Question

1. the graph below shows a parabola with vertex at (0, -1) that passes through (2, 7). which is the correct equation? a) y = (x)^2 - 1 b) y = 2x^2 - 1 c) y = 3x^2 - 1 d) y = -2x^2 - 1

Explanation:

Step1: Recall the vertex - form of a parabola

The vertex - form of a parabola is $y = a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. Given that the vertex is $(0, - 1)$, then $h = 0$ and $k=-1$, so the equation of the parabola is $y=a(x - 0)^2-1=ax^{2}-1$.

Step2: Substitute the point $(2,7)$ into the equation

Substitute $x = 2$ and $y = 7$ into $y=ax^{2}-1$. We get $7=a(2)^{2}-1$.

Step3: Solve for $a$

First, simplify the equation $7 = 4a-1$. Add 1 to both sides: $7 + 1=4a$, so $8 = 4a$. Then divide both sides by 4, we find $a = 2$.

Step4: Write the equation of the parabola

Substitute $a = 2$ into $y=ax^{2}-1$, the equation of the parabola is $y = 2x^{2}-1$.

Answer:

B. $y = 2x^{2}-1$