Graph y=-5x^2+100x+15

Math
y=-5×2+100x+15
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 -5×2+100x+15.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=-5,b=100,c=15
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=1002(-5)
Simplify the right side.
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Cancel the common factor of 100 and 2.
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Factor 2 out of 100.
d=2⋅502⋅-5
Cancel the common factors.
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Factor 2 out of 2⋅-5.
d=2⋅502(-5)
Cancel the common factor.
d=2⋅502⋅-5
Rewrite the expression.
d=50-5
d=50-5
d=50-5
Cancel the common factor of 50 and -5.
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Factor 5 out of 50.
d=5(10)-5
Move the negative one from the denominator of 10-1.
d=-1⋅10
d=-1⋅10
Multiply -1 by 10.
d=-10
d=-10
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raise 100 to the power of 2.
e=15-100004⋅-5
Multiply 4 by -5.
e=15-10000-20
Divide 10000 by -20.
e=15+500
Multiply -1 by -500.
e=15+500
e=15+500
Add 15 and 500.
e=515
e=515
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-5(x-10)2+515
-5(x-10)2+515
Set y equal to the new right side.
y=-5(x-10)2+515
y=-5(x-10)2+515
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-5
h=10
k=515
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(10,515)
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⋅-5
Simplify.
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Multiply 4 by -5.
1-20
Move the negative in front of the fraction.
-120
-120
-120
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.
(10,1029920)
(10,1029920)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=10
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=1030120
y=1030120
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (10,515)
Focus: (10,1029920)
Axis of Symmetry: x=10
Directrix: y=1030120
Direction: Opens Down
Vertex: (10,515)
Focus: (10,1029920)
Axis of Symmetry: x=10
Directrix: y=1030120
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 9 in the expression.
f(9)=-5(9)2+100(9)+15
Simplify the result.
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Simplify each term.
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Raise 9 to the power of 2.
f(9)=-5⋅81+100(9)+15
Multiply -5 by 81.
f(9)=-405+100(9)+15
Multiply 100 by 9.
f(9)=-405+900+15
f(9)=-405+900+15
Simplify by adding numbers.
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Add -405 and 900.
f(9)=495+15
Add 495 and 15.
f(9)=510
f(9)=510
The final answer is 510.
510
510
The y value at x=9 is 510.
y=510
Replace the variable x with 8 in the expression.
f(8)=-5(8)2+100(8)+15
Simplify the result.
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Simplify each term.
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Raise 8 to the power of 2.
f(8)=-5⋅64+100(8)+15
Multiply -5 by 64.
f(8)=-320+100(8)+15
Multiply 100 by 8.
f(8)=-320+800+15
f(8)=-320+800+15
Simplify by adding numbers.
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Add -320 and 800.
f(8)=480+15
Add 480 and 15.
f(8)=495
f(8)=495
The final answer is 495.
495
495
The y value at x=8 is 495.
y=495
Replace the variable x with 11 in the expression.
f(11)=-5(11)2+100(11)+15
Simplify the result.
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Simplify each term.
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Raise 11 to the power of 2.
f(11)=-5⋅121+100(11)+15
Multiply -5 by 121.
f(11)=-605+100(11)+15
Multiply 100 by 11.
f(11)=-605+1100+15
f(11)=-605+1100+15
Simplify by adding numbers.
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Add -605 and 1100.
f(11)=495+15
Add 495 and 15.
f(11)=510
f(11)=510
The final answer is 510.
510
510
The y value at x=11 is 510.
y=510
Replace the variable x with 12 in the expression.
f(12)=-5(12)2+100(12)+15
Simplify the result.
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Simplify each term.
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Raise 12 to the power of 2.
f(12)=-5⋅144+100(12)+15
Multiply -5 by 144.
f(12)=-720+100(12)+15
Multiply 100 by 12.
f(12)=-720+1200+15
f(12)=-720+1200+15
Simplify by adding numbers.
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Add -720 and 1200.
f(12)=480+15
Add 480 and 15.
f(12)=495
f(12)=495
The final answer is 495.
495
495
The y value at x=12 is 495.
y=495
Graph the parabola using its properties and the selected points.
xy84959510105151151012495
xy84959510105151151012495
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (10,515)
Focus: (10,1029920)
Axis of Symmetry: x=10
Directrix: y=1030120
xy84959510105151151012495
Graph y=-5x^2+100x+15

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