y=3×2-12x+16
Rewrite the equation in vertex form.
Complete the square for 3×2-12x+16.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=3,b=-12,c=16
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=-122(3)
Simplify the right side.
Cancel the common factor of 12 and 2.
Factor 2 out of 12.
d=-2⋅62⋅3
Cancel the common factors.
Factor 2 out of 2⋅3.
d=-2⋅62(3)
Cancel the common factor.
d=-2⋅62⋅3
Rewrite the expression.
d=-63
d=-63
d=-63
Cancel the common factor of 6 and 3.
Factor 3 out of 6.
d=-3⋅23
Cancel the common factors.
Factor 3 out of 3.
d=-3⋅23(1)
Cancel the common factor.
d=-3⋅23⋅1
Rewrite the expression.
d=-21
Divide 2 by 1.
d=-1⋅2
d=-1⋅2
d=-1⋅2
Multiply -1 by 2.
d=-2
d=-2
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise -12 to the power of 2.
e=16-1444⋅3
Multiply 4 by 3.
e=16-14412
Divide 144 by 12.
e=16-1⋅12
Multiply -1 by 12.
e=16-12
e=16-12
Subtract 12 from 16.
e=4
e=4
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
3(x-2)2+4
3(x-2)2+4
Set y equal to the new right side.
y=3(x-2)2+4
y=3(x-2)2+4
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=3
h=2
k=4
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(2,4)
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⋅3
Multiply 4 by 3.
112
112
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.
(2,4912)
(2,4912)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=2
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=4712
y=4712
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (2,4)
Focus: (2,4912)
Axis of Symmetry: x=2
Directrix: y=4712
Direction: Opens Up
Vertex: (2,4)
Focus: (2,4912)
Axis of Symmetry: x=2
Directrix: y=4712
Replace the variable x with 1 in the expression.
f(1)=3(1)2-12⋅1+16
Simplify the result.
Simplify each term.
One to any power is one.
f(1)=3⋅1-12⋅1+16
Multiply 3 by 1.
f(1)=3-12⋅1+16
Multiply -12 by 1.
f(1)=3-12+16
f(1)=3-12+16
Simplify by adding and subtracting.
Subtract 12 from 3.
f(1)=-9+16
Add -9 and 16.
f(1)=7
f(1)=7
The final answer is 7.
7
7
The y value at x=1 is 7.
y=7
Replace the variable x with 0 in the expression.
f(0)=3(0)2-12⋅0+16
Simplify the result.
Simplify each term.
Raising 0 to any positive power yields 0.
f(0)=3⋅0-12⋅0+16
Multiply 3 by 0.
f(0)=0-12⋅0+16
Multiply -12 by 0.
f(0)=0+0+16
f(0)=0+0+16
Simplify by adding zeros.
Add 0 and 0.
f(0)=0+16
Add 0 and 16.
f(0)=16
f(0)=16
The final answer is 16.
16
16
The y value at x=0 is 16.
y=16
Replace the variable x with 3 in the expression.
f(3)=3(3)2-12⋅3+16
Simplify the result.
Simplify each term.
Multiply 3 by (3)2 by adding the exponents.
Multiply 3 by (3)2.
Raise 3 to the power of 1.
f(3)=3(3)2-12⋅3+16
Use the power rule aman=am+n to combine exponents.
f(3)=31+2-12⋅3+16
f(3)=31+2-12⋅3+16
Add 1 and 2.
f(3)=33-12⋅3+16
f(3)=33-12⋅3+16
Raise 3 to the power of 3.
f(3)=27-12⋅3+16
Multiply -12 by 3.
f(3)=27-36+16
f(3)=27-36+16
Simplify by adding and subtracting.
Subtract 36 from 27.
f(3)=-9+16
Add -9 and 16.
f(3)=7
f(3)=7
The final answer is 7.
7
7
The y value at x=3 is 7.
y=7
Replace the variable x with 4 in the expression.
f(4)=3(4)2-12⋅4+16
Simplify the result.
Simplify each term.
Raise 4 to the power of 2.
f(4)=3⋅16-12⋅4+16
Multiply 3 by 16.
f(4)=48-12⋅4+16
Multiply -12 by 4.
f(4)=48-48+16
f(4)=48-48+16
Simplify by subtracting numbers.
Subtract 48 from 48.
f(4)=0+16
Add 0 and 16.
f(4)=16
f(4)=16
The final answer is 16.
16
16
The y value at x=4 is 16.
y=16
Graph the parabola using its properties and the selected points.
xy016172437416
xy016172437416
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (2,4)
Focus: (2,4912)
Axis of Symmetry: x=2
Directrix: y=4712
xy016172437416
Graph y=3x^2-12x+16
