Question

use transformations of $f(x)=x^2$ to graph the following function.$g(x)=4(x + 1)^2 - 2$use the graphing tool to graph the function.click to enlarge graph

Explanation:

Step1: Identify base function

Base function: $f(x)=x^2$, vertex at $(0,0)$

Step2: Horizontal shift left by 1

Transform to $f(x+1)=(x+1)^2$, vertex at $(-1,0)$

Step3: Vertical stretch by factor 4

Transform to $4(x+1)^2$, vertex remains $(-1,0)$, slope steepens

Step4: Vertical shift down by 2

Transform to $g(x)=4(x+1)^2-2$, vertex at $(-1,-2)$

Step5: Plot key points

• Vertex: $(-1,-2)$
• When $x=0$, $g(0)=4(0+1)^2-2=2$, point $(0,2)$
• When $x=-2$, $g(-2)=4(-2+1)^2-2=2$, point $(-2,2)$

Answer:

The graph of $g(x)=4(x+1)^2-2$ is a parabola with vertex at $(-1, -2)$, vertically stretched by a factor of 4, shifted left 1 unit and down 2 units from the base parabola $f(x)=x^2$. Key points to plot are $(-1,-2)$, $(0,2)$, and $(-2,2)$, with the parabola opening upwards.