Question

the graph of $y = x^2$ is the solid black graph below. which function represents the dotted graph?
(graph: a parabola $y = x^2$ (solid black) with vertex at (0,0), opening upwards. a dotted parabola with vertex at (-3,0), opening downwards.)
answer
$\circ$ $y = (-x)^2 + 3$ $\circ$ $y = -(x + 3)^2$
$\circ$ $y = (-x)^2 - 3$ $\circ$ $y = -(x - 3)^2$

Explanation:

Step1: Identify vertex of parent graph

The parent function $y=x^2$ has vertex at $(0,0)$.

Step2: Identify vertex of dotted graph

The dotted graph's vertex is at $(-3,0)$.

Step3: Check direction of opening

The dotted graph opens downward, so there is a vertical reflection (negative sign outside the squared term).

Step4: Apply transformation rules

For a parabola $y=a(x-h)^2+k$, vertex is $(h,k)$. Here, $h=-3$, $k=0$, $a=-1$. Substitute these values:
$y = -1(x - (-3))^2 + 0 = -(x+3)^2$

Step5: Verify against options

This matches the option $y=-(x+3)^2$.

Answer:

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