Question

the graph of f(x), shown below, resembles the graph of g(x) = x², but it has been changed somewhat. which of the following could be the equation of f(x)? a. f(x) = x² + 2 b. f(x) = 3x² + 2 c. f(x) = -x² + 2 d. f(x) = 0.3x² + 2

Explanation:

Step1: Analyze the vertex and direction

The graph of \( G(x)=x^2 \) has vertex at \((0,0)\) and opens upward. The graph of \( F(x) \) also has vertex at \((0,2)\) (since it's shifted up 2 units) and opens upward (same direction as \( G(x) \), so the coefficient of \( x^2 \) should be positive, eliminating option C).

Step2: Analyze the vertical stretch/compression

For a quadratic function \( y = ax^2 + k \), if \( |a|>1 \), it's a vertical stretch; if \( 0<|a|<1 \), it's a vertical compression. The graph of \( F(x) \) is narrower than \( G(x)=x^2 \) (since it's closer to the y - axis), which means \( |a|>1 \). Let's check the options:

• Option A: \( a = 1 \), same as \( G(x) \) in terms of stretch (but shifted up). But the graph of \( F(x) \) is narrower than \( G(x) \), so \( a

eq1 \).

• Option B: \( a = 3 \), \( |3|>1 \), vertical stretch, which would make the graph narrower than \( y = x^2 \).
• Option D: \( a=0.3 \), \( 0.3<1 \), vertical compression, which would make the graph wider than \( y = x^2 \), not narrower.

Answer:

B. \( F(x) = 3x^2 + 2 \)