y=5×2-10x+2

Rewrite the equation in vertex form.

Complete the square for 5×2-10x+2.

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

a=5,b=-10,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=-102(5)

Simplify the right side.

Cancel the common factor of 10 and 2.

Factor 2 out of 10.

d=-2⋅52⋅5

Cancel the common factors.

Factor 2 out of 2⋅5.

d=-2⋅52(5)

Cancel the common factor.

d=-2⋅52⋅5

Rewrite the expression.

d=-55

d=-55

d=-55

Cancel the common factor of 5.

Cancel the common factor.

d=-55

Divide 1 by 1.

d=-1⋅1

d=-1⋅1

Multiply -1 by 1.

d=-1

d=-1

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

Simplify each term.

Raise -10 to the power of 2.

e=2-1004⋅5

Multiply 4 by 5.

e=2-10020

Divide 100 by 20.

e=2-1⋅5

Multiply -1 by 5.

e=2-5

e=2-5

Subtract 5 from 2.

e=-3

e=-3

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

5(x-1)2-3

5(x-1)2-3

Set y equal to the new right side.

y=5(x-1)2-3

y=5(x-1)2-3

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

a=5

h=1

k=-3

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

Opens Up

Find the vertex (h,k).

(1,-3)

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.

(1,-5920)

(1,-5920)

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

x=1

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=-6120

y=-6120

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

Direction: Opens Up

Vertex: (1,-3)

Focus: (1,-5920)

Axis of Symmetry: x=1

Directrix: y=-6120

Direction: Opens Up

Vertex: (1,-3)

Focus: (1,-5920)

Axis of Symmetry: x=1

Directrix: y=-6120

Replace the variable x with 0 in the expression.

f(0)=5(0)2-10⋅0+2

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=5⋅0-10⋅0+2

Multiply 5 by 0.

f(0)=0-10⋅0+2

Multiply -10 by 0.

f(0)=0+0+2

f(0)=0+0+2

Simplify by adding zeros.

Add 0 and 0.

f(0)=0+2

Add 0 and 2.

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)2-10⋅-1+2

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=5⋅1-10⋅-1+2

Multiply 5 by 1.

f(-1)=5-10⋅-1+2

Multiply -10 by -1.

f(-1)=5+10+2

f(-1)=5+10+2

Simplify by adding numbers.

Add 5 and 10.

f(-1)=15+2

Add 15 and 2.

f(-1)=17

f(-1)=17

The final answer is 17.

17

17

The y value at x=-1 is 17.

y=17

Replace the variable x with 2 in the expression.

f(2)=5(2)2-10⋅2+2

Simplify the result.

Simplify each term.

Raise 2 to the power of 2.

f(2)=5⋅4-10⋅2+2

Multiply 5 by 4.

f(2)=20-10⋅2+2

Multiply -10 by 2.

f(2)=20-20+2

f(2)=20-20+2

Simplify by subtracting numbers.

Subtract 20 from 20.

f(2)=0+2

Add 0 and 2.

f(2)=2

f(2)=2

The final answer is 2.

2

2

The y value at x=2 is 2.

y=2

Replace the variable x with 3 in the expression.

f(3)=5(3)2-10⋅3+2

Simplify the result.

Simplify each term.

Raise 3 to the power of 2.

f(3)=5⋅9-10⋅3+2

Multiply 5 by 9.

f(3)=45-10⋅3+2

Multiply -10 by 3.

f(3)=45-30+2

f(3)=45-30+2

Simplify by adding and subtracting.

Subtract 30 from 45.

f(3)=15+2

Add 15 and 2.

f(3)=17

f(3)=17

The final answer is 17.

17

17

The y value at x=3 is 17.

y=17

Graph the parabola using its properties and the selected points.

xy-117021-322317

xy-117021-322317

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (1,-3)

Focus: (1,-5920)

Axis of Symmetry: x=1

Directrix: y=-6120

xy-117021-322317

Graph y=5x^2-10x+2