All Algebra II Resources
Example Questions
Example Question #631 : Exponents
Simplify:
In order to simplify this term, we will need to use the power rule of the exponents.
Apply this rule for the given problem.
Simplify the terms.
The answer is:
Example Question #1 : Solving Exponential Functions
What is the horizontal asymptote of the graph of the equation ?
The asymptote of this equation can be found by observing that regardless of . We are thus solving for the value of as approaches zero.
So the value that cannot exceed is , and the line is the asymptote.
Example Question #1 : Solving Exponential Functions
What is/are the asymptote(s) of the graph of the function
?
An exponential equation of the form has only one asymptote - a horizontal one at . In the given function, , so its one and only asymptote is .
Example Question #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
Find the vertical asymptote of the equation.
There are no vertical asymptotes.
To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.
Example Question #1 : Asymptotes
Determine the asymptotes, if any:
Factorize both the numerator and denominator.
Notice that one of the binomials will cancel.
The domain of this equation cannot include .
The simplified equation is:
Since the term canceled, the term will have a hole instead of an asymptote.
Set the denominator equal to zero.
Subtract one from both sides.
There will be an asymptote at only:
The answer is:
Example Question #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
Which of the choices represents asymptote(s), if any?
Factor the numerator and denominator.
Notice that the terms will cancel. The hole will be located at because this is a removable discontinuity.
The denominator cannot be equal to zero. Set the denominator to find the location where the x-variable cannot exist.
The asymptote is located at .
Example Question #1 : Asymptotes
Where is an asymptote located, if any?
Factor the numerator and denominator.
Rewrite the equation.
Notice that the will cancel. This means that the root of will be a hole instead of an asymptote.
Set the denominator equal to zero and solve for x.
An asymptote is located at:
The answer is:
Example Question #1 : Asymptotes
Consider the exponential function . Determine if there are any asymptotes and where they lie on the graph.
There are no asymptotes. goes to positive infinity in both the and directions.
There is one vertical asymptote at .
There is one horizontal asymptote at .
There is one vertical asymptote at .
There is one horizontal asymptote at .
For positive values, increases exponentially in the direction and goes to positive infinity, so there is no asymptote on the positive -axis. For negative values, as decreases, the term becomes closer and closer to zero so approaches as we move along the negative axis. As the graph below shows, this is forms a horizontal asymptote.
Example Question #1 : Solving Exponential Equations
Solve the equation for .
Begin by recognizing that both sides of the equation have a root term of .
Using the power rule, we can set the exponents equal to each other.
Example Question #1 : Solving And Graphing Exponential Equations
Solve the equation for .
Begin by recognizing that both sides of the equation have the same root term, .
We can use the power rule to combine exponents.
Set the exponents equal to each other.