y=5×2+7x-2

Rewrite the equation in vertex form.

Complete the square for 5×2+7x-2.

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

a=5,b=7,c=-2

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=72(5)

Multiply 2 by 5.

d=710

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

Simplify each term.

Raise 7 to the power of 2.

e=-2-494⋅5

Multiply 4 by 5.

e=-2-4920

e=-2-4920

To write -2 as a fraction with a common denominator, multiply by 2020.

e=-2⋅2020-4920

Combine -2 and 2020.

e=-2⋅2020-4920

Combine the numerators over the common denominator.

e=-2⋅20-4920

Simplify the numerator.

Multiply -2 by 20.

e=-40-4920

Subtract 49 from -40.

e=-8920

e=-8920

Move the negative in front of the fraction.

e=-8920

e=-8920

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

5(x+710)2-8920

5(x+710)2-8920

Set y equal to the new right side.

y=5(x+710)2-8920

y=5(x+710)2-8920

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

a=5

h=-710

k=-8920

Since the value of a is positive, the parabola opens up.

Opens Up

Find the vertex (h,k).

(-710,-8920)

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

Multiply 4 by 5.

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.

(-710,-225)

(-710,-225)

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

x=-710

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=-92

y=-92

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

Direction: Opens Up

Vertex: (-710,-8920)

Focus: (-710,-225)

Axis of Symmetry: x=-710

Directrix: y=-92

Direction: Opens Up

Vertex: (-710,-8920)

Focus: (-710,-225)

Axis of Symmetry: x=-710

Directrix: y=-92

Replace the variable x with -2 in the expression.

f(-2)=5(-2)2+7(-2)-2

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=5⋅4+7(-2)-2

Multiply 5 by 4.

f(-2)=20+7(-2)-2

Multiply 7 by -2.

f(-2)=20-14-2

f(-2)=20-14-2

Simplify by subtracting numbers.

Subtract 14 from 20.

f(-2)=6-2

Subtract 2 from 6.

f(-2)=4

f(-2)=4

The final answer is 4.

4

4

The y value at x=-2 is 4.

y=4

Replace the variable x with -3 in the expression.

f(-3)=5(-3)2+7(-3)-2

Simplify the result.

Simplify each term.

Raise -3 to the power of 2.

f(-3)=5⋅9+7(-3)-2

Multiply 5 by 9.

f(-3)=45+7(-3)-2

Multiply 7 by -3.

f(-3)=45-21-2

f(-3)=45-21-2

Simplify by subtracting numbers.

Subtract 21 from 45.

f(-3)=24-2

Subtract 2 from 24.

f(-3)=22

f(-3)=22

The final answer is 22.

22

22

The y value at x=-3 is 22.

y=22

Replace the variable x with 0 in the expression.

f(0)=5(0)2+7(0)-2

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=5⋅0+7(0)-2

Multiply 5 by 0.

f(0)=0+7(0)-2

Multiply 7 by 0.

f(0)=0+0-2

f(0)=0+0-2

Simplify by adding zeros.

Add 0 and 0.

f(0)=0-2

Subtract 2 from 0.

f(0)=-2

f(0)=-2

The final answer is -2.

-2

-2

The y value at x=0 is -2.

y=-2

Replace the variable x with 1 in the expression.

f(1)=5(1)2+7(1)-2

Simplify the result.

Simplify each term.

One to any power is one.

f(1)=5⋅1+7(1)-2

Multiply 5 by 1.

f(1)=5+7(1)-2

Multiply 7 by 1.

f(1)=5+7-2

f(1)=5+7-2

Simplify by adding and subtracting.

Add 5 and 7.

f(1)=12-2

Subtract 2 from 12.

f(1)=10

f(1)=10

The final answer is 10.

10

10

The y value at x=1 is 10.

y=10

Graph the parabola using its properties and the selected points.

xy-322-24-710-89200-2110

xy-322-24-710-89200-2110

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-710,-8920)

Focus: (-710,-225)

Axis of Symmetry: x=-710

Directrix: y=-92

xy-322-24-710-89200-2110

Graph y=5x^2+7x-2