Question

choose the 3 equations that are represented by the graph below.

choose all that apply

a \\( f(x) = 4(x - 3)^2 - 4 \\)
b \\( f(x) = 4(x - 4)(x + 2) \\)
c \\( f(x) = 4x^2 + 20x + 32 \\)
d \\( f(x) = 4(x + 3)^2 - 4 \\)
e \\( f(x) = 4(x + 4)(x + 2) \\)
f \\( f(x) = 4x^2 + 24x + 32 \\)

Explanation:

⚡ Using what you learned: graphing quadratic functions

Step 1: Identify key features from the graph

From the given graph of the parabola, we can identify the following key points:

• Vertex (lowest point): Located at \( (-3, -4) \).
• \(x\)-intercepts (roots): The graph crosses the \(x\)-axis at \( x = -4 \) and \( x = -2 \).
• \(y\)-intercept: The graph crosses the \(y\)-axis at \( (0, 32) \) (since \( f(0) = 32 \) when we project upward).

Step 2: Test Vertex Form

The vertex form of a quadratic function is:
\[ f(x) = a(x - h)^2 + k \]

Substituting the vertex \( (h, k) = (-3, -4) \):
\[ f(x) = a(x + 3)^2 - 4 \]

Using the point \( (-2, 0) \) to find \( a \):
\[ 0 = a(-2 + 3)^2 - 4 \]
\[ 4 = a(1)^2 \implies a = 4 \]

Thus, the vertex form is:
\[ f(x) = 4(x + 3)^2 - 4 \]

This matches option D.

Step 3: Test Factored Form

The factored form of a quadratic function is:
\[ f(x) = a(x - p)(x - q) \]

Substituting the \(x\)-intercepts \( p = -4 \) and \( q = -2 \), and using \( a = 4 \):
\[ f(x) = 4(x + 4)(x + 2) \]

This matches option E.

Step 4: Test Standard Form

Expand the factored form to find the standard form:
\[ f(x) = 4(x^2 + 2x + 4x + 8) \]
\[ f(x) = 4(x^2 + 6x + 8) \]
\[ f(x) = 4x^2 + 24x + 32 \]

This matches option F.

Answer:

The 3 correct equations are:

• D \( f(x) = 4(x + 3)^2 - 4 \)
• E \( f(x) = 4(x + 4)(x + 2) \)
• F \( f(x) = 4x^2 + 24x + 32 \)