Question

produce a rule for the function whose graph is shown. (hint: consider using the basic functions and the transformations of their graphs.)
$f(x) = \

$$\begin{cases} \\square & \\text{for } x < \\square \\\\ \\square & \\text{for } x \\geq \\square \\end{cases}$$

$

Explanation:

Step1: Identify split point

The graph splits at $x=3$.

Step2: Find rule for $x<3$

The curve is a downward-opening parabola with vertex at $(0,0)$, passing through $(2,-1)$. Use $f(x)=ax^2$:
$-1=a(2)^2 \implies a=-\frac{1}{4}$. So $f(x)=-\frac{1}{4}x^2$.

Step3: Find rule for $x\geq3$

The graph is a horizontal line at $y=2$, so $f(x)=2$.

Answer:

$$ f(x)=
LATEXBLOCK0
$$