Graph y=5x^2+7x-2

Math
y=5×2+7x-2
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+7x-2.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=5,b=7,c=-2
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=72(5)
Multiply 2 by 5.
d=710
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raise 7 to the power of 2.
e=-2-494⋅5
Multiply 4 by 5.
e=-2-4920
e=-2-4920
To write -2 as a fraction with a common denominator, multiply by 2020.
e=-2⋅2020-4920
Combine -2 and 2020.
e=-2⋅2020-4920
Combine the numerators over the common denominator.
e=-2⋅20-4920
Simplify the numerator.
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Multiply -2 by 20.
e=-40-4920
Subtract 49 from -40.
e=-8920
e=-8920
Move the negative in front of the fraction.
e=-8920
e=-8920
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
5(x+710)2-8920
5(x+710)2-8920
Set y equal to the new right side.
y=5(x+710)2-8920
y=5(x+710)2-8920
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=5
h=-710
k=-8920
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-710,-8920)
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
Multiply 4 by 5.
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.
(-710,-225)
(-710,-225)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-710
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=-92
y=-92
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-710,-8920)
Focus: (-710,-225)
Axis of Symmetry: x=-710
Directrix: y=-92
Direction: Opens Up
Vertex: (-710,-8920)
Focus: (-710,-225)
Axis of Symmetry: x=-710
Directrix: y=-92
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 -2 in the expression.
f(-2)=5(-2)2+7(-2)-2
Simplify the result.
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Simplify each term.
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Raise -2 to the power of 2.
f(-2)=5⋅4+7(-2)-2
Multiply 5 by 4.
f(-2)=20+7(-2)-2
Multiply 7 by -2.
f(-2)=20-14-2
f(-2)=20-14-2
Simplify by subtracting numbers.
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Subtract 14 from 20.
f(-2)=6-2
Subtract 2 from 6.
f(-2)=4
f(-2)=4
The final answer is 4.
4
4
The y value at x=-2 is 4.
y=4
Replace the variable x with -3 in the expression.
f(-3)=5(-3)2+7(-3)-2
Simplify the result.
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Simplify each term.
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Raise -3 to the power of 2.
f(-3)=5⋅9+7(-3)-2
Multiply 5 by 9.
f(-3)=45+7(-3)-2
Multiply 7 by -3.
f(-3)=45-21-2
f(-3)=45-21-2
Simplify by subtracting numbers.
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Subtract 21 from 45.
f(-3)=24-2
Subtract 2 from 24.
f(-3)=22
f(-3)=22
The final answer is 22.
22
22
The y value at x=-3 is 22.
y=22
Replace the variable x with 0 in the expression.
f(0)=5(0)2+7(0)-2
Simplify the result.
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Simplify each term.
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Raising 0 to any positive power yields 0.
f(0)=5⋅0+7(0)-2
Multiply 5 by 0.
f(0)=0+7(0)-2
Multiply 7 by 0.
f(0)=0+0-2
f(0)=0+0-2
Simplify by adding zeros.
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Add 0 and 0.
f(0)=0-2
Subtract 2 from 0.
f(0)=-2
f(0)=-2
The final answer is -2.
-2
-2
The y value at x=0 is -2.
y=-2
Replace the variable x with 1 in the expression.
f(1)=5(1)2+7(1)-2
Simplify the result.
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Simplify each term.
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One to any power is one.
f(1)=5⋅1+7(1)-2
Multiply 5 by 1.
f(1)=5+7(1)-2
Multiply 7 by 1.
f(1)=5+7-2
f(1)=5+7-2
Simplify by adding and subtracting.
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Add 5 and 7.
f(1)=12-2
Subtract 2 from 12.
f(1)=10
f(1)=10
The final answer is 10.
10
10
The y value at x=1 is 10.
y=10
Graph the parabola using its properties and the selected points.
xy-322-24-710-89200-2110
xy-322-24-710-89200-2110
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-710,-8920)
Focus: (-710,-225)
Axis of Symmetry: x=-710
Directrix: y=-92
xy-322-24-710-89200-2110
Graph y=5x^2+7x-2

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