Question

H(r) = (r + 1)(r +8)

1) What are the zeros of the function?
Write the smaller r first, and the larger r second.
smaller r =
larger r =
2) What is the vertex of the parabola?​

Answer 1

1) The solutions are:

smaller r = -8

larger r = -1

2) The vertex is
`((-9)/(2),(-49)/(4))`.

Explanation:

1) A zero of a function is an x-value that makes the function value 0.

To find the zeros of the function, first, we see that the linear factors of
`h(r)` are

`(r + 1)` and
`(r +8)`.

If we set
`h(r)=0` and solve for x, we get

`\left(r\:+\:1\right)\left(r\:+8\right)=\:0`

Using the Zero factor principle, If ab = 0, then either a = 0 or b = 0, or both a and b are 0.

`r+1=0:\quad r=-1\\r+8=0:\quad r=-8`

The solutions are:

smaller r = -8

larger r = -1

2) The vertex of a parabola is the highest or lowest point, also known as the maximum or minimum of a parabola.

To find the vertex of the function
`h(r) = (r + 1)(r +8)`, first we need to find the standard equation of a parabola, which is,

`y=ax^2+bx+c`

`\mathrm{Apply\:FOIL\:method}:\quad \left(a+b\right)\left(c+d\right)=ac+ad+bc+bd\\\\a=r,\:b=1,\:c=r,\:d=8\\\\rr+8r+1\cdot \:r+1\cdot \:8\\\\r^2+9r+8`

Next, we find the r-coordinate of the vertex with the formula
`-(b)/(2a)`.

We know from the standard equation of a parabola that

`a=1\\b=9\\c=8`

So,

`r=-(9)/(2\cdot 1) = -(9)/(2)`

And the h-coordinate is

`h(-(9)/(2))=(-(9)/(2))^2+9(-(9)/(2))+8\\\\=(9^2)/(2^2)-(81)/(2)+8\\\\=(9^2)/(4)-(162)/(4)+(32)/(4)\\\\=(-49)/(4)`

The vertex is
`((-9)/(2),(-49)/(4))`.