Functions and Graphs - Algebra 2
Card 1 of 2484
Reflect the line 
across 
, and shift the line down three units. What is the new equation?
Reflect the line
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The equations with an existing 
variable is incorrect because they either represent lines with slopes or vertical lines.
After the line 
is reflected across 
, the line becomes 
.
Shifting this line down three units mean that the line will have a vertical translation down three.
Subtract the equation 
by three.
The result is: 
The equations with an existing
After the line
Shifting this line down three units mean that the line will have a vertical translation down three.
Subtract the equation
The result is:
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Give the equation of the vertical asymptote of the graph of the equation 
.
Give the equation of the vertical asymptote of the graph of the equation
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Define 
. As an exponential function, this has a graph that has no vertical asymptote, as 
is defined for all real values of 
. In terms of 
:
,
The graph of 
is a transformation of that of 
- a horizontal shift ( 
), a vertical stretch ( 
), and a vertical shift ( 
) of the graph of 
; none of these transformations changes the status of the function as one whose graph has no vertical asymptote.
Define
The graph of
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Give the equation of the horizontal asymptote of the graph of the equation
Give the equation of the horizontal asymptote of the graph of the equation
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Define 
in terms of ,
It can be restated as the following:
The graph of 
has as its horizontal asymptote the line of the equation 
. The graph of is a transformation of that of —a right shift of 2 units 
, a vertical stretch 
, and an upward shift of 5 units 
. The right shift and the vertical stretch do not affect the position of the horizontal asymptote, but the upward shift moves the asymptote to the line of the equation 
. This is the correct response.
Define
It can be restated as the following:
The graph of
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Determine where the graphs of the following equations will intersect.
Determine where the graphs of the following equations will intersect.
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We can solve the system of equations using the substitution method.
Solve for 
in the second equation.
Substitute this value of 
into the first equation.
Now we can solve for 
.
Solve for 
using the first equation with this new value of 
.
The solution is the ordered pair 
.
We can solve the system of equations using the substitution method.
Solve for
Substitute this value of
Now we can solve for
Solve for
The solution is the ordered pair
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Which of the following graphs correctly depicts the graph of the inequality 
Which of the following graphs correctly depicts the graph of the inequality
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Let's start by looking at the given equation:
The inequality is written in slope-intercept form; therefore, the slope is equal to 
and the y-intercept is equal to 
.
All of the graphs depict a line with slope of 
and y-intercept 
. Next, we need to decide if we should shade above or below the line. To do this, we can determine if the statement is true using the origin 
. If the origin satisfies the inequality, we will know to shade below the line. Substitute the values into the given equation and solve.
Because this statement is true, the origin must be included in the shaded region, so we shade below the line.
Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.
Let's start by looking at the given equation:
The inequality is written in slope-intercept form; therefore, the slope is equal to
All of the graphs depict a line with slope of
Because this statement is true, the origin must be included in the shaded region, so we shade below the line.
Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.
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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 
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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.
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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.
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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.
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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.
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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.
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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 
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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.
Tap to reveal answer
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 
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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.
Tap to reveal answer
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.
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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.
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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.
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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.
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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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Given the above circle inequality, is the shading on the graph inside or outside the circle?
Given the above circle inequality, is the shading on the graph inside or outside the circle?
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Check the center of the circle to see if that point satisfies the inequality. When evaluating the function at the center (-2,4), we see that it does not satisfy the equation, so it cannot be in the shaded region of the graph. Therefore the shading is outside of the circle.
Check the center of the circle to see if that point satisfies the inequality. When evaluating the function at the center (-2,4), we see that it does not satisfy the equation, so it cannot be in the shaded region of the graph. Therefore the shading is outside of the circle.
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All inputs are valid. There is nothing you can put in for x that won't work.
All inputs are valid. There is nothing you can put in for x that won't work.
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True or false: The graph of 
has as a horizontal asymptote the graph of the equation 
.
True or false: The graph of
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is a rational function in simplest form whose denominator has a polynomial with degree greater than that of the polynomial in its numerator (2 and 1, respectively). The graph of such a function has as its horizontal asymptote the line of the equation 
.
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All inputs are valid. There is nothing you can put in for x that won't work.
All inputs are valid. There is nothing you can put in for x that won't work.
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You cannot take the square root of a negative number.
You cannot take the square root of a negative number.
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The function includes all possible y-values (outputs). There is nothing you can put in for y that won't work.
The function includes all possible y-values (outputs). There is nothing you can put in for y that won't work.
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