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 parabola

The vertex form of a parabola 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 \((0, 0)\).

Step2: Identify the vertex of the translated graph

From the graph, we can see that the vertex of the parabola is at the point \((1, 3)\). So, \( h = 1 \) and \( k = 3 \). Since there is no vertical stretch or compression (the parabola opens with the same width as \( y = x^2 \)), \( a = 1 \).

Step3: Substitute h, k, and a into vertex form

Substitute \( a = 1 \), \( h = 1 \), and \( k = 3 \) into the vertex form formula \( f(x)=a(x - h)^2 + k \). We get \( f(x)=(x - 1)^2 + 3 \).

Answer:

\( f(x)=(x - 1)^2 + 3 \)