y=3×2+48
Rewrite the equation in vertex form.
Complete the square for 3×2+48.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=3,b=0,c=48
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=02(3)
Simplify the right side.
Cancel the common factor of 0 and 2.
Factor 2 out of 0.
d=2(0)2(3)
Cancel the common factors.
Cancel the common factor.
d=2⋅02⋅3
Rewrite the expression.
d=03
d=03
d=03
Cancel the common factor of 0 and 3.
Factor 3 out of 0.
d=3(0)3
Cancel the common factors.
Factor 3 out of 3.
d=3⋅03⋅1
Cancel the common factor.
d=3⋅03⋅1
Rewrite the expression.
d=01
Divide 0 by 1.
d=0
d=0
d=0
d=0
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raising 0 to any positive power yields 0.
e=48-04⋅3
Multiply 4 by 3.
e=48-012
Divide 0 by 12.
e=48-0
Multiply -1 by 0.
e=48+0
e=48+0
Add 48 and 0.
e=48
e=48
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
3(x+0)2+48
3(x+0)2+48
Set y equal to the new right side.
y=3(x+0)2+48
y=3(x+0)2+48
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=3
h=0
k=48
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(0,48)
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⋅3
Multiply 4 by 3.
112
112
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.
(0,57712)
(0,57712)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=0
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=57512
y=57512
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (0,48)
Focus: (0,57712)
Axis of Symmetry: x=0
Directrix: y=57512
Direction: Opens Up
Vertex: (0,48)
Focus: (0,57712)
Axis of Symmetry: x=0
Directrix: y=57512
Replace the variable x with -1 in the expression.
f(-1)=3(-1)2+48
Simplify the result.
Simplify each term.
Raise -1 to the power of 2.
f(-1)=3⋅1+48
Multiply 3 by 1.
f(-1)=3+48
f(-1)=3+48
Add 3 and 48.
f(-1)=51
The final answer is 51.
51
51
The y value at x=-1 is 51.
y=51
Replace the variable x with -2 in the expression.
f(-2)=3(-2)2+48
Simplify the result.
Simplify each term.
Raise -2 to the power of 2.
f(-2)=3⋅4+48
Multiply 3 by 4.
f(-2)=12+48
f(-2)=12+48
Add 12 and 48.
f(-2)=60
The final answer is 60.
60
60
The y value at x=-2 is 60.
y=60
Replace the variable x with 1 in the expression.
f(1)=3(1)2+48
Simplify the result.
Simplify each term.
One to any power is one.
f(1)=3⋅1+48
Multiply 3 by 1.
f(1)=3+48
f(1)=3+48
Add 3 and 48.
f(1)=51
The final answer is 51.
51
51
The y value at x=1 is 51.
y=51
Replace the variable x with 2 in the expression.
f(2)=3(2)2+48
Simplify the result.
Simplify each term.
Raise 2 to the power of 2.
f(2)=3⋅4+48
Multiply 3 by 4.
f(2)=12+48
f(2)=12+48
Add 12 and 48.
f(2)=60
The final answer is 60.
60
60
The y value at x=2 is 60.
y=60
Graph the parabola using its properties and the selected points.
xy-260-151048151260
xy-260-151048151260
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (0,48)
Focus: (0,57712)
Axis of Symmetry: x=0
Directrix: y=57512
xy-260-151048151260
Graph y=3x^2+48
