Question

4.4.1 study: transformation of parent functions
what is the equation of the
function graphed?
\\( \bigcirc \\ f(x) = x^2 + 3 \\)
\\( \bigcirc \\ f(x) = 3x^2 \\)
\\( \bigcirc \\ f(x) = (x + 3)^2 \\)
\\( \bigcirc \\ f(x) = (x - 3)^2 \\)

Explanation:

Step1: Recall Parent Function Transformations

The parent function for a parabola is \( y = x^2 \), with vertex at \((0,0)\) and standard width. Transformations: vertical stretch/compression is \( y = ax^2 \) (stretch if \(|a|>1\), compress if \(0<|a|<1\)), vertical shift is \( y = x^2 + k \) (up if \(k>0\), down if \(k<0\)), horizontal shift is \( y=(x - h)^2 \) (right if \(h>0\), left if \(h<0\)).

Step2: Analyze the Graph

The graph shown has its vertex at \((0,0)\) (same as parent \(y = x^2\)), so no horizontal or vertical shift (eliminates \(F(x)=(x + 3)^2\) and \(F(x)=(x - 3)^2\) which shift left/right, and \(F(x)=x^2+3\) which shifts up). The parabola appears narrower than \(y = x^2\), indicating a vertical stretch. For \(F(x)=3x^2\), \(a = 3>1\), which stretches the graph vertically (makes it narrower), matching the graph.

Answer:

\( F(x) = 3x^2 \) (the second option: \( F(x) = 3x^2 \))