Graph f(x)=3x^2+30x-5

Math
f(x)=3×2+30x-5
Find the properties of the given parabola.
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Rewrite the equation in vertex form.
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Complete the square for 3×2+30x-5.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=3,b=30,c=-5
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=302(3)
Simplify the right side.
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Cancel the common factor of 30 and 2.
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Factor 2 out of 30.
d=2⋅152⋅3
Cancel the common factors.
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Factor 2 out of 2⋅3.
d=2⋅152(3)
Cancel the common factor.
d=2⋅152⋅3
Rewrite the expression.
d=153
d=153
d=153
Cancel the common factor of 15 and 3.
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Factor 3 out of 15.
d=3⋅53
Cancel the common factors.
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Factor 3 out of 3.
d=3⋅53(1)
Cancel the common factor.
d=3⋅53⋅1
Rewrite the expression.
d=51
Divide 5 by 1.
d=5
d=5
d=5
d=5
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raise 30 to the power of 2.
e=-5-9004⋅3
Multiply 4 by 3.
e=-5-90012
Divide 900 by 12.
e=-5-1⋅75
Multiply -1 by 75.
e=-5-75
e=-5-75
Subtract 75 from -5.
e=-80
e=-80
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
3(x+5)2-80
3(x+5)2-80
Set y equal to the new right side.
y=3(x+5)2-80
y=3(x+5)2-80
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=3
h=-5
k=-80
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-5,-80)
Find p, the distance from the vertex to the focus.
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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⋅3
Multiply 4 by 3.
112
112
Find the focus.
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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.
(-5,-95912)
(-5,-95912)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-5
Find the directrix.
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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=-96112
y=-96112
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-5,-80)
Focus: (-5,-95912)
Axis of Symmetry: x=-5
Directrix: y=-96112
Direction: Opens Up
Vertex: (-5,-80)
Focus: (-5,-95912)
Axis of Symmetry: x=-5
Directrix: y=-96112
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.
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Replace the variable x with -6 in the expression.
f(-6)=3(-6)2+30(-6)-5
Simplify the result.
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Simplify each term.
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Raise -6 to the power of 2.
f(-6)=3⋅36+30(-6)-5
Multiply 3 by 36.
f(-6)=108+30(-6)-5
Multiply 30 by -6.
f(-6)=108-180-5
f(-6)=108-180-5
Simplify by subtracting numbers.
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Subtract 180 from 108.
f(-6)=-72-5
Subtract 5 from -72.
f(-6)=-77
f(-6)=-77
The final answer is -77.
-77
-77
The y value at x=-6 is -77.
y=-77
Replace the variable x with -7 in the expression.
f(-7)=3(-7)2+30(-7)-5
Simplify the result.
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Simplify each term.
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Raise -7 to the power of 2.
f(-7)=3⋅49+30(-7)-5
Multiply 3 by 49.
f(-7)=147+30(-7)-5
Multiply 30 by -7.
f(-7)=147-210-5
f(-7)=147-210-5
Simplify by subtracting numbers.
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Subtract 210 from 147.
f(-7)=-63-5
Subtract 5 from -63.
f(-7)=-68
f(-7)=-68
The final answer is -68.
-68
-68
The y value at x=-7 is -68.
y=-68
Replace the variable x with -4 in the expression.
f(-4)=3(-4)2+30(-4)-5
Simplify the result.
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Simplify each term.
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Raise -4 to the power of 2.
f(-4)=3⋅16+30(-4)-5
Multiply 3 by 16.
f(-4)=48+30(-4)-5
Multiply 30 by -4.
f(-4)=48-120-5
f(-4)=48-120-5
Simplify by subtracting numbers.
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Subtract 120 from 48.
f(-4)=-72-5
Subtract 5 from -72.
f(-4)=-77
f(-4)=-77
The final answer is -77.
-77
-77
The y value at x=-4 is -77.
y=-77
Replace the variable x with -3 in the expression.
f(-3)=3(-3)2+30(-3)-5
Simplify the result.
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Simplify each term.
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Raise -3 to the power of 2.
f(-3)=3⋅9+30(-3)-5
Multiply 3 by 9.
f(-3)=27+30(-3)-5
Multiply 30 by -3.
f(-3)=27-90-5
f(-3)=27-90-5
Simplify by subtracting numbers.
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Subtract 90 from 27.
f(-3)=-63-5
Subtract 5 from -63.
f(-3)=-68
f(-3)=-68
The final answer is -68.
-68
-68
The y value at x=-3 is -68.
y=-68
Graph the parabola using its properties and the selected points.
xy-7-68-6-77-5-80-4-77-3-68
xy-7-68-6-77-5-80-4-77-3-68
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-5,-80)
Focus: (-5,-95912)
Axis of Symmetry: x=-5
Directrix: y=-96112
xy-7-68-6-77-5-80-4-77-3-68
Graph f(x)=3x^2+30x-5

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