercepts: approximately (-4.08, 0) and (0.08, 0)

Step 5: y-intercept
Set x = 0: y = 1 → (0, 1)

Step 6: Axis of Symmetry
x = -b/(2a) = -(-12)/(2*(-3)) = 12 / (-6) = -2

Step 7: Vertex
Plug x = -2 into equation:
y = -3(-2)² -12(-2) + 1 = -3(4) + 24 + 1 = -12 + 24 + 1 = 13
→ Vertex: (-2, 13)

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Final Answer:

Problem 3:
a = 1, b = -4, c = 8
Opens up
Vertex is a min
x-intercepts: none
y-intercept: (0, 8)
Axis of Symmetry: x = 2
Vertex: (2, 4)

Problem 4:
a = 2, b = 8, c = 0
Opens up
Vertex is a min
x-intercepts: (0, 0) and (-4, 0)
y-intercept: (0, 0)
Axis of Symmetry: x = -2
Vertex: (-2, -8)

Problem 5:
a = -3, b = -12, c = 1
Opens down
Vertex is a max
x-intercepts: approx (-4.08, 0) and (0.08, 0)
y-intercept: (0, 1)
Axis of Symmetry: x = -2
Vertex: (-2, 13)