# Graph f(x)=x^2+20x

f(x)=x2+20x
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for x2+20x.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,b=20,c=0
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=202(1)
Cancel the common factor of 20 and 2.
Factor 2 out of 20.
d=2⋅102⋅1
Cancel the common factors.
Factor 2 out of 2⋅1.
d=2⋅102(1)
Cancel the common factor.
d=2⋅102⋅1
Rewrite the expression.
d=101
Divide 10 by 1.
d=10
d=10
d=10
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise 20 to the power of 2.
e=0-4004⋅1
Multiply 4 by 1.
e=0-4004
Divide 400 by 4.
e=0-1⋅100
Multiply -1 by 100.
e=0-100
e=0-100
Subtract 100 from 0.
e=-100
e=-100
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(x+10)2-100
(x+10)2-100
Set y equal to the new right side.
y=(x+10)2-100
y=(x+10)2-100
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=1
h=-10
k=-100
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-10,-100)
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.
(-10,-3994)
(-10,-3994)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-10
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=-4014
y=-4014
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-10,-100)
Focus: (-10,-3994)
Axis of Symmetry: x=-10
Directrix: y=-4014
Direction: Opens Up
Vertex: (-10,-100)
Focus: (-10,-3994)
Axis of Symmetry: x=-10
Directrix: y=-4014
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 -11 in the expression.
f(-11)=(-11)2+20(-11)
Simplify the result.
Simplify each term.
Raise -11 to the power of 2.
f(-11)=121+20(-11)
Multiply 20 by -11.
f(-11)=121-220
f(-11)=121-220
Subtract 220 from 121.
f(-11)=-99
-99
-99
The y value at x=-11 is -99.
y=-99
Replace the variable x with -12 in the expression.
f(-12)=(-12)2+20(-12)
Simplify the result.
Simplify each term.
Raise -12 to the power of 2.
f(-12)=144+20(-12)
Multiply 20 by -12.
f(-12)=144-240
f(-12)=144-240
Subtract 240 from 144.
f(-12)=-96
-96
-96
The y value at x=-12 is -96.
y=-96
Replace the variable x with -9 in the expression.
f(-9)=(-9)2+20(-9)
Simplify the result.
Simplify each term.
Raise -9 to the power of 2.
f(-9)=81+20(-9)
Multiply 20 by -9.
f(-9)=81-180
f(-9)=81-180
Subtract 180 from 81.
f(-9)=-99
-99
-99
The y value at x=-9 is -99.
y=-99
Replace the variable x with -8 in the expression.
f(-8)=(-8)2+20(-8)
Simplify the result.
Simplify each term.
Raise -8 to the power of 2.
f(-8)=64+20(-8)
Multiply 20 by -8.
f(-8)=64-160
f(-8)=64-160
Subtract 160 from 64.
f(-8)=-96
-96
-96
The y value at x=-8 is -96.
y=-96
Graph the parabola using its properties and the selected points.
xy-12-96-11-99-10-100-9-99-8-96
xy-12-96-11-99-10-100-9-99-8-96
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-10,-100)
Focus: (-10,-3994)
Axis of Symmetry: x=-10
Directrix: y=-4014
xy-12-96-11-99-10-100-9-99-8-96
Graph f(x)=x^2+20x