Graph y=-5/4x^2-5/2x+15/4

Math
y=-54×2-52x+154
Simplify each term.
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Combine x2 and 54.
y=-x2⋅54-52x+154
Move 5 to the left of x2.
y=-5⋅x24-52x+154
Combine x and 52.
y=-5×24-x⋅52+154
Move 5 to the left of x.
y=-5×24-5×2+154
y=-5×24-5×2+154
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 -5×24-5×2+154.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=-54,b=-52,c=154
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=-522(-54)
Simplify the right side.
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Multiply the numerator by the reciprocal of the denominator.
d=-(52⋅12⋅(-1(54)))
Cancel the common factor of 1 and -1.
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Rewrite 1 as -1(-1).
d=-(52⋅-1⋅-12⋅(-1(54)))
Move the negative in front of the fraction.
d=-(52⋅(-12(54)))
d=-(52⋅(-12(54)))
Combine 2 and 54.
d=-(52⋅(-12⋅54))
Multiply 2 by 5.
d=-(52⋅(-1104))
Cancel the common factor of 10 and 4.
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Factor 2 out of 10.
d=-(52⋅(-12(5)4))
Cancel the common factors.
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Factor 2 out of 4.
d=-(52⋅(-12⋅52⋅2))
Cancel the common factor.
d=-(52⋅(-12⋅52⋅2))
Rewrite the expression.
d=-(52⋅(-152))
d=-(52⋅(-152))
d=-(52⋅(-152))
Multiply the numerator by the reciprocal of the denominator.
d=-(52⋅(-(1(25))))
Multiply 25 by 1.
d=-(52⋅(-25))
Cancel the common factor of 5.
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Move the leading negative in -25 into the numerator.
d=-(52⋅-25)
Cancel the common factor.
d=-(52⋅-25)
Rewrite the expression.
d=-(12⋅-2)
d=-(12⋅-2)
Cancel the common factor of 2.
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Factor 2 out of -2.
d=-(12⋅(2(-1)))
Cancel the common factor.
d=-(12⋅(2⋅-1))
Rewrite the expression.
d=1
d=1
Multiply -1 by -1.
d=1
d=1
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Simplify the numerator.
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Apply the product rule to -52.
e=154-(-1)2(52)24⋅(-1(54))
Raise -1 to the power of 2.
e=154-1(52)24⋅(-1(54))
Apply the product rule to 52.
e=154-1(5222)4⋅(-1(54))
Raise 5 to the power of 2.
e=154-1(2522)4⋅(-1(54))
Raise 2 to the power of 2.
e=154-1(254)4⋅(-1(54))
e=154-1(254)4⋅(-1(54))
Simplify the denominator.
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Multiply 4 by -1.
e=154-1(254)-4(54)
Combine -4 and 54.
e=154-1(254)-4⋅54
e=154-1(254)-4⋅54
Multiply -4 by 5.
e=154-1(254)-204
Multiply 254 by 1.
e=154-254-204
Divide -20 by 4.
e=154-254-5
Multiply the numerator by the reciprocal of the denominator.
e=154-(254⋅1-5)
Cancel the common factor of 5.
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Factor 5 out of 25.
e=154-(5(5)4⋅1-5)
Factor 5 out of -5.
e=154-(5⋅54⋅15⋅-1)
Cancel the common factor.
e=154-(5⋅54⋅15⋅-1)
Rewrite the expression.
e=154-(54⋅1-1)
e=154-(54⋅1-1)
Multiply 54 and 1-1.
e=154-54⋅-1
Multiply 4 by -1.
e=154-5-4
Move the negative in front of the fraction.
e=154+54
Multiply –54.
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Multiply -1 by -1.
e=154+1(54)
Multiply 54 by 1.
e=154+54
e=154+54
e=154+54
Combine the numerators over the common denominator.
e=15+54
Add 15 and 5.
e=204
Divide 20 by 4.
e=5
e=5
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-54⋅(x+1)2+5
-54⋅(x+1)2+5
Set y equal to the new right side.
y=-54⋅(x+1)2+5
y=-54⋅(x+1)2+5
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-54
h=-1
k=5
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(-1,5)
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⋅(-54)
Simplify.
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Cancel the common factor of 1 and -1.
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Rewrite 1 as -1(-1).
-1(-1)4⋅(-54)
Move the negative in front of the fraction.
-14(54)
-14(54)
Combine 4 and 54.
-14⋅54
Simplify the expression.
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Multiply 4 by 5.
-1204
Divide 20 by 4.
-15
-15
-15
-15
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.
(-1,245)
(-1,245)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-1
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=265
y=265
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (-1,5)
Focus: (-1,245)
Axis of Symmetry: x=-1
Directrix: y=265
Direction: Opens Down
Vertex: (-1,5)
Focus: (-1,245)
Axis of Symmetry: x=-1
Directrix: y=265
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 -3 in the expression.
f(-3)=-5(-3)24-5(-3)2+154
Simplify the result.
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Combine fractions with similar denominators.
f(-3)=-5(-3)2+154+-5⋅-32
Simplify the expression.
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Raise -3 to the power of 2.
f(-3)=-5⋅9+154+-5⋅-32
Multiply -5 by 9.
f(-3)=-45+154+-5⋅-32
Add -45 and 15.
f(-3)=-304+-5⋅-32
f(-3)=-304+-5⋅-32
Cancel the common factor of -30 and 4.
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Factor 2 out of -30.
f(-3)=2(-15)4+-5⋅-32
Cancel the common factors.
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Factor 2 out of 4.
f(-3)=2⋅-152⋅2+-5⋅-32
Cancel the common factor.
f(-3)=2⋅-152⋅2+-5⋅-32
Rewrite the expression.
f(-3)=-152+-5⋅-32
f(-3)=-152+-5⋅-32
f(-3)=-152+-5⋅-32
Simplify the expression.
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Move the negative in front of the fraction.
f(-3)=-152+-5⋅-32
Multiply -5 by -3.
f(-3)=-152+152
f(-3)=-152+152
Add -152 and 152.
f(-3)=0
The final answer is 0.
0
0
The y value at x=-3 is 0.
y=0
Replace the variable x with -2 in the expression.
f(-2)=-5(-2)24-5(-2)2+154
Simplify the result.
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Combine fractions with similar denominators.
f(-2)=-5(-2)2+154+-5⋅-22
Simplify the expression.
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Raise -2 to the power of 2.
f(-2)=-5⋅4+154+-5⋅-22
Multiply -5 by 4.
f(-2)=-20+154+-5⋅-22
Add -20 and 15.
f(-2)=-54+-5⋅-22
Move the negative in front of the fraction.
f(-2)=-54+-5⋅-22
Multiply -5 by -2.
f(-2)=-54+102
f(-2)=-54+102
Reduce the expression 102 by cancelling the common factors.
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Factor 2 out of 10.
f(-2)=-54+2⋅52
Factor 2 out of 2.
f(-2)=-54+2⋅52(1)
Cancel the common factor.
f(-2)=-54+2⋅52⋅1
Rewrite the expression.
f(-2)=-54+51
f(-2)=-54+51
Divide 5 by 1.
f(-2)=-54+5
To write 5 as a fraction with a common denominator, multiply by 44.
f(-2)=-54+5⋅44
Combine 5 and 44.
f(-2)=-54+5⋅44
Combine the numerators over the common denominator.
f(-2)=-5+5⋅44
Simplify the numerator.
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Multiply 5 by 4.
f(-2)=-5+204
Add -5 and 20.
f(-2)=154
f(-2)=154
The final answer is 154.
154
154
The y value at x=-2 is 154.
y=154
Replace the variable x with 1 in the expression.
f(1)=-5(1)24-5(1)2+154
Simplify the result.
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Combine fractions with similar denominators.
f(1)=-5(1)2+154+-5⋅12
Simplify the expression.
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One to any power is one.
f(1)=-5⋅1+154+-5⋅12
Multiply -5 by 1.
f(1)=-5+154+-5⋅12
Add -5 and 15.
f(1)=104+-5⋅12
f(1)=104+-5⋅12
Cancel the common factor of 10 and 4.
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Factor 2 out of 10.
f(1)=2(5)4+-5⋅12
Cancel the common factors.
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Factor 2 out of 4.
f(1)=2⋅52⋅2+-5⋅12
Cancel the common factor.
f(1)=2⋅52⋅2+-5⋅12
Rewrite the expression.
f(1)=52+-5⋅12
f(1)=52+-5⋅12
f(1)=52+-5⋅12
Simplify the expression.
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Multiply -5 by 1.
f(1)=52+-52
Move the negative in front of the fraction.
f(1)=52-52
Combine the numerators over the common denominator.
f(1)=5-52
f(1)=5-52
Subtract 5 from 5.
f(1)=02
Divide 0 by 2.
f(1)=0
The final answer is 0.
0
0
The y value at x=1 is 0.
y=0
Replace the variable x with 0 in the expression.
f(0)=-5(0)24-5(0)2+154
Simplify the result.
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Combine fractions with similar denominators.
f(0)=-5(0)2+154+-5⋅02
Simplify the expression.
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Raising 0 to any positive power yields 0.
f(0)=-5⋅0+154+-5⋅02
Multiply -5 by 0.
f(0)=0+154+-5⋅02
Add 0 and 15.
f(0)=154+-5⋅02
Multiply -5 by 0.
f(0)=154+02
f(0)=154+02
Reduce the expression 02 by cancelling the common factors.
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Factor 2 out of 0.
f(0)=154+2(0)2
Factor 2 out of 2.
f(0)=154+2⋅02⋅1
Cancel the common factor.
f(0)=154+2⋅02⋅1
Rewrite the expression.
f(0)=154+01
f(0)=154+01
Simplify the expression.
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Divide 0 by 1.
f(0)=154+0
Add 154 and 0.
f(0)=154
f(0)=154
The final answer is 154.
154
154
The y value at x=0 is 154.
y=154
Graph the parabola using its properties and the selected points.
xy-30-2154-15015410
xy-30-2154-15015410
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (-1,5)
Focus: (-1,245)
Axis of Symmetry: x=-1
Directrix: y=265
xy-30-2154-15015410
Graph y=-5/4x^2-5/2x+15/4

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