Question

My name is nessalovetrillo i am studying to be able to test out of my algebra class so this is a study guide please see picture of the help

Answer 1

Equation of the Parabola.

If the vertex of a parabola is located at the point (h, k), then the equation of the parabola is written as:

`y=a(x-h)^2+k`

Where a is the leading coefficient. To find the coordinates of the vertex, given its equation, we use the 'square completion' technique.

We are given the equation:

`y=x^2-4x-21`

We need to transform this equation so it can be expressed like the general equation given above. Adding and subtracting 4:

`y=x^2-4x+4-21-4`

Rearranging:

`y=(x^2-4x+4)-25`

Factoring:

`y=(x-2)^2-25`

The vertex of the parabola is located at (2, -25).

To find the x-intercepts, we set y=0 and solve the equation:

`\begin{gathered} (x-2)^2-25=0 \\ \text{Add 25:} \\ (x-2)^2=25 \end{gathered}`

Taking the square root (it has two signs):

`x-2=\pm5`

Solving for x, we get two possible answers:

x = 5 + 2 = 7

x = -5 + 2 = -3

X-inercepts: -3, 7