f(x)=45×2-180

Complete the square for 45×2-180.

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

a=45,b=0,c=-180

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=02(45)

Simplify the right side.

Cancel the common factor of 0 and 2.

Factor 2 out of 0.

d=2(0)2(45)

Cancel the common factors.

Cancel the common factor.

d=2⋅02⋅45

Rewrite the expression.

d=045

d=045

d=045

Cancel the common factor of 0 and 45.

Factor 45 out of 0.

d=45(0)45

Cancel the common factors.

Factor 45 out of 45.

d=45⋅045⋅1

Cancel the common factor.

d=45⋅045⋅1

Rewrite the expression.

d=01

Divide 0 by 1.

d=0

d=0

d=0

d=0

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

Simplify each term.

Raising 0 to any positive power yields 0.

e=-180-04⋅45

Multiply 4 by 45.

e=-180-0180

Divide 0 by 180.

e=-180-0

Multiply -1 by 0.

e=-180+0

e=-180+0

Add -180 and 0.

e=-180

e=-180

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

45(x+0)2-180

45(x+0)2-180

Set y equal to the new right side.

y=45(x+0)2-180

y=45(x+0)2-180

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

a=45

h=0

k=-180

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

Opens Up

Find the vertex (h,k).

(0,-180)

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⋅45

Multiply 4 by 45.

1180

1180

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.

(0,-32399180)

(0,-32399180)

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

x=0

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=-32401180

y=-32401180

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

Direction: Opens Up

Vertex: (0,-180)

Focus: (0,-32399180)

Axis of Symmetry: x=0

Directrix: y=-32401180

Graph f(x)=45x^2-180