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 y=(x - 2)^2 - 1. (type an equation using x and y as the variables. use integers or decimals for any numbers in the equation.)

Explanation:

Step1: Identify the base - function

The base - function is $y = x^{2}$, which is a parabola with vertex at the origin $(0,0)$.

Step2: Analyze horizontal shift

The vertex of the given graph is at $(2, - 1)$. For a parabola $y=a(x - h)^{2}+k$, a horizontal shift of $h$ units occurs. Here, $h = 2$, so we replace $x$ with $x - 2$ in the base - function to get $y=(x - 2)^{2}$.

Step3: Analyze vertical shift

The vertical shift is $k=-1$. We add $k$ to the function after the horizontal - shift. So the function becomes $y=(x - 2)^{2}-1$.

Answer:

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