t2+23t+19

Combine 23 and x.

y=x2+2×3+19

Rewrite the equation in vertex form.

Complete the square for x2+2×3+19.

Use the form ax2+bx+c, to find the values of a, b, and c.

a=1,b=23,c=19

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=232(1)

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

d=23⋅12⋅1

Cancel the common factor of 2.

Factor 2 out of 2⋅1.

d=23⋅12(1)

Cancel the common factor.

d=23⋅12⋅1

Rewrite the expression.

d=13

d=13

d=13

Find the value of e using the formula e=c-b24a.

Simplify each term.

Simplify the numerator.

Apply the product rule to 23.

e=19-22324⋅1

Raise 2 to the power of 2.

e=19-4324⋅1

Raise 3 to the power of 2.

e=19-494⋅1

e=19-494⋅1

Multiply 4 by 1.

e=19-494

Multiply the numerator by the reciprocal of the denominator.

e=19-(49⋅14)

Cancel the common factor of 4.

Cancel the common factor.

e=19-(49⋅14)

Rewrite the expression.

e=19-19

e=19-19

e=19-19

Combine the numerators over the common denominator.

e=1-19

Subtract 1 from 1.

e=09

Divide 0 by 9.

e=0

e=0

Substitute the values of a, d, and e into the vertex form a(x+d)2+e.

(x+13)2+0

(x+13)2+0

Set y equal to the new right side.

y=(x+13)2+0

y=(x+13)2+0

Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.

a=1

h=-13

k=0

Since the value of a is positive, the parabola opens up.

Opens Up

Find the vertex (h,k).

(-13,0)

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⋅1

Cancel the common factor of 1.

Cancel the common factor.

14⋅1

Rewrite the expression.

14

14

14

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.

(-13,14)

(-13,14)

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

x=-13

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=-14

y=-14

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: (-13,0)

Focus: (-13,14)

Axis of Symmetry: x=-13

Directrix: y=-14

Direction: Opens Up

Vertex: (-13,0)

Focus: (-13,14)

Axis of Symmetry: x=-13

Directrix: y=-14

Replace the variable x with -1 in the expression.

f(-1)=(-1)2+2(-1)3+19

Simplify the result.

Multiply 2 by -1.

f(-1)=(-1)2+-23+19

Simplify each term.

Raise -1 to the power of 2.

f(-1)=1+-23+19

Move the negative in front of the fraction.

f(-1)=1-23+19

f(-1)=1-23+19

Simplify the expression.

Write 1 as a fraction with a common denominator.

f(-1)=33-23+19

Combine the numerators over the common denominator.

f(-1)=3-23+19

f(-1)=3-23+19

Subtract 2 from 3.

f(-1)=13+19

To write 13 as a fraction with a common denominator, multiply by 33.

f(-1)=13⋅33+19

Write each expression with a common denominator of 9, by multiplying each by an appropriate factor of 1.

Multiply 13 and 33.

f(-1)=33⋅3+19

Multiply 3 by 3.

f(-1)=39+19

f(-1)=39+19

Combine the numerators over the common denominator.

f(-1)=3+19

Add 3 and 1.

f(-1)=49

The final answer is 49.

49

49

The y value at x=-1 is 49.

y=49

Replace the variable x with -2 in the expression.

f(-2)=(-2)2+2(-2)3+19

Simplify the result.

Multiply 2 by -2.

f(-2)=(-2)2+-43+19

Simplify each term.

Raise -2 to the power of 2.

f(-2)=4+-43+19

Move the negative in front of the fraction.

f(-2)=4-43+19

f(-2)=4-43+19

To write 4 as a fraction with a common denominator, multiply by 33.

f(-2)=4⋅33-43+19

Combine 4 and 33.

f(-2)=4⋅33-43+19

Combine the numerators over the common denominator.

f(-2)=4⋅3-43+19

Simplify the numerator.

Multiply 4 by 3.

f(-2)=12-43+19

Subtract 4 from 12.

f(-2)=83+19

f(-2)=83+19

To write 83 as a fraction with a common denominator, multiply by 33.

f(-2)=83⋅33+19

Write each expression with a common denominator of 9, by multiplying each by an appropriate factor of 1.

Multiply 83 and 33.

f(-2)=8⋅33⋅3+19

Multiply 3 by 3.

f(-2)=8⋅39+19

f(-2)=8⋅39+19

Combine the numerators over the common denominator.

f(-2)=8⋅3+19

Simplify the numerator.

Multiply 8 by 3.

f(-2)=24+19

Add 24 and 1.

f(-2)=259

f(-2)=259

The final answer is 259.

259

259

The y value at x=-2 is 259.

y=259

Replace the variable x with 1 in the expression.

f(1)=(1)2+2(1)3+19

Simplify the result.

Multiply 2 by 1.

f(1)=(1)2+23+19

One to any power is one.

f(1)=1+23+19

Find the common denominator.

Write 1 as a fraction with denominator 1.

f(1)=11+23+19

Multiply 11 by 99.

f(1)=11⋅99+23+19

Multiply 11 and 99.

f(1)=99+23+19

Multiply 23 by 33.

f(1)=99+23⋅33+19

Combine.

f(1)=99+2⋅33⋅3+19

Multiply 3 by 3.

f(1)=99+2⋅39+19

f(1)=99+2⋅39+19

Combine fractions.

Combine fractions with similar denominators.

f(1)=9+2⋅3+19

Multiply 2 by 3.

f(1)=9+6+19

f(1)=9+6+19

Simplify the numerator.

Add 9 and 6.

f(1)=15+19

Add 15 and 1.

f(1)=169

f(1)=169

The final answer is 169.

169

169

The y value at x=1 is 169.

y=169

Replace the variable x with 2 in the expression.

f(2)=(2)2+2(2)3+19

Simplify the result.

Multiply 2 by 2.

f(2)=(2)2+43+19

Raise 2 to the power of 2.

f(2)=4+43+19

Find the common denominator.

Write 4 as a fraction with denominator 1.

f(2)=41+43+19

Multiply 41 by 99.

f(2)=41⋅99+43+19

Multiply 41 and 99.

f(2)=4⋅99+43+19

Multiply 43 by 33.

f(2)=4⋅99+43⋅33+19

Combine.

f(2)=4⋅99+4⋅33⋅3+19

Multiply 3 by 3.

f(2)=4⋅99+4⋅39+19

f(2)=4⋅99+4⋅39+19

Combine fractions.

Combine fractions with similar denominators.

f(2)=4⋅9+4⋅3+19

Multiply.

Multiply 4 by 9.

f(2)=36+4⋅3+19

Multiply 4 by 3.

f(2)=36+12+19

f(2)=36+12+19

f(2)=36+12+19

Simplify the numerator.

Add 36 and 12.

f(2)=48+19

Add 48 and 1.

f(2)=499

f(2)=499

The final answer is 499.

499

499

The y value at x=2 is 499.

y=499

Graph the parabola using its properties and the selected points.

xy-2259-149-13011692499

xy-2259-149-13011692499

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-13,0)

Focus: (-13,14)

Axis of Symmetry: x=-13

Directrix: y=-14

xy-2259-149-13011692499

Graph t^2+2/3t+1/9