Question

write an equation for the function whose graph is shown to the right. the graph shows a transformation of a common function. an equation for the function of the given graph is (type an equation using x and y as the variables. use integers or decimals for any numbers in the equation.)

Explanation:

Step1: Identify the common - function

The graph appears to be a transformation of the quadratic function \(y = x^{2}\).

Step2: Consider the general form of a quadratic function transformation

The general form of a quadratic function is \(y=a(x - h)^{2}+k\), where \((h,k)\) is the vertex of the parabola.

Step3: Determine the vertex

From the graph, the vertex of the parabola is \((h,k)=(2, - 1)\). So the function becomes \(y=a(x - 2)^{2}-1\).

Step4: Find the value of \(a\)

We can use a point on the graph. Let's assume the graph passes through the point \((3,0)\) (by observing the graph). Substitute \(x = 3\) and \(y = 0\) into \(y=a(x - 2)^{2}-1\):
\[

$$\begin{align*} 0&=a(3 - 2)^{2}-1\\ 0&=a\times1^{2}-1\\ a&=1 \end{align*}$$

\]

Answer:

\(y=(x - 2)^{2}-1\)