Question

graphing transformations of quadratic functions
the graph of the function $f(x) = x^2$ is shown below. draw the graph of $g$ by reflecting the graph of $f$ about the $x$-axis, and then shift up 3 and left 4. lastly, write the formula for the function $g(x)$ below.

write the formula for $g(x)$ below. be sure to use proper function notation.

Explanation:

Step1: Reflect over x-axis

$h(x) = -f(x) = -x^2$

Step2: Shift left 4 units

$k(x) = h(x+4) = -(x+4)^2$

Step3: Shift up 3 units

$g(x) = k(x)+3 = -(x+4)^2 + 3$

Answer:

$g(x) = -(x+4)^2 + 3$

(Note for graphing: The vertex of $g(x)$ is at $(-4, 3)$, it opens downward, with the same width as $f(x)=x^2$)