Question

1. the graph shows the height of the flying disk with respect to time. what is the equation of the function? write the equation in vertex form. then write the equation in the form y = ax^2 + bx + c.

Explanation:

Step1: Write vertex - form equation

The vertex - form of a quadratic function is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex. The vertex of the parabola is $(2,10)$, so $h = 2$ and $k = 10$. Then the equation is $y=a(x - 2)^2+10$.

Step2: Find the value of $a$

The graph passes through the point $(0,4)$. Substitute $x = 0$ and $y = 4$ into $y=a(x - 2)^2+10$:
\[

$$\begin{align*} 4&=a(0 - 2)^2+10\\ 4&=4a+10\\ 4a&=4 - 10\\ 4a&=- 6\\ a&=-\frac{3}{2} \end{align*}$$

\]
So the vertex - form equation is $y=-\frac{3}{2}(x - 2)^2+10$.

Step3: Expand to standard form

\[

$$\begin{align*} y&=-\frac{3}{2}(x - 2)^2+10\\ &=-\frac{3}{2}(x^{2}-4x + 4)+10\\ &=-\frac{3}{2}x^{2}+6x-6 + 10\\ &=-\frac{3}{2}x^{2}+6x + 4 \end{align*}$$

\]

Answer:

Vertex - form: $y=-\frac{3}{2}(x - 2)^2+10$; Standard - form: $y=-\frac{3}{2}x^{2}+6x + 4$