y=12×2-3x+112
Combine 12 and x2.
y=x22-3x+112
Rewrite the equation in vertex form.
Complete the square for x22-3x+112.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=12,b=-3,c=112
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=-32(12)
Simplify the right side.
Combine 2 and 12.
d=-322
Simplify the expression.
Divide 2 by 2.
d=-31
Divide 3 by 1.
d=-1⋅3
Multiply -1 by 3.
d=-3
d=-3
d=-3
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise -3 to the power of 2.
e=112-94(12)
Combine 4 and 12.
e=112-942
Divide 4 by 2.
e=112-92
e=112-92
Combine the numerators over the common denominator.
e=11-92
Subtract 9 from 11.
e=22
Divide 2 by 2.
e=1
e=1
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
12⋅(x-3)2+1
12⋅(x-3)2+1
Set y equal to the new right side.
y=12⋅(x-3)2+1
y=12⋅(x-3)2+1
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=12
h=3
k=1
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(3,1)
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⋅12
Simplify.
Combine 4 and 12.
142
Divide 4 by 2.
12
12
12
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.
(3,32)
(3,32)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=3
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=12
y=12
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (3,1)
Focus: (3,32)
Axis of Symmetry: x=3
Directrix: y=12
Direction: Opens Up
Vertex: (3,1)
Focus: (3,32)
Axis of Symmetry: x=3
Directrix: y=12
Replace the variable x with 1 in the expression.
f(1)=(1)22-3⋅1+112
Simplify the result.
Simplify each term.
One to any power is one.
f(1)=12-3⋅1+112
Multiply -3 by 1.
f(1)=12-3+112
f(1)=12-3+112
Combine fractions.
Combine fractions with similar denominators.
f(1)=-3+1+112
Simplify the expression.
Add 1 and 11.
f(1)=-3+122
Divide 12 by 2.
f(1)=-3+6
Add -3 and 6.
f(1)=3
f(1)=3
f(1)=3
The final answer is 3.
3
3
The y value at x=1 is 3.
y=3
Replace the variable x with 2 in the expression.
f(2)=(2)22-3⋅2+112
Simplify the result.
Simplify each term.
Cancel the common factor of (2)2 and 2.
Factor 2 out of (2)2.
f(2)=2⋅22-3⋅2+112
Cancel the common factors.
Factor 2 out of 2.
f(2)=2⋅22(1)-3⋅2+112
Cancel the common factor.
f(2)=2⋅22⋅1-3⋅2+112
Rewrite the expression.
f(2)=21-3⋅2+112
Divide 2 by 1.
f(2)=2-3⋅2+112
f(2)=2-3⋅2+112
f(2)=2-3⋅2+112
Multiply -3 by 2.
f(2)=2-6+112
f(2)=2-6+112
Find the common denominator.
Write 2 as a fraction with denominator 1.
f(2)=21-6+112
Multiply 21 by 22.
f(2)=21⋅22-6+112
Multiply 21 and 22.
f(2)=2⋅22-6+112
Write -6 as a fraction with denominator 1.
f(2)=2⋅22+-61+112
Multiply -61 by 22.
f(2)=2⋅22+-61⋅22+112
Multiply -61 and 22.
f(2)=2⋅22+-6⋅22+112
f(2)=2⋅22+-6⋅22+112
Combine fractions.
Combine fractions with similar denominators.
f(2)=2⋅2-6⋅2+112
Multiply.
Multiply 2 by 2.
f(2)=4-6⋅2+112
Multiply -6 by 2.
f(2)=4-12+112
f(2)=4-12+112
f(2)=4-12+112
Simplify the numerator.
Subtract 12 from 4.
f(2)=-8+112
Add -8 and 11.
f(2)=32
f(2)=32
The final answer is 32.
32
32
The y value at x=2 is 32.
y=32
Replace the variable x with 5 in the expression.
f(5)=(5)22-3⋅5+112
Simplify the result.
Simplify each term.
Raise 5 to the power of 2.
f(5)=252-3⋅5+112
Multiply -3 by 5.
f(5)=252-15+112
f(5)=252-15+112
Combine fractions.
Combine fractions with similar denominators.
f(5)=-15+25+112
Simplify the expression.
Add 25 and 11.
f(5)=-15+362
Divide 36 by 2.
f(5)=-15+18
Add -15 and 18.
f(5)=3
f(5)=3
f(5)=3
The final answer is 3.
3
3
The y value at x=5 is 3.
y=3
Replace the variable x with 4 in the expression.
f(4)=(4)22-3⋅4+112
Simplify the result.
Simplify each term.
Raise 4 to the power of 2.
f(4)=162-3⋅4+112
Divide 16 by 2.
f(4)=8-3⋅4+112
Multiply -3 by 4.
f(4)=8-12+112
f(4)=8-12+112
Find the common denominator.
Write 8 as a fraction with denominator 1.
f(4)=81-12+112
Multiply 81 by 22.
f(4)=81⋅22-12+112
Multiply 81 and 22.
f(4)=8⋅22-12+112
Write -12 as a fraction with denominator 1.
f(4)=8⋅22+-121+112
Multiply -121 by 22.
f(4)=8⋅22+-121⋅22+112
Multiply -121 and 22.
f(4)=8⋅22+-12⋅22+112
f(4)=8⋅22+-12⋅22+112
Combine fractions.
Combine fractions with similar denominators.
f(4)=8⋅2-12⋅2+112
Multiply.
Multiply 8 by 2.
f(4)=16-12⋅2+112
Multiply -12 by 2.
f(4)=16-24+112
f(4)=16-24+112
f(4)=16-24+112
Simplify the numerator.
Subtract 24 from 16.
f(4)=-8+112
Add -8 and 11.
f(4)=32
f(4)=32
The final answer is 32.
32
32
The y value at x=4 is 32.
y=32
Graph the parabola using its properties and the selected points.
xy132323143253
xy132323143253
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (3,1)
Focus: (3,32)
Axis of Symmetry: x=3
Directrix: y=12
xy132323143253
Graph y=1/2x^2-3x+11/2
