Question

1. write in vertex form, a parabola that has a vertex at (- 2, 1) and has a y-intercept of (0,9).

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Given the vertex $(-2, 1)$, so $h=-2$ and $k = 1$. Substituting these values, we get $y=a(x + 2)^2+1$.

Step2: Use y - intercept to find a

The y - intercept is $(0,9)$, which means when $x = 0$, $y=9$. Substitute $x = 0$ and $y = 9$ into the equation $y=a(x + 2)^2+1$:
$$9=a(0 + 2)^2+1$$
$$9 = 4a+1$$
Subtract 1 from both sides:
$$8=4a$$
Divide both sides by 4:
$$a = 2$$

Step3: Write the final equation

Substitute $a = 2$ back into the vertex form equation $y=a(x + 2)^2+1$. We get $y = 2(x + 2)^2+1$.

Answer:

The vertex form of the parabola is $y = 2(x + 2)^2+1$