Graph y=1/3x^2+4

y=13×2+4
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Combine 13 and x2.
y=x23+4
Complete the square for x23+4.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=13,b=0,c=4
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(13)
Simplify the right side.
Cancel the common factor of 0 and 2.
Factor 2 out of 0.
d=2(0)2(13)
Cancel the common factors.
Cancel the common factor.
d=2⋅02(13)
Rewrite the expression.
d=013
d=013
d=013
Multiply the numerator by the reciprocal of the denominator.
d=0⋅3
Multiply 0 by 3.
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=4-04(13)
Combine 4 and 13.
e=4-043
Multiply the numerator by the reciprocal of the denominator.
e=4-(0(34))
Multiply 0 by 34.
e=4-0
Multiply -1 by 0.
e=4+0
e=4+0
e=4
e=4
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
13⋅(x+0)2+4
13⋅(x+0)2+4
Set y equal to the new right side.
y=13⋅(x+0)2+4
y=13⋅(x+0)2+4
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=13
h=0
k=4
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(0,4)
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⋅13
Simplify.
Combine 4 and 13.
143
Multiply the numerator by the reciprocal of the denominator.
1(34)
Multiply 34 by 1.
34
34
34
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.
(0,194)
(0,194)
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=134
y=134
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (0,4)
Focus: (0,194)
Axis of Symmetry: x=0
Directrix: y=134
Direction: Opens Up
Vertex: (0,4)
Focus: (0,194)
Axis of Symmetry: x=0
Directrix: y=134
Select a few x values, and plug them into the equation to find the corresponding y values. The x values should be selected around the vertex.
Replace the variable x with -1 in the expression.
f(-1)=(-1)23+4
Simplify the result.
Raise -1 to the power of 2.
f(-1)=13+4
To write 4 as a fraction with a common denominator, multiply by 33.
f(-1)=13+4⋅33
Combine 4 and 33.
f(-1)=13+4⋅33
Combine the numerators over the common denominator.
f(-1)=1+4⋅33
Simplify the numerator.
Multiply 4 by 3.
f(-1)=1+123
f(-1)=133
f(-1)=133
133
133
The y value at x=-1 is 133.
y=133
Replace the variable x with -2 in the expression.
f(-2)=(-2)23+4
Simplify the result.
Raise -2 to the power of 2.
f(-2)=43+4
To write 4 as a fraction with a common denominator, multiply by 33.
f(-2)=43+4⋅33
Combine 4 and 33.
f(-2)=43+4⋅33
Combine the numerators over the common denominator.
f(-2)=4+4⋅33
Simplify the numerator.
Multiply 4 by 3.
f(-2)=4+123
f(-2)=163
f(-2)=163
163
163
The y value at x=-2 is 163.
y=163
Replace the variable x with 3 in the expression.
f(3)=(3)23+4
Simplify the result.
Cancel the common factor of (3)2 and 3.
Factor 3 out of (3)2.
f(3)=3⋅33+4
Cancel the common factors.
Factor 3 out of 3.
f(3)=3⋅33(1)+4
Cancel the common factor.
f(3)=3⋅33⋅1+4
Rewrite the expression.
f(3)=31+4
Divide 3 by 1.
f(3)=3+4
f(3)=3+4
f(3)=3+4
f(3)=7
7
7
The y value at x=3 is 7.
y=7
Replace the variable x with 1 in the expression.
f(1)=(1)23+4
Simplify the result.
One to any power is one.
f(1)=13+4
To write 4 as a fraction with a common denominator, multiply by 33.
f(1)=13+4⋅33
Combine 4 and 33.
f(1)=13+4⋅33
Combine the numerators over the common denominator.
f(1)=1+4⋅33
Simplify the numerator.
Multiply 4 by 3.
f(1)=1+123
f(1)=133
f(1)=133
133
133
The y value at x=1 is 133.
y=133
Graph the parabola using its properties and the selected points.
xy-2163-113304113337
xy-2163-113304113337
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (0,4)
Focus: (0,194)
Axis of Symmetry: x=0
Directrix: y=134
xy-2163-113304113337
Graph y=1/3x^2+4