Question

transforming parent functions
what is the equation of the
function graphed?
\\( f(x) = x^2 + 3 \\)
\\( f(x) = 3x^2 \\)
\\( f(x) = (x + 3)^2 \\)
\\( f(x) = x^2 - 3 \\)

Explanation:

Step1: Recall parent function \( y = x^2 \)

The parent function \( y = x^2 \) has its vertex at \( (0,0) \).

Step2: Analyze the vertex of the graphed function

The graphed function \( F(x) \) has its vertex at \( (0, 3) \) (since it's on the y - axis, \( x = 0 \), and the y - coordinate is 3).

Step3: Analyze each option

• For \( F(x)=x^{2}+3 \): The transformation of the parent function \( y = x^2 \) is a vertical shift up by 3 units. The vertex of \( y=x^{2}+3 \) is \( (0,3) \), which matches the vertex of the graphed function.
• For \( F(x) = 3x^{2}\): This is a vertical stretch of the parent function \( y=x^{2}\). The vertex of \( y = 3x^{2}\) is \( (0,0) \), which does not match the vertex \( (0,3) \) of the graphed function.
• For \( F(x)=(x + 3)^{2}\): This is a horizontal shift of the parent function \( y=x^{2}\) to the left by 3 units. The vertex of \( y=(x + 3)^{2}\) is \( (-3,0) \), which does not match the vertex \( (0,3) \) of the graphed function.
• For \( F(x)=x^{2}-3 \): This is a vertical shift down by 3 units of the parent function \( y=x^{2}\). The vertex of \( y=x^{2}-3 \) is \( (0,-3) \), which does not match the vertex \( (0,3) \) of the graphed function.

Answer:

\( \boldsymbol{F(x)=x^{2}+3} \) (corresponding to the first option: \( F(x)=x^{2}+3 \))