Question

in the cartesian plane on the right, functions f and g are represented by parabolas. the intersection points a and b of the parabolas are located on the x - axis. the rule of function g is $g(x)=-\frac{2}{3}(x - 10)^2+24$ point c(10, - 18) is the vertex of the parabola representing function f. what is the initial value of function f? the initial value of function f equals:

Explanation:

Step1: Identify vertex form of $f(x)$

The vertex of $f$ is $(10, -18)$, so its general vertex form is $f(x)=a(x-10)^2 - 18$, where $a$ is a constant.

Step2: Find intersection points of $g(x)$ and x-axis

Set $g(x)=0$:
$$0=\frac{-2}{3}(x-10)^2 + 24$$
Rearrange:
$$\frac{2}{3}(x-10)^2=24$$
$$(x-10)^2=24\times\frac{3}{2}=36$$
$$x-10=\pm6$$
$$x=10+6=16 \text{ or } x=10-6=4$$
So points A and B are $(4,0)$ and $(16,0)$.

Step3: Solve for $a$ using point A

Substitute $(4,0)$ into $f(x)=a(x-10)^2 - 18$:
$$0=a(4-10)^2 - 18$$
$$0=a(-6)^2 - 18$$
$$36a=18$$
$$a=\frac{18}{36}=\frac{1}{2}$$

Step4: Write full rule of $f(x)$

$$f(x)=\frac{1}{2}(x-10)^2 - 18$$

Step5: Calculate initial value (x=0)

$$f(0)=\frac{1}{2}(0-10)^2 - 18=\frac{1}{2}(100)-18=50-18=32$$

Answer:

32