Question

how does the value of a in the function affect its graph when compared to the graph of the quadratic parent function?
( g(x) = 10x^2 )

in what ways is the graph of ( g(x) ) different from the graph of the parent function? select all that apply

a. the graph of ( g(x) ) is narrower
b. the graph of ( g(x) ) opens downward
c. the graph of ( g(x) ) is wider
d. the graph of ( g(x) ) opens upward

Explanation:

Step1: Recall the parent quadratic function

The parent quadratic function is \( y = x^2 \), which opens upward (since the coefficient of \( x^2 \) is positive) and has a "width" determined by the coefficient of \( x^2 \).

Step2: Analyze the coefficient of \( x^2 \) in \( g(x) = 10x^2 \)

For a quadratic function \( y = ax^2 \):

• If \( |a|> 1 \), the graph is narrower than the graph of \( y=x^2 \). Here, \( a = 10 \), and \( |10|=10>1 \), so the graph of \( g(x) \) is narrower than the parent function's graph.
• If \( a>0 \), the graph opens upward. Since \( 10>0 \), the graph of \( g(x) \) opens upward.

Now, let's analyze each option:

• Option A: Since \( |10| > 1 \), the graph of \( g(x) \) is narrower. This is correct.
• Option B: The coefficient \( 10 \) is positive, so the graph opens upward, not downward. This is incorrect.
• Option C: Since \( |10|>1 \), the graph is narrower, not wider. This is incorrect.
• Option D: Since the coefficient \( 10 \) is positive, the graph opens upward. This is correct.

Answer:

A. The graph of \( g(x) \) is narrower, D. The graph of \( g(x) \) opens upward