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 base function

The graph is a parabola opening upwards, so the base function is $y = x^2$.

Step2: Identify horizontal shift

The vertex of the base $y=x^2$ is at $(0,0)$. The vertex of the given graph is at $(-3, 1)$, so there is a left shift by 3 units: replace $x$ with $x+3$, giving $y=(x+3)^2$.

Step3: Identify vertical shift

The vertex is shifted up by 1 unit, so add 1 to the function: $y=(x+3)^2 + 1$.

Step4: Verify shape

The parabola has the same width as $y=x^2$, so no vertical stretch/compression or reflection is needed.

Answer:

$y=(x+3)^2 + 1$