Question

the graph shown is a translation of the graph of $f(x)=x^2$. write the function in vertex form.
$f(x)=\square$
(type your answer in vertex form.)

Explanation:

Step1: Recall vertex form of a parabola

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For the parent function \( f(x)=x^2 \), the vertex is at \((0,0)\).

Step2: Identify the vertex of the translated graph

From the graph, the vertex of the parabola is at \((3, 4)\). Since it's a translation of \( f(x)=x^2 \) (so \( a = 1 \), no vertical stretch or compression), we substitute \( h = 3 \), \( k = 4 \), and \( a = 1 \) into the vertex form.

Step3: Write the function in vertex form

Substituting the values into \( f(x) = a(x - h)^2 + k \), we get \( f(x)=(x - 3)^2 + 4 \).

Answer:

\( f(x)=(x - 3)^2 + 4 \)