Question

cool-down: matching equations and graphs
here are graphs that represent three quadratic functions, defined by

• $f(x) = x^2 - 4$
• $g(x) = 1 - x^2$
• $h(x) = x^2 + 4$

1. match each equation to a graph that represents it. explain how you know.

Explanation:

Step1: Analyze the vertex of each quadratic function

For a quadratic function in the form \( y = ax^2 + k \) (when \( b = 0 \)), the vertex is at \( (0, k) \), and the parabola opens up if \( a>0 \), down if \( a < 0 \).

• For \( f(x)=x^{2}-4 \), \( a = 1>0 \) (opens up), vertex at \( (0, - 4) \).
• For \( g(x)=1 - x^{2}=-x^{2}+1 \), \( a=-1 < 0 \) (opens down), vertex at \( (0,1) \).
• For \( h(x)=x^{2}+4 \), \( a = 1>0 \) (opens up), vertex at \( (0,4) \).

Step2: Match the functions to the graphs

• Graph A: Opens up, vertex at \( (0,4) \), so it matches \( h(x)=x^{2}+4 \).
• Graph B: Opens up, vertex at \( (0, - 4) \), so it matches \( f(x)=x^{2}-4 \).
• Graph C: Opens down, vertex at \( (0,1) \), so it matches \( g(x)=1 - x^{2} \).

Answer:

• \( f(x)=x^{2}-4 \) matches Graph B.
• \( g(x)=1 - x^{2} \) matches Graph C.
• \( h(x)=x^{2}+4 \) matches Graph A.