# Graph f(x)=-4x^2-8x f(x)=-4×2-8x
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for -4×2-8x.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=-4,b=-8,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=-82(-4)
Simplify the right side.
Cancel the common factor of 8 and 2.
Factor 2 out of 8.
d=-2⋅42⋅-4
Cancel the common factors.
Factor 2 out of 2⋅-4.
d=-2⋅42(-4)
Cancel the common factor.
d=-2⋅42⋅-4
Rewrite the expression.
d=-4-4
d=-4-4
d=-4-4
Cancel the common factor of 4 and -4.
Factor 4 out of 4.
d=-4(1)-4
Move the negative one from the denominator of 1-1.
d=-(-1⋅1)
d=-(-1⋅1)
Multiply.
Multiply -1 by 1.
d=1
Multiply -1 by -1.
d=1
d=1
d=1
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise -8 to the power of 2.
e=0-644⋅-4
Multiply 4 by -4.
e=0-64-16
Divide 64 by -16.
e=0+4
Multiply -1 by -4.
e=0+4
e=0+4
Add 0 and 4.
e=4
e=4
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-4(x+1)2+4
-4(x+1)2+4
Set y equal to the new right side.
y=-4(x+1)2+4
y=-4(x+1)2+4
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-4
h=-1
k=4
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(-1,4)
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⋅-4
Simplify.
Multiply 4 by -4.
1-16
Move the negative in front of the fraction.
-116
-116
-116
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.
(-1,6316)
(-1,6316)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-1
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=6516
y=6516
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (-1,4)
Focus: (-1,6316)
Axis of Symmetry: x=-1
Directrix: y=6516
Direction: Opens Down
Vertex: (-1,4)
Focus: (-1,6316)
Axis of Symmetry: x=-1
Directrix: y=6516
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)=-4(-2)2-8⋅-2
Simplify the result.
Simplify each term.
Raise -2 to the power of 2.
f(-2)=-4⋅4-8⋅-2
Multiply -4 by 4.
f(-2)=-16-8⋅-2
Multiply -8 by -2.
f(-2)=-16+16
f(-2)=-16+16
Add -16 and 16.
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 -3 in the expression.
f(-3)=-4(-3)2-8⋅-3
Simplify the result.
Simplify each term.
Raise -3 to the power of 2.
f(-3)=-4⋅9-8⋅-3
Multiply -4 by 9.
f(-3)=-36-8⋅-3
Multiply -8 by -3.
f(-3)=-36+24
f(-3)=-36+24
Add -36 and 24.
f(-3)=-12
The final answer is -12.
-12
-12
The y value at x=-3 is -12.
y=-12
Replace the variable x with 0 in the expression.
f(0)=-4(0)2-8⋅0
Simplify the result.
Simplify each term.
Raising 0 to any positive power yields 0.
f(0)=-4⋅0-8⋅0
Multiply -4 by 0.
f(0)=0-8⋅0
Multiply -8 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)=-4(1)2-8⋅1
Simplify the result.
Simplify each term.
One to any power is one.
f(1)=-4⋅1-8⋅1
Multiply -4 by 1.
f(1)=-4-8⋅1
Multiply -8 by 1.
f(1)=-4-8
f(1)=-4-8
Subtract 8 from -4.
f(1)=-12
The final answer is -12.
-12
-12
The y value at x=1 is -12.
y=-12
Graph the parabola using its properties and the selected points.
xy-3-12-20-14001-12
xy-3-12-20-14001-12
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (-1,4)
Focus: (-1,6316)
Axis of Symmetry: x=-1
Directrix: y=6516
xy-3-12-20-14001-12
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