# Graph f(x)=7x^2+40x-12

f(x)=7×2+40x-12
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for 7×2+40x-12.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=7,b=40,c=-12
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=402(7)
Cancel the common factor of 40 and 2.
Factor 2 out of 40.
d=2⋅202⋅7
Cancel the common factors.
Factor 2 out of 2⋅7.
d=2⋅202(7)
Cancel the common factor.
d=2⋅202⋅7
Rewrite the expression.
d=207
d=207
d=207
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise 40 to the power of 2.
e=-12-16004⋅7
Multiply 4 by 7.
e=-12-160028
Cancel the common factor of 1600 and 28.
Factor 4 out of 1600.
e=-12-4(400)28
Cancel the common factors.
Factor 4 out of 28.
e=-12-4⋅4004⋅7
Cancel the common factor.
e=-12-4⋅4004⋅7
Rewrite the expression.
e=-12-4007
e=-12-4007
e=-12-4007
e=-12-4007
To write -12 as a fraction with a common denominator, multiply by 77.
e=-12⋅77-4007
Combine -12 and 77.
e=-12⋅77-4007
Combine the numerators over the common denominator.
e=-12⋅7-4007
Simplify the numerator.
Multiply -12 by 7.
e=-84-4007
Subtract 400 from -84.
e=-4847
e=-4847
Move the negative in front of the fraction.
e=-4847
e=-4847
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
7(x+207)2-4847
7(x+207)2-4847
Set y equal to the new right side.
y=7(x+207)2-4847
y=7(x+207)2-4847
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=7
h=-207
k=-4847
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-207,-4847)
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⋅7
Multiply 4 by 7.
128
128
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.
(-207,-193528)
(-207,-193528)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-207
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=-193728
y=-193728
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-207,-4847)
Focus: (-207,-193528)
Axis of Symmetry: x=-207
Directrix: y=-193728
Direction: Opens Up
Vertex: (-207,-4847)
Focus: (-207,-193528)
Axis of Symmetry: x=-207
Directrix: y=-193728
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)=7(-4)2+40(-4)-12
Simplify the result.
Simplify each term.
Raise -4 to the power of 2.
f(-4)=7⋅16+40(-4)-12
Multiply 7 by 16.
f(-4)=112+40(-4)-12
Multiply 40 by -4.
f(-4)=112-160-12
f(-4)=112-160-12
Simplify by subtracting numbers.
Subtract 160 from 112.
f(-4)=-48-12
Subtract 12 from -48.
f(-4)=-60
f(-4)=-60
-60
-60
The y value at x=-4 is -60.
y=-60
Replace the variable x with -2 in the expression.
f(-2)=7(-2)2+40(-2)-12
Simplify the result.
Simplify each term.
Raise -2 to the power of 2.
f(-2)=7⋅4+40(-2)-12
Multiply 7 by 4.
f(-2)=28+40(-2)-12
Multiply 40 by -2.
f(-2)=28-80-12
f(-2)=28-80-12
Simplify by subtracting numbers.
Subtract 80 from 28.
f(-2)=-52-12
Subtract 12 from -52.
f(-2)=-64
f(-2)=-64
-64
-64
The y value at x=-2 is -64.
y=-64
Replace the variable x with -1 in the expression.
f(-1)=7(-1)2+40(-1)-12
Simplify the result.
Simplify each term.
Raise -1 to the power of 2.
f(-1)=7⋅1+40(-1)-12
Multiply 7 by 1.
f(-1)=7+40(-1)-12
Multiply 40 by -1.
f(-1)=7-40-12
f(-1)=7-40-12
Simplify by subtracting numbers.
Subtract 40 from 7.
f(-1)=-33-12
Subtract 12 from -33.
f(-1)=-45
f(-1)=-45
-45
-45
The y value at x=-1 is -45.
y=-45
Graph the parabola using its properties and the selected points.
xy-4-60-207-4847-2-64-1-45
xy-4-60-207-4847-2-64-1-45
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-207,-4847)
Focus: (-207,-193528)
Axis of Symmetry: x=-207
Directrix: y=-193728
xy-4-60-207-4847-2-64-1-45
Graph f(x)=7x^2+40x-12
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