Trigonometry Examples

Graph 4x^2-24x+4x^2-20y=3
Step 1
Solve for .
Tap for more steps...
Step 1.1
Add and .
Step 1.2
Move all terms not containing to the right side of the equation.
Tap for more steps...
Step 1.2.1
Subtract from both sides of the equation.
Step 1.2.2
Add to both sides of the equation.
Step 1.3
Divide each term in by and simplify.
Tap for more steps...
Step 1.3.1
Divide each term in by .
Step 1.3.2
Simplify the left side.
Tap for more steps...
Step 1.3.2.1
Cancel the common factor of .
Tap for more steps...
Step 1.3.2.1.1
Cancel the common factor.
Step 1.3.2.1.2
Divide by .
Step 1.3.3
Simplify the right side.
Tap for more steps...
Step 1.3.3.1
Simplify each term.
Tap for more steps...
Step 1.3.3.1.1
Move the negative in front of the fraction.
Step 1.3.3.1.2
Cancel the common factor of and .
Tap for more steps...
Step 1.3.3.1.2.1
Factor out of .
Step 1.3.3.1.2.2
Cancel the common factors.
Tap for more steps...
Step 1.3.3.1.2.2.1
Factor out of .
Step 1.3.3.1.2.2.2
Cancel the common factor.
Step 1.3.3.1.2.2.3
Rewrite the expression.
Step 1.3.3.1.3
Cancel the common factor of and .
Tap for more steps...
Step 1.3.3.1.3.1
Factor out of .
Step 1.3.3.1.3.2
Cancel the common factors.
Tap for more steps...
Step 1.3.3.1.3.2.1
Factor out of .
Step 1.3.3.1.3.2.2
Cancel the common factor.
Step 1.3.3.1.3.2.3
Rewrite the expression.
Step 1.3.3.1.4
Move the negative in front of the fraction.
Step 2
Find the properties of the given parabola.
Tap for more steps...
Step 2.1
Rewrite the equation in vertex form.
Tap for more steps...
Step 2.1.1
Move .
Step 2.1.2
Complete the square for .
Tap for more steps...
Step 2.1.2.1
Use the form , to find the values of , , and .
Step 2.1.2.2
Consider the vertex form of a parabola.
Step 2.1.2.3
Find the value of using the formula .
Tap for more steps...
Step 2.1.2.3.1
Substitute the values of and into the formula .
Step 2.1.2.3.2
Simplify the right side.
Tap for more steps...
Step 2.1.2.3.2.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.2.3.2.2
Cancel the common factor of .
Tap for more steps...
Step 2.1.2.3.2.2.1
Move the leading negative in into the numerator.
Step 2.1.2.3.2.2.2
Factor out of .
Step 2.1.2.3.2.2.3
Cancel the common factor.
Step 2.1.2.3.2.2.4
Rewrite the expression.
Step 2.1.2.3.2.3
Multiply by .
Step 2.1.2.3.2.4
Combine and .
Step 2.1.2.3.2.5
Multiply by .
Step 2.1.2.3.2.6
Cancel the common factor of and .
Tap for more steps...
Step 2.1.2.3.2.6.1
Factor out of .
Step 2.1.2.3.2.6.2
Cancel the common factors.
Tap for more steps...
Step 2.1.2.3.2.6.2.1
Factor out of .
Step 2.1.2.3.2.6.2.2
Cancel the common factor.
Step 2.1.2.3.2.6.2.3
Rewrite the expression.
Step 2.1.2.3.2.6.2.4
Divide by .
Step 2.1.2.3.2.7
Move the negative in front of the fraction.
Step 2.1.2.4
Find the value of using the formula .
Tap for more steps...
Step 2.1.2.4.1
Substitute the values of , and into the formula .
Step 2.1.2.4.2
Simplify the right side.
Tap for more steps...
Step 2.1.2.4.2.1
Simplify each term.
Tap for more steps...
Step 2.1.2.4.2.1.1
Simplify the numerator.
Tap for more steps...
Step 2.1.2.4.2.1.1.1
Apply the product rule to .
Step 2.1.2.4.2.1.1.2
Raise to the power of .
Step 2.1.2.4.2.1.1.3
Apply the product rule to .
Step 2.1.2.4.2.1.1.4
Raise to the power of .
Step 2.1.2.4.2.1.1.5
Raise to the power of .
Step 2.1.2.4.2.1.1.6
Multiply by .
Step 2.1.2.4.2.1.2
Combine and .
Step 2.1.2.4.2.1.3
Multiply by .
Step 2.1.2.4.2.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.2.4.2.1.5
Cancel the common factor of .
Tap for more steps...
Step 2.1.2.4.2.1.5.1
Factor out of .
Step 2.1.2.4.2.1.5.2
Factor out of .
Step 2.1.2.4.2.1.5.3
Cancel the common factor.
Step 2.1.2.4.2.1.5.4
Rewrite the expression.
Step 2.1.2.4.2.1.6
Cancel the common factor of .
Tap for more steps...
Step 2.1.2.4.2.1.6.1
Factor out of .
Step 2.1.2.4.2.1.6.2
Cancel the common factor.
Step 2.1.2.4.2.1.6.3
Rewrite the expression.
Step 2.1.2.4.2.1.7
Multiply by .
Step 2.1.2.4.2.1.8
Multiply by .
Step 2.1.2.4.2.2
To write as a fraction with a common denominator, multiply by .
Step 2.1.2.4.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps...
Step 2.1.2.4.2.3.1
Multiply by .
Step 2.1.2.4.2.3.2
Multiply by .
Step 2.1.2.4.2.4
Combine the numerators over the common denominator.
Step 2.1.2.4.2.5
Simplify the numerator.
Tap for more steps...
Step 2.1.2.4.2.5.1
Multiply by .
Step 2.1.2.4.2.5.2
Subtract from .
Step 2.1.2.4.2.6
Move the negative in front of the fraction.
Step 2.1.2.5
Substitute the values of , , and into the vertex form .
Step 2.1.3
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.
Tap for more steps...
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.
Tap for more steps...
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.
Tap for more steps...
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.
Tap for more steps...
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
Select a few values, and plug them into the equation to find the corresponding values. The values should be selected around the vertex.
Tap for more steps...
Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
Tap for more steps...
Step 3.2.1
Combine the numerators over the common denominator.
Step 3.2.2
Simplify each term.
Tap for more steps...
Step 3.2.2.1
One to any power is one.
Step 3.2.2.2
Multiply by .
Step 3.2.2.3
Multiply by .
Step 3.2.3
Subtract from .
Step 3.2.4
Simplify each term.
Tap for more steps...
Step 3.2.4.1
Move the negative in front of the fraction.
Step 3.2.4.2
Move the negative in front of the fraction.
Step 3.2.5
To write as a fraction with a common denominator, multiply by .
Step 3.2.6
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps...
Step 3.2.6.1
Multiply by .
Step 3.2.6.2
Multiply by .
Step 3.2.7
Combine the numerators over the common denominator.
Step 3.2.8
Simplify the numerator.
Tap for more steps...
Step 3.2.8.1
Multiply by .
Step 3.2.8.2
Subtract from .
Step 3.2.9
Move the negative in front of the fraction.
Step 3.2.10
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.
Tap for more steps...
Step 3.5.1
Combine the numerators over the common denominator.
Step 3.5.2
Simplify each term.
Tap for more steps...
Step 3.5.2.1
Raising to any positive power yields .
Step 3.5.2.2
Multiply by .
Step 3.5.2.3
Multiply by .
Step 3.5.3
Add and .
Step 3.5.4
Simplify each term.
Tap for more steps...
Step 3.5.4.1
Divide by .
Step 3.5.4.2
Move the negative in front of the fraction.
Step 3.5.5
Subtract from .
Step 3.5.6
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.
Tap for more steps...
Step 3.8.1
Combine the numerators over the common denominator.
Step 3.8.2
Simplify each term.
Tap for more steps...
Step 3.8.2.1
Raise to the power of .
Step 3.8.2.2
Multiply by .
Step 3.8.2.3
Multiply by .
Step 3.8.3
Subtract from .
Step 3.8.4
Simplify each term.
Tap for more steps...
Step 3.8.4.1
Divide by .
Step 3.8.4.2
Move the negative in front of the fraction.
Step 3.8.5
Subtract from .
Step 3.8.6
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.
Tap for more steps...
Step 3.11.1
Combine the numerators over the common denominator.
Step 3.11.2
Simplify each term.
Tap for more steps...
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
Simplify the expression.
Tap for more steps...
Step 3.11.3.1
Subtract from .
Step 3.11.3.2
Move the negative in front of the fraction.
Step 3.11.4
To write as a fraction with a common denominator, multiply by .
Step 3.11.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps...
Step 3.11.5.1
Multiply by .
Step 3.11.5.2
Multiply by .
Step 3.11.6
Combine the numerators over the common denominator.
Step 3.11.7
Simplify the numerator.
Tap for more steps...
Step 3.11.7.1
Multiply by .
Step 3.11.7.2
Subtract from .
Step 3.11.8
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