Identify key features of quadratic equations by matching graphs to their equations.

Let’s solve this step by step. We have four quadratic equations and four graphs. We need to match each equation to its correct graph, then fill in the key features: form name, x-intercepts, y-intercept, vertex, and axis of symmetry.

We’ll go one equation at a time.

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Equation 1: y = (x + 2)(x - 4)

This is in factored form.

To find x-intercepts, set y = 0:
(x + 2)(x - 4) = 0 → x = -2 or x = 4

So x-intercepts are (-2, 0) and (4, 0)

To find y-intercept, set x = 0:
y = (0 + 2)(0 - 4) = (2)(-4) = -8 → (0, -8)

Vertex: The x-coordinate of the vertex is halfway between the x-intercepts:
(-2 + 4)/2 = 1 → x = 1
Plug into equation: y = (1+2)(1-4) = (3)(-3) = -9 → Vertex: (1, -9)

Axis of symmetry: x = 1

Now look at the graphs. Which graph has x-intercepts at -2 and 4? That’s the first graph on the top left — it crosses x-axis at -2 and 4, and goes down to about y=-9 at x=1. Perfect match.

→ So Equation 1 matches Graph 1.

Form: Factored Form
x-intercepts: -2, 4
y-intercept: -8
vertex: (1, -9)
axis of symmetry: x = 1

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Equation 2: y = (x - 2)² - 4

This is in vertex form: y = a(x - h)² + k → vertex is (h, k)

Here, vertex is (2, -4)

To find x-intercepts, set y = 0:
(x - 2)² - 4 = 0 → (x - 2)² = 4 → x - 2 = ±2 → x = 4 or x = 0

So x-intercepts: (0, 0) and (4, 0)

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

Axis of symmetry: x = 2 (from vertex)

Look at graphs: which one has vertex at (2, -4)? And passes through (0,0) and (4,0)? That’s the fourth graph (top right). It opens up, vertex near (2,-4), crosses origin and (4,0).

→ Equation 2 matches Graph 4.

Form: Vertex Form
x-intercepts: 0, 4
y-intercept: 0
vertex: (2, -4)
axis of symmetry: x = 2

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Equation 3: y = (1/2)x² + x - 4

This is in standard form: y = ax² + bx + c

a = 1/2, b = 1, c = -4

Since a > 0, parabola opens upward.

Find vertex: x = -b/(2a) = -1 / (2 * 1/2) = -1 / 1 = -1

Then y = (1/2)(-1)² + (-1) - 4 = (1/2)(1) -1 -4 = 0.5 - 5 = -4.5 → Vertex: (-1, -4.5)

x-intercepts: Solve (1/2)x² + x - 4 = 0 → Multiply by 2: x² + 2x - 8 = 0 → Factor: (x+4)(x-2)=0 → x = -4, 2

So x-intercepts: (-4, 0), (2, 0)

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

Axis of symmetry: x = -1

Which graph matches? Look for vertex at (-1, -4.5), x-intercepts at -4 and 2. That’s the second graph (top middle-left). It crosses x at -4 and 2, vertex around (-1, -4.5). Yes.

→ Equation 3 matches Graph 2.

Form: Standard Form
x-intercepts: -4, 2
y-intercept: -4
vertex: (-1, -4.5)
axis of symmetry: x = -1

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Equation 4: y = -(x + 2)² + 4

This is in vertex form, but with a negative sign → opens downward.

Vertex: (h,k) = (-2, 4) because it’s (x + 2) = (x - (-2))

Set y = 0 to find x-intercepts:
-(x + 2)² + 4 = 0 → (x + 2)² = 4 → x + 2 = ±2 → x = 0 or x = -4

x-intercepts: (-4, 0), (0, 0)

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

Axis of symmetry: x = -2

Which graph? Opens downward, vertex at (-2, 4), crosses x at -4 and 0. That’s the third graph (top middle-right). Yes!

→ Equation 4 matches Graph 3.

Form: Vertex Form
x-intercepts: -4, 0
y-intercept: 0
vertex: (-2, 4)
axis of symmetry: x = -2

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Now let’s organize all answers by column (graph order from left to right):

Column 1 (Graph 1): Matches Equation 1: y = (x+2)(x-4)
Form: Factored Form
x-intercepts: -2, 4
y-intercept: -8
vertex: (1, -9)
axis of symmetry: x = 1

Column 2 (Graph 2): Matches Equation 3: y = (1/2)x² + x - 4
Form: Standard Form
x-intercepts: -4, 2
y-intercept: -4
vertex: (-1, -4.5)
axis of symmetry: x = -1

Column 3 (Graph 3): Matches Equation 4: y = -(x+2)² + 4
Form: Vertex Form
x-intercepts: -4, 0
y-intercept: 0
vertex: (-2, 4)
axis of symmetry: x = -2

Column 4 (Graph 4): Matches Equation 2: y = (x-2)² - 4
Form: Vertex Form
x-intercepts: 0, 4
y-intercept: 0
vertex: (2, -4)
axis of symmetry: x = 2

Final Answer:

Column 1:
Name the form: Factored Form
x-intercept(s): -2, 4
y-intercept(s): -8
vertex: (1, -9)
Axis of symmetry: x = 1

Column 2:
Name the form: Standard Form
x-intercept(s): -4, 2
y-intercept(s): -4
vertex: (-1, -4.5)
Axis of symmetry: x = -1

Column 3:
Name the form: Vertex Form
x-intercept(s): -4, 0
y-intercept(s): 0
vertex: (-2, 4)
Axis of symmetry: x = -2

Column 4:
Name the form: Vertex Form
x-intercept(s): 0, 4
y-intercept(s): 0
vertex: (2, -4)
Axis of symmetry: x = 2