-0.5×2+0.5x+15

Rewrite the equation in vertex form.

Complete the square for -0.5×2+0.5x+15.

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

a=-0.5,b=0.5,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=0.52(-0.5)

Simplify the right side.

Cancel the common factor of 0.5 and -0.5.

Rewrite 0.5 as -1(-0.5).

d=-1⋅-0.52(-0.5)

Cancel the common factor.

d=-1⋅-0.52⋅-0.5

Rewrite the expression.

d=-12

d=-12

Move the negative in front of the fraction.

d=-12

d=-12

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

Simplify each term.

Cancel the common factor of (0.5)2 and -0.5.

Rewrite 0.5 as -1(-0.5).

e=15-(-1⋅-0.5)24(-0.5)

Apply the product rule to -1(-0.5).

e=15-(-1)2(-0.5)24(-0.5)

Raise -1 to the power of 2.

e=15-1(-0.5)24(-0.5)

Multiply (-0.5)2 by 1.

e=15-(-0.5)24(-0.5)

Factor -0.5 out of (-0.5)2.

e=15–0.5⋅-0.54⋅-0.5

Cancel the common factors.

Factor -0.5 out of 4⋅-0.5.

e=15–0.5⋅-0.5-0.5(-8⋅-0.5)

Cancel the common factor.

e=15–0.5⋅-0.5-0.5(-8⋅-0.5)

Rewrite the expression.

e=15–0.5-8⋅-0.5

e=15–0.5-8⋅-0.5

e=15–0.5-8⋅-0.5

Cancel the common factor of -0.5.

Cancel the common factor.

e=15–0.5-8⋅-0.5

Rewrite the expression.

e=15-1-8

e=15-1-8

Move the negative in front of the fraction.

e=15+18

Multiply –18.

Multiply -1 by -1.

e=15+1(18)

Multiply 18 by 1.

e=15+18

e=15+18

e=15+18

To write 15 as a fraction with a common denominator, multiply by 88.

e=15⋅88+18

Combine 15 and 88.

e=15⋅88+18

Combine the numerators over the common denominator.

e=15⋅8+18

Simplify the numerator.

Multiply 15 by 8.

e=120+18

Add 120 and 1.

e=1218

e=1218

e=1218

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

-0.5(x-12)2+1218

-0.5(x-12)2+1218

Set y equal to the new right side.

y=-0.5(x-12)2+1218

y=-0.5(x-12)2+1218

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

a=-0.5

h=12

k=1218

Since the value of a is negative, the parabola opens down.

Opens Down

Find the vertex (h,k).

(12,1218)

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⋅-0.5

Simplify.

Multiply 4 by -0.5.

1-2

Move the negative in front of the fraction.

-12

Divide 1 by 2.

-1⋅0.5

Multiply -1 by 0.5.

-0.5

-0.5

-0.5

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.

(12,1178)

(12,1178)

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

x=12

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=1258

y=1258

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

Direction: Opens Down

Vertex: (12,1218)

Focus: (12,1178)

Axis of Symmetry: x=12

Directrix: y=1258

Direction: Opens Down

Vertex: (12,1218)

Focus: (12,1178)

Axis of Symmetry: x=12

Directrix: y=1258

Replace the variable x with -1 in the expression.

f(-1)=-0.5(-1)2+0.5(-1)+15

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=-0.5⋅1+0.5(-1)+15

Multiply -0.5 by 1.

f(-1)=-0.5+0.5(-1)+15

Multiply 0.5 by -1.

f(-1)=-0.5-0.5+15

f(-1)=-0.5-0.5+15

Simplify by adding and subtracting.

Subtract 0.5 from -0.5.

f(-1)=-1+15

Add -1 and 15.

f(-1)=14

f(-1)=14

The final answer is 14.

14

14

The y value at x=-1 is 14.

y=14

Replace the variable x with -2 in the expression.

f(-2)=-0.5(-2)2+0.5(-2)+15

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=-0.5⋅4+0.5(-2)+15

Multiply -0.5 by 4.

f(-2)=-2+0.5(-2)+15

Multiply 0.5 by -2.

f(-2)=-2-1+15

f(-2)=-2-1+15

Simplify by adding and subtracting.

Subtract 1 from -2.

f(-2)=-3+15

Add -3 and 15.

f(-2)=12

f(-2)=12

The final answer is 12.

12

12

The y value at x=-2 is 12.

y=12

Replace the variable x with 1 in the expression.

f(1)=-0.5(1)2+0.5(1)+15

Simplify the result.

Simplify each term.

One to any power is one.

f(1)=-0.5⋅1+0.5(1)+15

Multiply -0.5 by 1.

f(1)=-0.5+0.5(1)+15

Multiply 0.5 by 1.

f(1)=-0.5+0.5+15

f(1)=-0.5+0.5+15

Simplify by adding numbers.

Add -0.5 and 0.5.

f(1)=0+15

Add 0 and 15.

f(1)=15

f(1)=15

The final answer is 15.

15

15

The y value at x=1 is 15.

y=15

Replace the variable x with 2 in the expression.

f(2)=-0.5(2)2+0.5(2)+15

Simplify the result.

Simplify each term.

Raise 2 to the power of 2.

f(2)=-0.5⋅4+0.5(2)+15

Multiply -0.5 by 4.

f(2)=-2+0.5(2)+15

Multiply 0.5 by 2.

f(2)=-2+1+15

f(2)=-2+1+15

Simplify by adding numbers.

Add -2 and 1.

f(2)=-1+15

Add -1 and 15.

f(2)=14

f(2)=14

The final answer is 14.

14

14

The y value at x=2 is 14.

y=14

Graph the parabola using its properties and the selected points.

xy-212-114121218115214

xy-212-114121218115214

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (12,1218)

Focus: (12,1178)

Axis of Symmetry: x=12

Directrix: y=1258

xy-212-114121218115214

Graph -0.5x^2+0.5x+15