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

g(x)=4(x-4)2
Find the properties of the given parabola.
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
Select a few x values, and plug them into the equation to find the corresponding y values. The x values should be selected around the vertex.
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
Subtract 96 from 36.
f(3)=-60+64
f(3)=4
f(3)=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
Subtract 64 from 16.
f(2)=-48+64
f(2)=16
f(2)=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
Subtract 160 from 100.
f(5)=-60+64
f(5)=4
f(5)=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
Subtract 192 from 144.
f(6)=-48+64
f(6)=16
f(6)=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