Hi Brandi;

*Find an equation of the line containing the pair of points (-4, -7) and (0, 4)*

Our first priority is to establish slope.

Slope is change-of-y divided by change-of-x...

m=(y-y_{1})/(x-x_{1})

m=(-7-4)/(-4-0)

m=-11/-4

A negative number divided by a negative number has a positive result...

m=11/4

The equation in slope-intercept form is...

y=mx+b

m is the slope.

b is the y-intercept, the value of y when x=0.

The slope is (11/4).

In this circumstance, the y-intercept is provided as (0,4).

**y=(11/4)x+4**

Let's use the other equation to verify our results...

-7=(11/4)(-4)+4

-7=-11+4

-7=-7

*Determine the vertex of the parabola:*

*y= 4x^2-80x+406*

The vertex is the value of x at the point of the parabola in which the change of slope is zero. Henceforth, we will take the first derivative of the equation and set this equal to zero...

0=8x-80

80=8x

10=x

Let's establish the value of y...

y=4x^{2}-80x+406

y=[(4)(10^{2})]-[(80)(10)]+406

y=[(4)(100)]-[(80)(10)]+406

y=400-800+406

y=-400+406

y=6

**The vertex is (10,6).**

The x value of the vertex can also be established by applying the equation -b/2a to the original formula.

y= 4x^2-80x+406

-b/2a

x=-[-80/(2)(4)]

x=80/8

x=10