y25-x249=1
Add x249 to both sides of the equation.
y25=1+x249
Multiply both sides of the equation by 25.
25⋅y25=25⋅(1+x249)
Simplify both sides of the equation.
Cancel the common factor of 25.
Cancel the common factor.
25⋅y25=25⋅(1+x249)
Rewrite the expression.
y=25⋅(1+x249)
y=25⋅(1+x249)
Simplify 25⋅(1+x249).
Apply the distributive property.
y=25⋅1+25×249
Multiply 25 by 1.
y=25+25×249
Combine 25 and x249.
y=25+25×249
y=25+25×249
y=25+25×249
Reorder 25 and 25×249.
y=25×249+25
y=25×249+25
Rewrite the equation in vertex form.
Complete the square for 25×249+25.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=2549,b=0,c=25
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(2549)
Simplify the right side.
Cancel the common factor of 0 and 2.
Factor 2 out of 0.
d=2(0)2(2549)
Cancel the common factors.
Cancel the common factor.
d=2⋅02(2549)
Rewrite the expression.
d=02549
d=02549
d=02549
Multiply the numerator by the reciprocal of the denominator.
d=0(4925)
Multiply 0 by 4925.
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=25-04(2549)
Combine 4 and 2549.
e=25-04⋅2549
Multiply 4 by 25.
e=25-010049
Multiply the numerator by the reciprocal of the denominator.
e=25-(0(49100))
Multiply 0 by 49100.
e=25-0
Multiply -1 by 0.
e=25+0
e=25+0
Add 25 and 0.
e=25
e=25
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
2549⋅(x+0)2+25
2549⋅(x+0)2+25
Set y equal to the new right side.
y=2549⋅(x+0)2+25
y=2549⋅(x+0)2+25
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=2549
h=0
k=25
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(0,25)
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⋅2549
Simplify.
Combine 4 and 2549.
14⋅2549
Multiply 4 by 25.
110049
Multiply the numerator by the reciprocal of the denominator.
1(49100)
Multiply 49100 by 1.
49100
49100
49100
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.
(0,2549100)
(0,2549100)
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=2451100
y=2451100
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (0,25)
Focus: (0,2549100)
Axis of Symmetry: x=0
Directrix: y=2451100
Direction: Opens Up
Vertex: (0,25)
Focus: (0,2549100)
Axis of Symmetry: x=0
Directrix: y=2451100
Replace the variable x with -1 in the expression.
f(-1)=25(-1)249+25
Simplify the result.
Simplify each term.
Raise -1 to the power of 2.
f(-1)=25⋅149+25
Multiply 25 by 1.
f(-1)=2549+25
f(-1)=2549+25
To write 25 as a fraction with a common denominator, multiply by 4949.
f(-1)=2549+25⋅4949
Combine 25 and 4949.
f(-1)=2549+25⋅4949
Combine the numerators over the common denominator.
f(-1)=25+25⋅4949
Simplify the numerator.
Multiply 25 by 49.
f(-1)=25+122549
Add 25 and 1225.
f(-1)=125049
f(-1)=125049
The final answer is 125049.
125049
125049
The y value at x=-1 is 125049.
y=125049
Replace the variable x with -2 in the expression.
f(-2)=25(-2)249+25
Simplify the result.
Simplify each term.
Raise -2 to the power of 2.
f(-2)=25⋅449+25
Multiply 25 by 4.
f(-2)=10049+25
f(-2)=10049+25
To write 25 as a fraction with a common denominator, multiply by 4949.
f(-2)=10049+25⋅4949
Combine 25 and 4949.
f(-2)=10049+25⋅4949
Combine the numerators over the common denominator.
f(-2)=100+25⋅4949
Simplify the numerator.
Multiply 25 by 49.
f(-2)=100+122549
Add 100 and 1225.
f(-2)=132549
f(-2)=132549
The final answer is 132549.
132549
132549
The y value at x=-2 is 132549.
y=132549
Replace the variable x with 1 in the expression.
f(1)=25(1)249+25
Simplify the result.
Simplify each term.
One to any power is one.
f(1)=25⋅149+25
Multiply 25 by 1.
f(1)=2549+25
f(1)=2549+25
To write 25 as a fraction with a common denominator, multiply by 4949.
f(1)=2549+25⋅4949
Combine 25 and 4949.
f(1)=2549+25⋅4949
Combine the numerators over the common denominator.
f(1)=25+25⋅4949
Simplify the numerator.
Multiply 25 by 49.
f(1)=25+122549
Add 25 and 1225.
f(1)=125049
f(1)=125049
The final answer is 125049.
125049
125049
The y value at x=1 is 125049.
y=125049
Replace the variable x with 2 in the expression.
f(2)=25(2)249+25
Simplify the result.
Simplify each term.
Raise 2 to the power of 2.
f(2)=25⋅449+25
Multiply 25 by 4.
f(2)=10049+25
f(2)=10049+25
To write 25 as a fraction with a common denominator, multiply by 4949.
f(2)=10049+25⋅4949
Combine 25 and 4949.
f(2)=10049+25⋅4949
Combine the numerators over the common denominator.
f(2)=100+25⋅4949
Simplify the numerator.
Multiply 25 by 49.
f(2)=100+122549
Add 100 and 1225.
f(2)=132549
f(2)=132549
The final answer is 132549.
132549
132549
The y value at x=2 is 132549.
y=132549
Graph the parabola using its properties and the selected points.
xy-2132549-112504902511250492132549
xy-2132549-112504902511250492132549
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (0,25)
Focus: (0,2549100)
Axis of Symmetry: x=0
Directrix: y=2451100
xy-2132549-112504902511250492132549
Graph y/25-(x^2)/49=1
