Quadratic Functions in Intercept Form Practice Worksheet

Let’s solve each quadratic function step by step. We’ll find:

- x-intercepts (where y = 0)
- Axis of symmetry (x = -b/(2a) or average of x-intercepts if in intercept form)
- Vertex (plug axis of symmetry into equation to get y-value)
- y-intercept (set x = 0 and solve for y)

We’ll do this for all 6 problems.

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Problem 1: y = ½(x + 4)(x – 2)

This is in intercept form: y = a(x – p)(x – q), so x-intercepts are at x = p and x = q.

→ x-intercepts: set y=0 → (x+4)=0 or (x–2)=0 → x = -4, 2
So points: (-4, 0) and (2, 0)

→ Axis of symmetry: halfway between x-intercepts → x = (-4 + 2)/2 = -1

→ Vertex: plug x = -1 into equation
y = ½(-1 + 4)(-1 – 2) = ½(3)(-3) = ½(-9) = -4.5
Vertex: (-1, -4.5)

→ y-intercept: set x = 0
y = ½(0 + 4)(0 – 2) = ½(4)(-2) = ½(-8) = -4
Point: (0, -4)

✔ Final for #1:
x-intercepts: (-4, 0), (2, 0)
Axis of Symmetry: x = -1
Vertex: (-1, -4.5)
y-intercept: (0, -4)

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Problem 2: y = ¼(x – 4)(x – 8)

→ x-intercepts: x = 4, 8 → (4, 0), (8, 0)

→ Axis of symmetry: (4 + 8)/2 = 6 → x = 6

→ Vertex: plug x = 6
y = ¼(6 – 4)(6 – 8) = ¼(2)(-2) = ¼(-4) = -1
Vertex: (6, -1)

→ y-intercept: x = 0
y = ¼(0 – 4)(0 – 8) = ¼(-4)(-8) = ¼(32) = 8
Point: (0, 8)

✔ Final for #2:
x-intercepts: (4, 0), (8, 0)
Axis of Symmetry: x = 6
Vertex: (6, -1)
y-intercept: (0, 8)

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Problem 3: y = -(x – 2)(x – 2)

Note: This is y = -(x – 2)² — it’s a perfect square, so only one x-intercept (vertex touches x-axis).

→ x-intercepts: set y=0 → -(x–2)² = 0 → x = 2 (double root)
Point: (2, 0) — but since it says “label the x-intercepts”, we write it once? Or twice? Usually, even if repeated, we list it as one point unless specified. But technically, multiplicity 2. For graphing, it just touches at x=2.

But problem says “label the x-intercepts” — so we can say (2, 0) — and note it’s a double root.

Actually, let’s follow standard: if factored as (x-p)(x-q), even if p=q, we still report both? No — usually we say “x-intercept at x=2” (one point). But the worksheet has two blanks: “___ , 0) ( ___ , 0)” — so maybe they expect two same values? Let’s check context.

Looking at other problems — when there are two different roots, two blanks. Here, same root — probably fill both with 2.

So: x-intercepts: (2, 0), (2, 0) — but that’s redundant. Alternatively, perhaps they want us to recognize it’s a repeated root and still write it twice? I think for consistency with format, we’ll put (2, 0) and (2, 0).

But actually, in most curricula, you’d say “x-intercept at x=2” — but since the worksheet has two slots, maybe they expect two numbers. Let me see problem 6 later — similar case.

Alternatively, perhaps they consider it as one intercept but written twice? I’ll go with (2, 0) and (2, 0) to match the blank structure.

Wait — no, better to think: the equation is y = -(x-2)(x-2), so roots are x=2 and x=2 — so yes, two identical roots. So we fill both blanks with 2.

→ x-intercepts: (2, 0), (2, 0)

→ Axis of symmetry: since both roots same, axis is at x=2

→ Vertex: plug x=2 → y = -(2-2)(2-2) = 0 → vertex (2, 0)

→ y-intercept: x=0 → y = -(0-2)(0-2) = -(-2)(-2) = -(4) = -4 → (0, -4)

✔ Final for #3:
x-intercepts: (2, 0), (2, 0)
Axis of Symmetry: x = 2
Vertex: (2, 0)
y-intercept: (0, -4)

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Problem 4: y = -⅔(x + 1)(x – 5)

→ x-intercepts: x = -1, 5 → (-1, 0), (5, 0)

→ Axis of symmetry: (-1 + 5)/2 = 4/2 = 2 → x = 2

→ Vertex: plug x=2
y = -⅔(2 + 1)(2 – 5) = -⅔(3)(-3) = -⅔(-9) = (2/3)*9 = 18/3 = 6
Wait: -⅔ * 3 * (-3) = first, 3*(-3) = -9; then -⅔ * (-9) = + (2/3)*9 = 18/3 = 6
Yes → vertex (2, 6)

→ y-intercept: x=0
y = -⅔(0 + 1)(0 – 5) = -⅔(1)(-5) = -⅔(-5) = +10/3 ≈ 3.333...
But let’s keep as fraction: 10/3 → point (0, 10/3)

✔ Final for #4:
x-intercepts: (-1, 0), (5, 0)
Axis of Symmetry: x = 2
Vertex: (2, 6)
y-intercept: (0, 10/3)

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Problem 5: y = 4(x + 2)(x + 5)

→ x-intercepts: x = -2, -5 → (-2, 0), (-5, 0)

→ Axis of symmetry: (-2 + (-5))/2 = -7/2 = -3.5 → x = -3.5

→ Vertex: plug x = -3.5
First, write as fractions: -3.5 = -7/2
y = 4(-7/2 + 2)(-7/2 + 5) = 4(-7/2 + 4/2)(-7/2 + 10/2) = 4(-3/2)(3/2)
= 4 * (-9/4) = -9
Vertex: (-3.5, -9) or (-7/2, -9)

→ y-intercept: x=0
y = 4(0 + 2)(0 + 5) = 4(2)(5) = 40 → (0, 40)

✔ Final for #5:
x-intercepts: (-5, 0), (-2, 0) [order doesn’t matter, but usually left to right]
Axis of Symmetry: x = -3.5
Vertex: (-3.5, -9)
y-intercept: (0, 40)

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Problem 6: y = -(x – 3)(x – 3)

Same as problem 3 — repeated root.

→ x-intercepts: x=3, 3 → (3, 0), (3, 0)

→ Axis of symmetry: x=3

→ Vertex: plug x=3 → y=0 → (3, 0)

→ y-intercept: x=0 → y = -(0-3)(0-3) = -(-3)(-3) = -(9) = -9 → (0, -9)

✔ Final for #6:
x-intercepts: (3, 0), (3, 0)
Axis of Symmetry: x = 3
Vertex: (3, 0)
y-intercept: (0, -9)

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Now, compiling all answers clearly for each problem.

Final Answer:

1)
x-intercepts: (-4, 0), (2, 0)
Axis of Symmetry: x = -1
Vertex: (-1, -4.5)
y-intercept: (0, -4)

2)
x-intercepts: (4, 0), (8, 0)
Axis of Symmetry: x = 6
Vertex: (6, -1)
y-intercept: (0, 8)

3)
x-intercepts: (2, 0), (2, 0)
Axis of Symmetry: x = 2
Vertex: (2, 0)
y-intercept: (0, -4)

4)
x-intercepts: (-1, 0), (5, 0)
Axis of Symmetry: x = 2
Vertex: (2, 6)
y-intercept: (0, 10/3)

5)
x-intercepts: (-5, 0), (-2, 0)
Axis of Symmetry: x = -3.5
Vertex: (-3.5, -9)
y-intercept: (0, 40)

6)
x-intercepts: (3, 0), (3, 0)
Axis of Symmetry: x = 3
Vertex: (3, 0)
y-intercept: (0, -9)