Enter a problem...
Trigonometry Examples
Step 1
Step 1.1
Move all terms not containing to the right side of the equation.
Step 1.1.1
Add to both sides of the equation.
Step 1.1.2
Subtract from both sides of the equation.
Step 1.2
Divide each term in by and simplify.
Step 1.2.1
Divide each term in by .
Step 1.2.2
Simplify the left side.
Step 1.2.2.1
Cancel the common factor of .
Step 1.2.2.1.1
Cancel the common factor.
Step 1.2.2.1.2
Divide by .
Step 1.2.3
Simplify the right side.
Step 1.2.3.1
Move the negative in front of the fraction.
Step 2
Step 2.1
Rewrite the equation in vertex form.
Step 2.1.1
Complete the square for .
Step 2.1.1.1
Use the form , to find the values of , , and .
Step 2.1.1.2
Consider the vertex form of a parabola.
Step 2.1.1.3
Find the value of using the formula .
Step 2.1.1.3.1
Substitute the values of and into the formula .
Step 2.1.1.3.2
Simplify the right side.
Step 2.1.1.3.2.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.3.2.2
Combine and .
Step 2.1.1.3.2.3
Multiply by .
Step 2.1.1.3.2.4
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.3.2.5
Multiply by .
Step 2.1.1.3.2.6
Cancel the common factor of .
Step 2.1.1.3.2.6.1
Move the leading negative in into the numerator.
Step 2.1.1.3.2.6.2
Cancel the common factor.
Step 2.1.1.3.2.6.3
Rewrite the expression.
Step 2.1.1.3.2.7
Combine and .
Step 2.1.1.3.2.8
Move the negative in front of the fraction.
Step 2.1.1.4
Find the value of using the formula .
Step 2.1.1.4.1
Substitute the values of , and into the formula .
Step 2.1.1.4.2
Simplify the right side.
Step 2.1.1.4.2.1
Simplify each term.
Step 2.1.1.4.2.1.1
Simplify the numerator.
Step 2.1.1.4.2.1.1.1
Apply the product rule to .
Step 2.1.1.4.2.1.1.2
Raise to the power of .
Step 2.1.1.4.2.1.1.3
Apply the product rule to .
Step 2.1.1.4.2.1.1.4
Raise to the power of .
Step 2.1.1.4.2.1.1.5
Raise to the power of .
Step 2.1.1.4.2.1.1.6
Multiply by .
Step 2.1.1.4.2.1.2
Combine and .
Step 2.1.1.4.2.1.3
Multiply by .
Step 2.1.1.4.2.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.1.4.2.1.5
Cancel the common factor of .
Step 2.1.1.4.2.1.5.1
Factor out of .
Step 2.1.1.4.2.1.5.2
Cancel the common factor.
Step 2.1.1.4.2.1.5.3
Rewrite the expression.
Step 2.1.1.4.2.1.6
Multiply by .
Step 2.1.1.4.2.1.7
Multiply by .
Step 2.1.1.4.2.2
Subtract from .
Step 2.1.1.5
Substitute the values of , , and into the vertex form .
Step 2.1.2
Set equal to the new right side.
Step 2.2
Use the vertex form, , to determine the values of , , and .
Step 2.3
Since the value of is positive, the parabola opens up.
Opens Up
Step 2.4
Find the vertex .
Step 2.5
Find , the distance from the vertex to the focus.
Step 2.5.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 2.5.2
Substitute the value of into the formula.
Step 2.5.3
Simplify.
Step 2.5.3.1
Combine and .
Step 2.5.3.2
Multiply by .
Step 2.5.3.3
Multiply the numerator by the reciprocal of the denominator.
Step 2.5.3.4
Multiply by .
Step 2.6
Find the focus.
Step 2.6.1
The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Step 2.6.2
Substitute the known values of , , and into the formula and simplify.
Step 2.7
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Step 2.8
Find the directrix.
Step 2.8.1
The directrix of a parabola is the horizontal line found by subtracting from the y-coordinate of the vertex if the parabola opens up or down.
Step 2.8.2
Substitute the known values of and into the formula and simplify.
Step 2.9
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 3
Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
Step 3.2.1
Combine the numerators over the common denominator.
Step 3.2.2
Simplify each term.
Step 3.2.2.1
Raising to any positive power yields .
Step 3.2.2.2
Multiply by .
Step 3.2.2.3
Multiply by .
Step 3.2.3
Simplify the expression.
Step 3.2.3.1
Add and .
Step 3.2.3.2
Divide by .
Step 3.2.4
The final answer is .
Step 3.3
The value at is .
Step 3.4
Replace the variable with in the expression.
Step 3.5
Simplify the result.
Step 3.5.1
Combine the numerators over the common denominator.
Step 3.5.2
Simplify each term.
Step 3.5.2.1
Raise to the power of .
Step 3.5.2.2
Multiply by .
Step 3.5.2.3
Multiply by .
Step 3.5.3
Add and .
Step 3.5.4
The final answer is .
Step 3.6
The value at is .
Step 3.7
Replace the variable with in the expression.
Step 3.8
Simplify the result.
Step 3.8.1
Combine the numerators over the common denominator.
Step 3.8.2
Simplify each term.
Step 3.8.2.1
Multiply by by adding the exponents.
Step 3.8.2.1.1
Multiply by .
Step 3.8.2.1.1.1
Raise to the power of .
Step 3.8.2.1.1.2
Use the power rule to combine exponents.
Step 3.8.2.1.2
Add and .
Step 3.8.2.2
Raise to the power of .
Step 3.8.2.3
Multiply by .
Step 3.8.3
Subtract from .
Step 3.8.4
The final answer is .
Step 3.9
The value at is .
Step 3.10
Replace the variable with in the expression.
Step 3.11
Simplify the result.
Step 3.11.1
Combine the numerators over the common denominator.
Step 3.11.2
Simplify each term.
Step 3.11.2.1
Raise to the power of .
Step 3.11.2.2
Multiply by .
Step 3.11.2.3
Multiply by .
Step 3.11.3
Subtract from .
Step 3.11.4
The final answer is .
Step 3.12
The value at is .
Step 3.13
Graph the parabola using its properties and the selected points.
Step 4
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 5