Algebra Examples

Graph f(x)=x^2-4
Step 1
Find the properties of the given parabola.
Tap for more steps...
Step 1.1
Rewrite the equation in vertex form.
Tap for more steps...
Step 1.1.1
Complete the square for .
Tap for more steps...
Step 1.1.1.1
Use the form , to find the values of , , and .
Step 1.1.1.2
Consider the vertex form of a parabola.
Step 1.1.1.3
Find the value of using the formula .
Tap for more steps...
Step 1.1.1.3.1
Substitute the values of and into the formula .
Step 1.1.1.3.2
Cancel the common factor of and .
Tap for more steps...
Step 1.1.1.3.2.1
Factor out of .
Step 1.1.1.3.2.2
Cancel the common factors.
Tap for more steps...
Step 1.1.1.3.2.2.1
Factor out of .
Step 1.1.1.3.2.2.2
Cancel the common factor.
Step 1.1.1.3.2.2.3
Rewrite the expression.
Step 1.1.1.3.2.2.4
Divide by .
Step 1.1.1.4
Find the value of using the formula .
Tap for more steps...
Step 1.1.1.4.1
Substitute the values of , and into the formula .
Step 1.1.1.4.2
Simplify the right side.
Tap for more steps...
Step 1.1.1.4.2.1
Simplify each term.
Tap for more steps...
Step 1.1.1.4.2.1.1
Raising to any positive power yields .
Step 1.1.1.4.2.1.2
Multiply by .
Step 1.1.1.4.2.1.3
Divide by .
Step 1.1.1.4.2.1.4
Multiply by .
Step 1.1.1.4.2.2
Add and .
Step 1.1.1.5
Substitute the values of , , and into the vertex form .
Step 1.1.2
Set equal to the new right side.
Step 1.2
Use the vertex form, , to determine the values of , , and .
Step 1.3
Since the value of is positive, the parabola opens up.
Opens Up
Step 1.4
Find the vertex .
Step 1.5
Find , the distance from the vertex to the focus.
Tap for more steps...
Step 1.5.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 1.5.2
Substitute the value of into the formula.
Step 1.5.3
Cancel the common factor of .
Tap for more steps...
Step 1.5.3.1
Cancel the common factor.
Step 1.5.3.2
Rewrite the expression.
Step 1.6
Find the focus.
Tap for more steps...
Step 1.6.1
The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Step 1.6.2
Substitute the known values of , , and into the formula and simplify.
Step 1.7
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Step 1.8
Find the directrix.
Tap for more steps...
Step 1.8.1
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 1.8.2
Substitute the known values of and into the formula and simplify.
Step 1.9
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 2
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 2.1
Replace the variable with in the expression.
Step 2.2
Simplify the result.
Tap for more steps...
Step 2.2.1
Raise to the power of .
Step 2.2.2
Subtract from .
Step 2.2.3
The final answer is .
Step 2.3
The value at is .
Step 2.4
Replace the variable with in the expression.
Step 2.5
Simplify the result.
Tap for more steps...
Step 2.5.1
Raise to the power of .
Step 2.5.2
Subtract from .
Step 2.5.3
The final answer is .
Step 2.6
The value at is .
Step 2.7
Replace the variable with in the expression.
Step 2.8
Simplify the result.
Tap for more steps...
Step 2.8.1
One to any power is one.
Step 2.8.2
Subtract from .
Step 2.8.3
The final answer is .
Step 2.9
The value at is .
Step 2.10
Replace the variable with in the expression.
Step 2.11
Simplify the result.
Tap for more steps...
Step 2.11.1
Raise to the power of .
Step 2.11.2
Subtract from .
Step 2.11.3
The final answer is .
Step 2.12
The value at is .
Step 2.13
Graph the parabola using its properties and the selected points.
Step 3
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 4