-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
