Question

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Explanation:

⚡ Using what you learned: Zeros and Multiplicity

Step 1: Analyze the first graph

The first graph passes through the origin \( (0,0) \). At this point, the curve flattens out as it crosses the x-axis, which is a characteristic shape of a zero with an odd multiplicity greater than 1 (specifically, a cubic behavior, multiplicity of 3).

Looking at the grid:

• When \( x = 1 \), \( y \approx 2 \).
• When \( x = -1 \), \( y \approx -2 \).

This matches the cubic function:

$$ y = 2x^3 $$

Step 2: Analyze the second graph

The second graph is a parabola opening upwards with its vertex below the x-axis.

• The vertex is located at \( (0, -16) \).
• The x-intercepts (zeros) are at \( x = -2 \) and \( x = 2 \).

This matches a quadratic function of the form:

$$ y = a(x - 2)(x + 2) = a(x^2 - 4) $$

Using the y-intercept \( (0, -16) \):

$$ -16 = a(0^2 - 4) \implies a = 4 $$

Thus, the equation is:

$$ y = 4x^2 - 16 $$

Answer:

• First Graph (top): Matches the cubic function \( y = 2x^3 \)
• Second Graph (bottom): Matches the quadratic function \( y = 4x^2 - 16 \)