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

f(x)=45×2-180
Rewrite the equation in vertex form.
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
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 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⋅45
Multiply 4 by 45.
1180
1180
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.
(0,-32399180)
(0,-32399180)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=0
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=-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