y=-5×2+100x+15

Rewrite the equation in vertex form.

Complete the square for -5×2+100x+15.

Use the form ax2+bx+c, to find the values of a, b, and c.

a=-5,b=100,c=15

Consider the vertex form of a parabola.

a(x+d)2+e

Substitute the values of a and b into the formula d=b2a.

d=1002(-5)

Simplify the right side.

Cancel the common factor of 100 and 2.

Factor 2 out of 100.

d=2⋅502⋅-5

Cancel the common factors.

Factor 2 out of 2⋅-5.

d=2⋅502(-5)

Cancel the common factor.

d=2⋅502⋅-5

Rewrite the expression.

d=50-5

d=50-5

d=50-5

Cancel the common factor of 50 and -5.

Factor 5 out of 50.

d=5(10)-5

Move the negative one from the denominator of 10-1.

d=-1⋅10

d=-1⋅10

Multiply -1 by 10.

d=-10

d=-10

Find the value of e using the formula e=c-b24a.

Simplify each term.

Raise 100 to the power of 2.

e=15-100004⋅-5

Multiply 4 by -5.

e=15-10000-20

Divide 10000 by -20.

e=15+500

Multiply -1 by -500.

e=15+500

e=15+500

Add 15 and 500.

e=515

e=515

Substitute the values of a, d, and e into the vertex form a(x+d)2+e.

-5(x-10)2+515

-5(x-10)2+515

Set y equal to the new right side.

y=-5(x-10)2+515

y=-5(x-10)2+515

Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.

a=-5

h=10

k=515

Since the value of a is negative, the parabola opens down.

Opens Down

Find the vertex (h,k).

(10,515)

Find p, the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

14a

Substitute the value of a into the formula.

14⋅-5

Simplify.

Multiply 4 by -5.

1-20

Move the negative in front of the fraction.

-120

-120

-120

Find the focus.

The focus of a parabola can be found by adding p to the y-coordinate k if the parabola opens up or down.

(h,k+p)

Substitute the known values of h, p, and k into the formula and simplify.

(10,1029920)

(10,1029920)

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

x=10

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting p from the y-coordinate k of the vertex if the parabola opens up or down.

y=k-p

Substitute the known values of p and k into the formula and simplify.

y=1030120

y=1030120

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: (10,515)

Focus: (10,1029920)

Axis of Symmetry: x=10

Directrix: y=1030120

Direction: Opens Down

Vertex: (10,515)

Focus: (10,1029920)

Axis of Symmetry: x=10

Directrix: y=1030120

Replace the variable x with 9 in the expression.

f(9)=-5(9)2+100(9)+15

Simplify the result.

Simplify each term.

Raise 9 to the power of 2.

f(9)=-5⋅81+100(9)+15

Multiply -5 by 81.

f(9)=-405+100(9)+15

Multiply 100 by 9.

f(9)=-405+900+15

f(9)=-405+900+15

Simplify by adding numbers.

Add -405 and 900.

f(9)=495+15

Add 495 and 15.

f(9)=510

f(9)=510

The final answer is 510.

510

510

The y value at x=9 is 510.

y=510

Replace the variable x with 8 in the expression.

f(8)=-5(8)2+100(8)+15

Simplify the result.

Simplify each term.

Raise 8 to the power of 2.

f(8)=-5⋅64+100(8)+15

Multiply -5 by 64.

f(8)=-320+100(8)+15

Multiply 100 by 8.

f(8)=-320+800+15

f(8)=-320+800+15

Simplify by adding numbers.

Add -320 and 800.

f(8)=480+15

Add 480 and 15.

f(8)=495

f(8)=495

The final answer is 495.

495

495

The y value at x=8 is 495.

y=495

Replace the variable x with 11 in the expression.

f(11)=-5(11)2+100(11)+15

Simplify the result.

Simplify each term.

Raise 11 to the power of 2.

f(11)=-5⋅121+100(11)+15

Multiply -5 by 121.

f(11)=-605+100(11)+15

Multiply 100 by 11.

f(11)=-605+1100+15

f(11)=-605+1100+15

Simplify by adding numbers.

Add -605 and 1100.

f(11)=495+15

Add 495 and 15.

f(11)=510

f(11)=510

The final answer is 510.

510

510

The y value at x=11 is 510.

y=510

Replace the variable x with 12 in the expression.

f(12)=-5(12)2+100(12)+15

Simplify the result.

Simplify each term.

Raise 12 to the power of 2.

f(12)=-5⋅144+100(12)+15

Multiply -5 by 144.

f(12)=-720+100(12)+15

Multiply 100 by 12.

f(12)=-720+1200+15

f(12)=-720+1200+15

Simplify by adding numbers.

Add -720 and 1200.

f(12)=480+15

Add 480 and 15.

f(12)=495

f(12)=495

The final answer is 495.

495

495

The y value at x=12 is 495.

y=495

Graph the parabola using its properties and the selected points.

xy84959510105151151012495

xy84959510105151151012495

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (10,515)

Focus: (10,1029920)

Axis of Symmetry: x=10

Directrix: y=1030120

xy84959510105151151012495

Graph y=-5x^2+100x+15