# Graph g(x)=x^2-x-6 g(x)=x2-x-6
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for x2-x-6.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,b=-1,c=-6
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=-12(1)
Cancel the common factor of 1.
Cancel the common factor.
d=-12⋅1
Rewrite the expression.
d=-12
d=-12
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise -1 to the power of 2.
e=-6-14⋅1
Multiply 4 by 1.
e=-6-14
e=-6-14
To write -6 as a fraction with a common denominator, multiply by 44.
e=-6⋅44-14
Combine -6 and 44.
e=-6⋅44-14
Combine the numerators over the common denominator.
e=-6⋅4-14
Simplify the numerator.
Multiply -6 by 4.
e=-24-14
Subtract 1 from -24.
e=-254
e=-254
Move the negative in front of the fraction.
e=-254
e=-254
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(x-12)2-254
(x-12)2-254
Set y equal to the new right side.
y=(x-12)2-254
y=(x-12)2-254
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=1
h=12
k=-254
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(12,-254)
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⋅1
Cancel the common factor of 1.
Cancel the common factor.
14⋅1
Rewrite the expression.
14
14
14
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.
(12,-6)
(12,-6)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=12
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=-132
y=-132
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (12,-254)
Focus: (12,-6)
Axis of Symmetry: x=12
Directrix: y=-132
Direction: Opens Up
Vertex: (12,-254)
Focus: (12,-6)
Axis of Symmetry: x=12
Directrix: y=-132
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)=(-1)2-(-1)-6
Simplify the result.
Simplify each term.
Raise -1 to the power of 2.
f(-1)=1-(-1)-6
Multiply -1 by -1.
f(-1)=1+1-6
f(-1)=1+1-6
Simplify by adding and subtracting.
Add 1 and 1.
f(-1)=2-6
Subtract 6 from 2.
f(-1)=-4
f(-1)=-4
The final answer is -4.
-4
-4
The y value at x=-1 is -4.
y=-4
Replace the variable x with -2 in the expression.
f(-2)=(-2)2-(-2)-6
Simplify the result.
Simplify each term.
Raise -2 to the power of 2.
f(-2)=4-(-2)-6
Multiply -1 by -2.
f(-2)=4+2-6
f(-2)=4+2-6
Simplify by adding and subtracting.
Add 4 and 2.
f(-2)=6-6
Subtract 6 from 6.
f(-2)=0
f(-2)=0
The final answer is 0.
0
0
The y value at x=-2 is 0.
y=0
Replace the variable x with 1 in the expression.
f(1)=(1)2-(1)-6
Simplify the result.
Simplify each term.
One to any power is one.
f(1)=1-(1)-6
Multiply -1 by 1.
f(1)=1-1-6
f(1)=1-1-6
Simplify by subtracting numbers.
Subtract 1 from 1.
f(1)=0-6
Subtract 6 from 0.
f(1)=-6
f(1)=-6
The final answer is -6.
-6
-6
The y value at x=1 is -6.
y=-6
Replace the variable x with 2 in the expression.
f(2)=(2)2-(2)-6
Simplify the result.
Simplify each term.
Raise 2 to the power of 2.
f(2)=4-(2)-6
Multiply -1 by 2.
f(2)=4-2-6
f(2)=4-2-6
Simplify by subtracting numbers.
Subtract 2 from 4.
f(2)=2-6
Subtract 6 from 2.
f(2)=-4
f(2)=-4
The final answer is -4.
-4
-4
The y value at x=2 is -4.
y=-4
Graph the parabola using its properties and the selected points.
xy-20-1-412-2541-62-4
xy-20-1-412-2541-62-4
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (12,-254)
Focus: (12,-6)
Axis of Symmetry: x=12
Directrix: y=-132
xy-20-1-412-2541-62-4
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