Functions and Graphs - Algebra II
Card 0 of 2484
For the function 
, please state the end behavior, find any local maxima and local minima, and then state whether the graph symmetry is even, odd, or neither even nor odd.
For the function 
To get started on this problem, it helps to use a graphing calculator or other graphing tool to visualize the function. The graph of 
is below:
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting smaller and smaller (more and more negative.) Therefore, as x approaches negative infinity, y also approaches negative infinity. Next, look at x=2, then x=3, and so on, and you can see that as x gets bigger and bigger, so too does y. Therefore as x approaches infinity, y also approaches infinity. Mathematically, this is written like:
As 
and as 
.
Next, the question asks to identify any local minima and maxima. It helps to think of these as "peaks" and "valleys." Looking at the graph, it appears that these exist at the points (0, 1) and (2, -3). We can check this algebraically by plugging in these x values and seeing that the associated y values come out of the function.
This confirms that the point (0, 1) is a local maxima (peak) and the point (2, -3) is a local minima (valley).
Finally, the question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. The right side of this graph has a local minima, while the left side does not, therefore, this graph is not even. In order to have odd symmetry, the graph must have symmetry over the line y=x. An easy way to spot this is to see if the graph looks the same right side up as it does upside down. This does not, therefore, the graph has no symmetry. Algebraically, a function has even symmetry if f(x)=f(-x), and a function has odd symmetry if -f(x)=f(-x).
To get started on this problem, it helps to use a graphing calculator or other graphing tool to visualize the function. The graph of 
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting smaller and smaller (more and more negative.) Therefore, as x approaches negative infinity, y also approaches negative infinity. Next, look at x=2, then x=3, and so on, and you can see that as x gets bigger and bigger, so too does y. Therefore as x approaches infinity, y also approaches infinity. Mathematically, this is written like:
As 
Next, the question asks to identify any local minima and maxima. It helps to think of these as "peaks" and "valleys." Looking at the graph, it appears that these exist at the points (0, 1) and (2, -3). We can check this algebraically by plugging in these x values and seeing that the associated y values come out of the function.
This confirms that the point (0, 1) is a local maxima (peak) and the point (2, -3) is a local minima (valley).
Finally, the question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. The right side of this graph has a local minima, while the left side does not, therefore, this graph is not even. In order to have odd symmetry, the graph must have symmetry over the line y=x. An easy way to spot this is to see if the graph looks the same right side up as it does upside down. This does not, therefore, the graph has no symmetry. Algebraically, a function has even symmetry if f(x)=f(-x), and a function has odd symmetry if -f(x)=f(-x).
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Below is the graph of 
Select the choice that correctly indicates this graph's end behavior.
Below is the graph of
Select the choice that correctly indicates this graph's end behavior.
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting smaller and smaller (more and more negative.) Therefore, as x approaches negative infinity, y also approaches negative infinity. Then start again at the origin, this time moving right. You can see that as x gets bigger and bigger, so too does y. Therefore as x approaches infinity, y also approaches infinity. Mathematically, this is written like:
As and as .
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting smaller and smaller (more and more negative.) Therefore, as x approaches negative infinity, y also approaches negative infinity. Then start again at the origin, this time moving right. You can see that as x gets bigger and bigger, so too does y. Therefore as x approaches infinity, y also approaches infinity. Mathematically, this is written like:
As
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Below is the graph of 
Select the choice that correctly indicates whether this graph has even symmetry, odd symmetry, or neither.
Below is the graph of
Select the choice that correctly indicates whether this graph has even symmetry, odd symmetry, or neither.
The question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. Quadrant I (x and y both positive) has a piece of the graph, while Quadrant II (x negative, y positive) has no part of the graph. Because these are not matching, this graph is not even. In order to have odd symmetry, the graph must have symmetry over the line y=x. An easy way to spot this is to see if the graph looks the same right side up as it does upside down. This does have this quality, so it has odd symmetry. Algebraically, a function has even symmetry if f(x)=f(-x), and a function has odd symmetry if -f(x)=f(-x). You can plug in several test values of x to see this for yourself.
The question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. Quadrant I (x and y both positive) has a piece of the graph, while Quadrant II (x negative, y positive) has no part of the graph. Because these are not matching, this graph is not even. In order to have odd symmetry, the graph must have symmetry over the line y=x. An easy way to spot this is to see if the graph looks the same right side up as it does upside down. This does have this quality, so it has odd symmetry. Algebraically, a function has even symmetry if f(x)=f(-x), and a function has odd symmetry if -f(x)=f(-x). You can plug in several test values of x to see this for yourself.
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Below is the graph of 
Select the choice that correctly indicates whether this graph has even symmetry, odd symmetry, or neither.
Below is the graph of
Select the choice that correctly indicates whether this graph has even symmetry, odd symmetry, or neither.
The question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. We can see that what is on the left side of the line x=0 is an exact match of what is on the right side of the line x=0. Therefore, this graph has even symmetry. Algebraically, a function has even symmetry if f(x)=f(-x). You can plug in several test values of x to see this for yourself.
The question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. We can see that what is on the left side of the line x=0 is an exact match of what is on the right side of the line x=0. Therefore, this graph has even symmetry. Algebraically, a function has even symmetry if f(x)=f(-x). You can plug in several test values of x to see this for yourself.
Compare your answer with the correct one above
Below is the graph of 
Select the choice that correctly indicates this graph's end behavior.
Below is the graph of
Select the choice that correctly indicates this graph's end behavior.
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting bigger and bigger (more and more positive.) Therefore, as x approaches negative infinity, y approaches infinity. Then start again at the origin, this time moving right. You can see that as x gets bigger and bigger, so too does y. Therefore as x approaches infinity, y also approaches infinity. Mathematically, this is written like:
As 
and as .
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting bigger and bigger (more and more positive.) Therefore, as x approaches negative infinity, y approaches infinity. Then start again at the origin, this time moving right. You can see that as x gets bigger and bigger, so too does y. Therefore as x approaches infinity, y also approaches infinity. Mathematically, this is written like:
As 
Compare your answer with the correct one above
Below is the graph of 
Select the choice that correctly indicates this graph's end behavior.
Below is the graph of
Select the choice that correctly indicates this graph's end behavior.
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting bigger and bigger (more and more positive.) Therefore, as x approaches negative infinity, y approaches infinity. Then start again at the origin, this time moving right. You can see that as x gets bigger and bigger, y gets more and more negative. Therefore as x approaches infinity, y approaches negative infinity. Mathematically, this is written like:
As and as 
.
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting bigger and bigger (more and more positive.) Therefore, as x approaches negative infinity, y approaches infinity. Then start again at the origin, this time moving right. You can see that as x gets bigger and bigger, y gets more and more negative. Therefore as x approaches infinity, y approaches negative infinity. Mathematically, this is written like:
As 
Compare your answer with the correct one above
Below is the graph of 
Select the choice that correctly indicates whether this graph has even symmetry, odd symmetry, or neither.
Below is the graph of
Select the choice that correctly indicates whether this graph has even symmetry, odd symmetry, or neither.
The question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. Quadrant I (x and y both positive) has no piece of this graph, while Quadrant II (x negative, y positive) has a part of the graph. Because these are not matching, this graph is not even. In order to have odd symmetry, the graph must have symmetry over the line y=x. An easy way to spot this is to see if the graph looks the same right side up as it does upside down. This does have this quality, so it has odd symmetry. Algebraically, a function has even symmetry if f(x)=f(-x), and a function has odd symmetry if -f(x)=f(-x). You can plug in several test values of x to see this for yourself.
The question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. Quadrant I (x and y both positive) has no piece of this graph, while Quadrant II (x negative, y positive) has a part of the graph. Because these are not matching, this graph is not even. In order to have odd symmetry, the graph must have symmetry over the line y=x. An easy way to spot this is to see if the graph looks the same right side up as it does upside down. This does have this quality, so it has odd symmetry. Algebraically, a function has even symmetry if f(x)=f(-x), and a function has odd symmetry if -f(x)=f(-x). You can plug in several test values of x to see this for yourself.
Compare your answer with the correct one above
Below is the graph of 
Select the choice that correctly indicates whether this graph has even symmetry, odd symmetry, or neither.
Below is the graph of
Select the choice that correctly indicates whether this graph has even symmetry, odd symmetry, or neither.
The question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. Quadrant I (x and y both positive) has a piece of the graph as does Quadrant II (x negative, y positive); however, these pieces are not mirror images of one another. Therefore, this graph is not even. In order to have odd symmetry, the graph must have symmetry over the line y=x. An easy way to spot this is to see if the graph looks the same right side up as it does upside down. In the original graph, the graph flattens above the origin, but if we flip this upside down, it flattens below the origin. While it has odd symmetry around the point (0, 5), it does not have symmetry around the origin, and therefore the function is not odd. Therefore, this graph does not have symmetry. Algebraically, a function has even symmetry if f(x)=f(-x), and a function has odd symmetry if -f(x)=f(-x). You can plug in several test values of x to see that neither of these are satisfied.
The question asks us to determine whether the graph has even, odd, or no symmetry. In order for a graph to have even symmetry, it must produce the same image when reflected over the y-axis. Quadrant I (x and y both positive) has a piece of the graph as does Quadrant II (x negative, y positive); however, these pieces are not mirror images of one another. Therefore, this graph is not even. In order to have odd symmetry, the graph must have symmetry over the line y=x. An easy way to spot this is to see if the graph looks the same right side up as it does upside down. In the original graph, the graph flattens above the origin, but if we flip this upside down, it flattens below the origin. While it has odd symmetry around the point (0, 5), it does not have symmetry around the origin, and therefore the function is not odd. Therefore, this graph does not have symmetry. Algebraically, a function has even symmetry if f(x)=f(-x), and a function has odd symmetry if -f(x)=f(-x). You can plug in several test values of x to see that neither of these are satisfied.
Compare your answer with the correct one above
Below is the graph of 
Select the choice that correctly indicates this graph's end behavior.
Below is the graph of
Select the choice that correctly indicates this graph's end behavior.
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting smaller and smaller (more and more negative.) Therefore, as x approaches negative infinity, y also approaches negative infinity. Then start again at the origin, this time moving right. You can see that as x gets bigger and bigger, so too does y. Therefore as x approaches infinity, y also approaches infinity. Mathematically, this is written like:
As and as .
When identifying end behavior, you want to ask yourself "As x gets infinitely big/small, what happens to y?" If you start at x=0, then move left to where x=-1, you can see that the values of y are getting smaller and smaller (more and more negative.) Therefore, as x approaches negative infinity, y also approaches negative infinity. Then start again at the origin, this time moving right. You can see that as x gets bigger and bigger, so too does y. Therefore as x approaches infinity, y also approaches infinity. Mathematically, this is written like:
As
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Find the domain:
Find the domain:
To find the domain, find all areas of the number line where the fraction is defined.
because the denominator of a fraction must be nonzero.
Factor by finding two numbers that sum to -2 and multiply to 1. These numbers are -1 and -1.
To find the domain, find all areas of the number line where the fraction is defined.
Factor by finding two numbers that sum to -2 and multiply to 1. These numbers are -1 and -1.
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What is the domain of the function 
?
What is the domain of the function
The domain is the set of x-values that make the function defined.
This function is defined everywhere except at 
, since division by zero is undefined.
The domain is the set of x-values that make the function defined.
This function is defined everywhere except at
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is a sine curve. What are the domain and range of this function?
The domain includes the values that go into a function (the x-values) and the range are the values that come out (the 
or y-values). A sine curve represent a wave the repeats at a regular frequency. Based upon this graph, the maximum 
is equal to 1, while the minimum is equal to –1. The x-values span all real numbers, as there is no limit to the input fo a sine function. The domain of the function is all real numbers and the range is 
.
The domain includes the values that go into a function (the x-values) and the range are the values that come out (the
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Give the domain of the function below.
Give the domain of the function below.
The domain is the set of possible value for the 
variable. We can find the impossible values of 
by setting the denominator of the fractional function equal to zero, as this would yield an impossible equation.
Now we can solve for 
.
There is no real value of 
that will fit this equation; any real value squared will be a positive number.
The radicand is always positive, and 
is defined for all real values of 
. This makes the domain of 
the set of all real numbers.
The domain is the set of possible value for the
Now we can solve for
There is no real value of
The radicand is always positive, and
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If 
, which of these values of 
is NOT in the domain of this equation?
If
Using 
as the input (
) value for this equation generates an output (
) value that contradicts the stated condition of 
.
Therefore 
is not a valid value for 
and not in the equation's domain:
Using
Therefore
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What is the domain of the function?
What is the domain of the function?
In order for the function to be real, the value inside of the square root must be greater than or equal to zero. The domain refers to the possible values of the independent variable (x-value) that allow this to be true.
For this term to be real, 
must be greater than or equal to zero.
In order for the function to be real, the value inside of the square root must be greater than or equal to zero. The domain refers to the possible values of the independent variable (x-value) that allow this to be true.
For this term to be real,
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Give the domain of the function:
Give the domain of the function:
The domain of a rational function is the set of all values of 
for which the denominator is not equal to 0 (the value of the numerator is irrelevant), so we set the denominator to 0 and solve for 
to find the excluded values.
This is a quadratic function, so we factor the expression as 
, replacing the question marks with two numbers whose product is 8 and whose sum is 
. These numbers are 
, so
becomes
So either
, in which case 
,
or 
, in which case 
.
Therefore, 2 and 4 are the only numbers excluded from the domain.
The domain of a rational function is the set of all values of
This is a quadratic function, so we factor the expression as
becomes
So either
or
Therefore, 2 and 4 are the only numbers excluded from the domain.
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Give the domain of the function:
Give the domain of the function:
The domain of a rational function is the set of all values of 
for which the denominator is not equal to 0, so we set the denominator to 0 and solve for 
.
This is a quadratic function, so we factor the expression as 
, replacing the question marks with two numbers whose product is 9 and whose sum is 
. These numbers are 
, so
becomes
,
or 
.
This means that 
, or 
.
Therefore, 3 is the only number excluded from the domain.
The domain of a rational function is the set of all values of
This is a quadratic function, so we factor the expression as
becomes
or
This means that
Therefore, 3 is the only number excluded from the domain.
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Which of the following is NOT a function?
Which of the following is NOT a function?
A function has to pass the vertical line test, which means that a vertical line can only cross the function one time. To put it another way, for any given value of 
, there can only be one value of 
. For the function 
, there is one 
value for two possible 
values. For instance, if 
, then 
. But if 
, 
as well. This function fails the vertical line test. The other functions listed are a line,
, the top half of a right facing parabola, 
, a cubic equation, 
, and a semicircle, 
. These will all pass the vertical line test.
A function has to pass the vertical line test, which means that a vertical line can only cross the function one time. To put it another way, for any given value of
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What is the range of the function?
What is the range of the function?
This function is a parabola that has been shifted up five units. The standard parabola has a range that goes from 0 (inclusive) to positive infinity. If the vertex has been moved up by 5, this means that its minimum has been shifted up by five. The first term is inclusive, which means you need a "\[" for the beginning.
Minimum: 5 inclusive, maximum: infinity
Range: 
This function is a parabola that has been shifted up five units. The standard parabola has a range that goes from 0 (inclusive) to positive infinity. If the vertex has been moved up by 5, this means that its minimum has been shifted up by five. The first term is inclusive, which means you need a "\[" for the beginning.
Minimum: 5 inclusive, maximum: infinity
Range:
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What is the domain of the function?
What is the domain of the function?
The domain represents the acceptable 
values for this function. Based on the members of the function, the only limit that you have is the non-allowance of a negative number (because of the square root). The square and the linear terms are fine with any numbers. You cannot have any negative values, otherwise the square root will not be a real number.
Minimum: 0 inclusive, maximum: infinity
Domain: 
The domain represents the acceptable
Minimum: 0 inclusive, maximum: infinity
Domain:
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