f(x)=3×2-12x+17

Rewrite the equation in vertex form.

Complete the square for 3×2-12x+17.

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

a=3,b=-12,c=17

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=17-1444⋅3

Multiply 4 by 3.

e=17-14412

Divide 144 by 12.

e=17-1⋅12

Multiply -1 by 12.

e=17-12

e=17-12

Subtract 12 from 17.

e=5

e=5

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

3(x-2)2+5

3(x-2)2+5

Set y equal to the new right side.

y=3(x-2)2+5

y=3(x-2)2+5

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

a=3

h=2

k=5

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

Opens Up

Find the vertex (h,k).

(2,5)

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.

(2,6112)

(2,6112)

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=5912

y=5912

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

Direction: Opens Up

Vertex: (2,5)

Focus: (2,6112)

Axis of Symmetry: x=2

Directrix: y=5912

Direction: Opens Up

Vertex: (2,5)

Focus: (2,6112)

Axis of Symmetry: x=2

Directrix: y=5912

Replace the variable x with 1 in the expression.

f(1)=3(1)2-12⋅1+17

Simplify the result.

Simplify each term.

One to any power is one.

f(1)=3⋅1-12⋅1+17

Multiply 3 by 1.

f(1)=3-12⋅1+17

Multiply -12 by 1.

f(1)=3-12+17

f(1)=3-12+17

Simplify by adding and subtracting.

Subtract 12 from 3.

f(1)=-9+17

Add -9 and 17.

f(1)=8

f(1)=8

The final answer is 8.

8

8

The y value at x=1 is 8.

y=8

Replace the variable x with 0 in the expression.

f(0)=3(0)2-12⋅0+17

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=3⋅0-12⋅0+17

Multiply 3 by 0.

f(0)=0-12⋅0+17

Multiply -12 by 0.

f(0)=0+0+17

f(0)=0+0+17

Simplify by adding zeros.

Add 0 and 0.

f(0)=0+17

Add 0 and 17.

f(0)=17

f(0)=17

The final answer is 17.

17

17

The y value at x=0 is 17.

y=17

Replace the variable x with 3 in the expression.

f(3)=3(3)2-12⋅3+17

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+17

Use the power rule aman=am+n to combine exponents.

f(3)=31+2-12⋅3+17

f(3)=31+2-12⋅3+17

Add 1 and 2.

f(3)=33-12⋅3+17

f(3)=33-12⋅3+17

Raise 3 to the power of 3.

f(3)=27-12⋅3+17

Multiply -12 by 3.

f(3)=27-36+17

f(3)=27-36+17

Simplify by adding and subtracting.

Subtract 36 from 27.

f(3)=-9+17

Add -9 and 17.

f(3)=8

f(3)=8

The final answer is 8.

8

8

The y value at x=3 is 8.

y=8

Replace the variable x with 4 in the expression.

f(4)=3(4)2-12⋅4+17

Simplify the result.

Simplify each term.

Raise 4 to the power of 2.

f(4)=3⋅16-12⋅4+17

Multiply 3 by 16.

f(4)=48-12⋅4+17

Multiply -12 by 4.

f(4)=48-48+17

f(4)=48-48+17

Simplify by subtracting numbers.

Subtract 48 from 48.

f(4)=0+17

Add 0 and 17.

f(4)=17

f(4)=17

The final answer is 17.

17

17

The y value at x=4 is 17.

y=17

Graph the parabola using its properties and the selected points.

xy017182538417

xy017182538417

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (2,5)

Focus: (2,6112)

Axis of Symmetry: x=2

Directrix: y=5912

xy017182538417

Graph f(x)=3x^2-12x+17