Question

the graph shown is a translation of the graph of $f(x)=2x^2$. write the function for the graph in vertex form.
$f(x)=\boxed{2(x - 1)^2 - 2}$
(type your answer in vertex form.)

Explanation:

Step1: Recall Vertex Form of Quadratic

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the vertical stretch/compression and direction.

Step2: Identify Transformations from \(f(x)=2x^2\)

The parent function is \( f(x) = 2x^2 \) (vertex at \((0, 0)\), \(a = 2\)). The graph is translated. From the vertex form \( f(x) = 2(x - 1)^2 - 2 \), we see:

• Horizontal translation: \(h = 1\) (shift right 1 unit, since \(x - h\) becomes \(x - 1\)).
• Vertical translation: \(k = -2\) (shift down 2 units).
• The vertical stretch factor \(a = 2\) remains the same as the parent function.

Step3: Verify Vertex Form

Using the vertex form formula with \(a = 2\), \(h = 1\), \(k = -2\), we substitute into \( f(x) = a(x - h)^2 + k \) to get \( f(x) = 2(x - 1)^2 - 2 \), which matches the given answer.

Answer:

\( f(x) = 2(x - 1)^2 - 2 \)