Question

how does the value of a in the function affect its graph when compared to the graph of the quadratic parent function?
$f(x)=0.9x^{2}$

in what ways is the graph of $f(x)$ different from the graph of the parent function? select all that apply.
a. the graph of $f(x)$ is wider.
b. the graph of $f(x)$ opens upward.
c. the graph of $f(x)$ is narrower.
d. the graph of $f(x)$ opens downward.

Explanation:

1. The parent quadratic function is \( y = x^2 \), where \( a = 1 \), and it opens upward (since \( a>0 \)) and has a certain width.
2. For the function \( f(x)=0.9x^2 \), the coefficient \( a = 0.9 \).

• When \( |a| < 1 \) in a quadratic function \( y = ax^2 \), the graph is wider than the parent function \( y=x^2 \) (because a smaller \( |a| \) makes the parabola "stretch" horizontally less, so it appears wider).
• Since \( a = 0.9>0 \), the graph opens upward (same direction as the parent function in terms of upward/downward, but the width is different).
• Option A: Since \( 0.9<1 \), the graph of \( f(x) \) is wider than \( y = x^2 \), so A is correct.
• Option B: The parent function \( y=x^2 \) also opens upward, so the direction of opening (upward) is not a difference from the parent function.
• Option C: A value of \( |a| < 1 \) makes the graph wider, not narrower, so C is incorrect.
• Option D: Since \( a = 0.9>0 \), the graph opens upward, not downward, so D is incorrect.

Answer:

A. The graph of \( f(x) \) is wider.