# Graph y=1/2x^2-9 y=12×2-9
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Combine 12 and x2.
y=x22-9
Complete the square for x22-9.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=12,b=0,c=-9
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=02(12)
Simplify the right side.
Cancel the common factor of 0 and 2.
Factor 2 out of 0.
d=2(0)2(12)
Cancel the common factors.
Cancel the common factor.
d=2⋅02(12)
Rewrite the expression.
d=012
d=012
d=012
Multiply the numerator by the reciprocal of the denominator.
d=0⋅2
Multiply 0 by 2.
d=0
d=0
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raising 0 to any positive power yields 0.
e=-9-04(12)
Combine 4 and 12.
e=-9-042
Divide 4 by 2.
e=-9-02
Divide 0 by 2.
e=-9-0
Multiply -1 by 0.
e=-9+0
e=-9+0
Add -9 and 0.
e=-9
e=-9
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
12⋅(x+0)2-9
12⋅(x+0)2-9
Set y equal to the new right side.
y=12⋅(x+0)2-9
y=12⋅(x+0)2-9
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=12
h=0
k=-9
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(0,-9)
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.
(0,-172)
(0,-172)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=0
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=-192
y=-192
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (0,-9)
Focus: (0,-172)
Axis of Symmetry: x=0
Directrix: y=-192
Direction: Opens Up
Vertex: (0,-9)
Focus: (0,-172)
Axis of Symmetry: x=0
Directrix: y=-192
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 -2 in the expression.
f(-2)=(-2)22-9
Simplify the result.
Cancel the common factor of (-2)2 and 2.
Rewrite -2 as -1(2).
f(-2)=(-1⋅2)22-9
Apply the product rule to -1(2).
f(-2)=(-1)2⋅222-9
Raise -1 to the power of 2.
f(-2)=1⋅222-9
Multiply 22 by 1.
f(-2)=222-9
Factor 2 out of 22.
f(-2)=2⋅22-9
Cancel the common factors.
Factor 2 out of 2.
f(-2)=2⋅22(1)-9
Cancel the common factor.
f(-2)=2⋅22⋅1-9
Rewrite the expression.
f(-2)=21-9
Divide 2 by 1.
f(-2)=2-9
f(-2)=2-9
f(-2)=2-9
Subtract 9 from 2.
f(-2)=-7
The final answer is -7.
-7
-7
The y value at x=-2 is -7.
y=-7
Replace the variable x with -1 in the expression.
f(-1)=(-1)22-9
Simplify the result.
Raise -1 to the power of 2.
f(-1)=12-9
To write -9 as a fraction with a common denominator, multiply by 22.
f(-1)=12-9⋅22
Combine -9 and 22.
f(-1)=12+-9⋅22
Combine the numerators over the common denominator.
f(-1)=1-9⋅22
Simplify the numerator.
Multiply -9 by 2.
f(-1)=1-182
Subtract 18 from 1.
f(-1)=-172
f(-1)=-172
Move the negative in front of the fraction.
f(-1)=-172
The final answer is -172.
-172
-172
The y value at x=-1 is -172.
y=-172
Replace the variable x with 2 in the expression.
f(2)=(2)22-9
Simplify the result.
Cancel the common factor of (2)2 and 2.
Factor 2 out of (2)2.
f(2)=2⋅22-9
Cancel the common factors.
Factor 2 out of 2.
f(2)=2⋅22(1)-9
Cancel the common factor.
f(2)=2⋅22⋅1-9
Rewrite the expression.
f(2)=21-9
Divide 2 by 1.
f(2)=2-9
f(2)=2-9
f(2)=2-9
Subtract 9 from 2.
f(2)=-7
The final answer is -7.
-7
-7
The y value at x=2 is -7.
y=-7
Replace the variable x with 1 in the expression.
f(1)=(1)22-9
Simplify the result.
One to any power is one.
f(1)=12-9
To write -9 as a fraction with a common denominator, multiply by 22.
f(1)=12-9⋅22
Combine -9 and 22.
f(1)=12+-9⋅22
Combine the numerators over the common denominator.
f(1)=1-9⋅22
Simplify the numerator.
Multiply -9 by 2.
f(1)=1-182
Subtract 18 from 1.
f(1)=-172
f(1)=-172
Move the negative in front of the fraction.
f(1)=-172
The final answer is -172.
-172
-172
The y value at x=1 is -172.
y=-172
Graph the parabola using its properties and the selected points.
xy-2-7-1-1720-91-1722-7
xy-2-7-1-1720-91-1722-7
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (0,-9)
Focus: (0,-172)
Axis of Symmetry: x=0
Directrix: y=-192
xy-2-7-1-1720-91-1722-7
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