g(x)=4(x-4)2

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

a=4

h=4

k=0

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

Opens Up

Find the vertex (h,k).

(4,0)

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⋅4

Multiply 4 by 4.

116

116

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.

(4,116)

(4,116)

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

x=4

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=-116

y=-116

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

Direction: Opens Up

Vertex: (4,0)

Focus: (4,116)

Axis of Symmetry: x=4

Directrix: y=-116

Direction: Opens Up

Vertex: (4,0)

Focus: (4,116)

Axis of Symmetry: x=4

Directrix: y=-116

Replace the variable x with 3 in the expression.

f(3)=4(3)2-32⋅3+64

Simplify the result.

Simplify each term.

Raise 3 to the power of 2.

f(3)=4⋅9-32⋅3+64

Multiply 4 by 9.

f(3)=36-32⋅3+64

Multiply -32 by 3.

f(3)=36-96+64

f(3)=36-96+64

Simplify by adding and subtracting.

Subtract 96 from 36.

f(3)=-60+64

Add -60 and 64.

f(3)=4

f(3)=4

The final answer is 4.

4

4

The y value at x=3 is 4.

y=4

Replace the variable x with 2 in the expression.

f(2)=4(2)2-32⋅2+64

Simplify the result.

Simplify each term.

Raise 2 to the power of 2.

f(2)=4⋅4-32⋅2+64

Multiply 4 by 4.

f(2)=16-32⋅2+64

Multiply -32 by 2.

f(2)=16-64+64

f(2)=16-64+64

Simplify by adding and subtracting.

Subtract 64 from 16.

f(2)=-48+64

Add -48 and 64.

f(2)=16

f(2)=16

The final answer is 16.

16

16

The y value at x=2 is 16.

y=16

Replace the variable x with 5 in the expression.

f(5)=4(5)2-32⋅5+64

Simplify the result.

Simplify each term.

Raise 5 to the power of 2.

f(5)=4⋅25-32⋅5+64

Multiply 4 by 25.

f(5)=100-32⋅5+64

Multiply -32 by 5.

f(5)=100-160+64

f(5)=100-160+64

Simplify by adding and subtracting.

Subtract 160 from 100.

f(5)=-60+64

Add -60 and 64.

f(5)=4

f(5)=4

The final answer is 4.

4

4

The y value at x=5 is 4.

y=4

Replace the variable x with 6 in the expression.

f(6)=4(6)2-32⋅6+64

Simplify the result.

Simplify each term.

Raise 6 to the power of 2.

f(6)=4⋅36-32⋅6+64

Multiply 4 by 36.

f(6)=144-32⋅6+64

Multiply -32 by 6.

f(6)=144-192+64

f(6)=144-192+64

Simplify by adding and subtracting.

Subtract 192 from 144.

f(6)=-48+64

Add -48 and 64.

f(6)=16

f(6)=16

The final answer is 16.

16

16

The y value at x=6 is 16.

y=16

Graph the parabola using its properties and the selected points.

xy216344054616

xy216344054616

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (4,0)

Focus: (4,116)

Axis of Symmetry: x=4

Directrix: y=-116

xy216344054616

Graph g(x)=4(x-4)^2