# Graph f(x)=x^2+4x-y

f(x)=x2+4x-y
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Isolate y to the left side of the equation.
Move all terms containing y to the left side of the equation.
Add y to both sides of the equation.
y+y=x2+4x
Add y and y.
2y=x2+4x
2y=x2+4x
Divide each term by 2 and simplify.
Divide each term in 2y=x2+4x by 2.
2y2=x22+4×2
Cancel the common factor of 2.
Cancel the common factor.
2y2=x22+4×2
Divide y by 1.
y=x22+4×2
y=x22+4×2
Cancel the common factor of 4 and 2.
Factor 2 out of 4x.
y=x22+2(2x)2
Cancel the common factors.
Factor 2 out of 2.
y=x22+2(2x)2(1)
Cancel the common factor.
y=x22+2(2x)2⋅1
Rewrite the expression.
y=x22+2×1
Divide 2x by 1.
y=x22+2x
y=x22+2x
y=x22+2x
y=x22+2x
y=x22+2x
Complete the square for x22+2x.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=12,b=2,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=22(12)
Simplify the right side.
Cancel the common factor of 2.
Cancel the common factor.
d=22(12)
Rewrite the expression.
d=112
d=112
Multiply the numerator by the reciprocal of the denominator.
d=1⋅2
Multiply 2 by 1.
d=2
d=2
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise 2 to the power of 2.
e=0-44(12)
Combine 4 and 12.
e=0-442
Divide 4 by 2.
e=0-42
Divide 4 by 2.
e=0-1⋅2
Multiply -1 by 2.
e=0-2
e=0-2
Subtract 2 from 0.
e=-2
e=-2
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
12⋅(x+2)2-2
12⋅(x+2)2-2
Set y equal to the new right side.
y=12⋅(x+2)2-2
y=12⋅(x+2)2-2
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=12
h=-2
k=-2
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-2,-2)
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.
(-2,-32)
(-2,-32)
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=-52
y=-52
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-2,-2)
Focus: (-2,-32)
Axis of Symmetry: x=-2
Directrix: y=-52
Direction: Opens Up
Vertex: (-2,-2)
Focus: (-2,-32)
Axis of Symmetry: x=-2
Directrix: y=-52
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 -4 in the expression.
f(-4)=(-4)22+2(-4)
Simplify the result.
Simplify each term.
Raise -4 to the power of 2.
f(-4)=162+2(-4)
Divide 16 by 2.
f(-4)=8+2(-4)
Multiply 2 by -4.
f(-4)=8-8
f(-4)=8-8
Subtract 8 from 8.
f(-4)=0
The final answer is 0.
0
0
The y value at x=-4 is 0.
y=0
Replace the variable x with -3 in the expression.
f(-3)=(-3)22+2(-3)
Simplify the result.
Simplify each term.
Raise -3 to the power of 2.
f(-3)=92+2(-3)
Multiply 2 by -3.
f(-3)=92-6
f(-3)=92-6
To write -6 as a fraction with a common denominator, multiply by 22.
f(-3)=92-6⋅22
Combine -6 and 22.
f(-3)=92+-6⋅22
Combine the numerators over the common denominator.
f(-3)=9-6⋅22
Simplify the numerator.
Multiply -6 by 2.
f(-3)=9-122
Subtract 12 from 9.
f(-3)=-32
f(-3)=-32
Move the negative in front of the fraction.
f(-3)=-32
The final answer is -32.
-32
-32
The y value at x=-3 is -32.
y=-32
Replace the variable x with 0 in the expression.
f(0)=(0)22+2(0)
Simplify the result.
Simplify each term.
Raising 0 to any positive power yields 0.
f(0)=02+2(0)
Divide 0 by 2.
f(0)=0+2(0)
Multiply 2 by 0.
f(0)=0+0
f(0)=0+0
Add 0 and 0.
f(0)=0
The final answer is 0.
0
0
The y value at x=0 is 0.
y=0
Replace the variable x with -1 in the expression.
f(-1)=(-1)22+2(-1)
Simplify the result.
Simplify each term.
Raise -1 to the power of 2.
f(-1)=12+2(-1)
Multiply 2 by -1.
f(-1)=12-2
f(-1)=12-2
To write -2 as a fraction with a common denominator, multiply by 22.
f(-1)=12-2⋅22
Combine -2 and 22.
f(-1)=12+-2⋅22
Combine the numerators over the common denominator.
f(-1)=1-2⋅22
Simplify the numerator.
Multiply -2 by 2.
f(-1)=1-42
Subtract 4 from 1.
f(-1)=-32
f(-1)=-32
Move the negative in front of the fraction.
f(-1)=-32
The final answer is -32.
-32
-32
The y value at x=-1 is -32.
y=-32
Graph the parabola using its properties and the selected points.
xy-40-3-32-2-2-1-3200
xy-40-3-32-2-2-1-3200
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-2,-2)
Focus: (-2,-32)
Axis of Symmetry: x=-2
Directrix: y=-52
xy-40-3-32-2-2-1-3200
Graph f(x)=x^2+4x-y

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