Question

match each quadratic function to its graph.
$f(x) = x^2 + 2$
$g(x) = -x^2$

Explanation:

Step1: Analyze \( f(x) = x^2 + 2 \)

The parent function is \( y = x^2 \), which opens upward (since the coefficient of \( x^2 \) is positive) with vertex at \( (0,0) \). For \( f(x) = x^2 + 2 \), we apply a vertical shift up by 2 units. So the vertex moves to \( (0, 2) \), and the parabola still opens upward.

Step2: Analyze \( g(x) = -x^2 \)

The parent function is \( y = x^2 \). The negative sign in front of \( x^2 \) reflects the parabola over the \( x \)-axis, so it opens downward, and the vertex remains at \( (0,0) \) (since there's no vertical shift).

Step3: Match to Graphs

• The graph with vertex at \( (0,2) \) and opening upward corresponds to \( f(x) = x^2 + 2 \) (the right graph, as it has a vertex above the origin and opens up).
• The graph with vertex at \( (0,0) \) and opening downward corresponds to \( g(x) = -x^2 \) (the left graph, as it opens down from the origin).

Answer:

• \( f(x) = x^2 + 2 \) matches the Right Graph (opening upward, vertex at \( (0,2) \))
• \( g(x) = -x^2 \) matches the Left Graph (opening downward, vertex at \( (0,0) \))