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

f(x)=3×2-12x+17
Find the properties of the given parabola.
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
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 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
Subtract 12 from 3.
f(1)=-9+17
f(1)=8
f(1)=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
f(0)=0+17
f(0)=17
f(0)=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
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
Subtract 36 from 27.
f(3)=-9+17
f(3)=8
f(3)=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
f(4)=17
f(4)=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

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