Question

1. \\( f(x) = (x + 5)^2 - 2 \\)\

transformations:

Explanation:

Step1: Identify parent function

Parent function: $g(x)=x^2$ (vertex at $(0,0)$)

Step2: Horizontal transformation

Shift left 5 units: $g(x+5)=(x+5)^2$ (vertex at $(-5,0)$)

Step3: Vertical transformation

Shift down 2 units: $f(x)=(x+5)^2-2$ (vertex at $(-5,-2)$)

Step4: Plot key points

• Vertex: $(-5,-2)$
• Additional points:

When $x=-3$, $f(-3)=(-3+5)^2-2=4-2=2$ → $(-3,2)$
When $x=-7$, $f(-7)=(-7+5)^2-2=4-2=2$ → $(-7,2)$
When $x=-4$, $f(-4)=(-4+5)^2-2=1-2=-1$ → $(-4,-1)$
When $x=-6$, $f(-6)=(-6+5)^2-2=1-2=-1$ → $(-6,-1)$

Answer:

Transformations: 5 units left, 2 units down from $y=x^2$; the graph is a parabola with vertex at $(-5,-2)$, passing through points $(-3,2)$, $(-7,2)$, $(-4,-1)$, $(-6,-1)$ (and symmetric across the line $x=-5$).