y=-54×2-52x+154

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

Rewrite the equation in vertex form.

Complete the square for -5×24-5×2+154.

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.

Multiply the numerator by the reciprocal of the denominator.

d=-(52⋅12⋅(-1(54)))

Cancel the common factor of 1 and -1.

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.

Factor 2 out of 10.

d=-(52⋅(-12(5)4))

Cancel the common factors.

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.

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.

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.

Simplify each term.

Simplify the numerator.

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.

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.

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.

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.

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.

Cancel the common factor of 1 and -1.

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.

Multiply 4 by 5.

-1204

Divide 20 by 4.

-15

-15

-15

-15

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.

(-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.

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

Replace the variable x with -3 in the expression.

f(-3)=-5(-3)24-5(-3)2+154

Simplify the result.

Combine fractions with similar denominators.

f(-3)=-5(-3)2+154+-5⋅-32

Simplify the expression.

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.

Factor 2 out of -30.

f(-3)=2(-15)4+-5⋅-32

Cancel the common factors.

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.

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.

Combine fractions with similar denominators.

f(-2)=-5(-2)2+154+-5⋅-22

Simplify the expression.

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.

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.

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.

Combine fractions with similar denominators.

f(1)=-5(1)2+154+-5⋅12

Simplify the expression.

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.

Factor 2 out of 10.

f(1)=2(5)4+-5⋅12

Cancel the common factors.

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.

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.

Combine fractions with similar denominators.

f(0)=-5(0)2+154+-5⋅02

Simplify the expression.

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.

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.

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