Functions as Graphs - Algebra 2
Card 0 of 112
True or false: The graph of 
has as a horizontal asymptote the graph of the equation 
.
True or false: The graph of
Tap to see back →
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 
.
Find the slope from the following equation.
Find the slope from the following equation.
Tap to see back →
To find the slope of an equation first get the equation in slope intercept form.
where,
represents the slope.
Thus
To find the slope of an equation first get the equation in slope intercept form.
where,
Thus
Tap to see back →
When determining how a the graph of a function will be translated, we know that anything that happens to x in the function will impact the graph horizontally, opposite of what is expressed in the function, whereas anything that is outside the function will impact the graph vertically the same as it is in the function notation.
For this graph:
The graph will move 3 spaces left, because that is the opposite sign of the what is connected to x directly.
Also, the graph will move down 2 spaces, because that is what is outside the function and the 2 is negative.
When determining how a the graph of a function will be translated, we know that anything that happens to x in the function will impact the graph horizontally, opposite of what is expressed in the function, whereas anything that is outside the function will impact the graph vertically the same as it is in the function notation.
For this graph:
The graph will move 3 spaces left, because that is the opposite sign of the what is connected to x directly.
Also, the graph will move down 2 spaces, because that is what is outside the function and the 2 is negative.
Which of the following is true of the relation graphed above?
Which of the following is true of the relation graphed above?
Tap to see back →
The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:
Also, it is seen to be symmetrical about the 
-axis. Consequently, for each 
in the domain, 
- the function is even.
The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:
Also, it is seen to be symmetrical about the
Which graph depicts a function?
Which graph depicts a function?
Tap to see back →
A function may only have one y-value for each x-value.
The vertical line test can be used to identify the function. If at any point on the graph, a straight vertical line intersects the curve at more than one point, the curve is not a function.
A function may only have one y-value for each x-value.
The vertical line test can be used to identify the function. If at any point on the graph, a straight vertical line intersects the curve at more than one point, the curve is not a function.
Which analysis can be performed to determine if an equation is a function?
Which analysis can be performed to determine if an equation is a function?
Tap to see back →
The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one 
(or 
) value for each value of 
. The vertical line test determines how many 
(or 
) values are present for each value of 
. If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function.
The horizontal line test can be used to determine if a function is one-to-one, that is, if only one 
value exists for each 
(or 
) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.
Example of a function: 
Example of an equation that is not a function: 
The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one
The horizontal line test can be used to determine if a function is one-to-one, that is, if only one
Example of a function:
Example of an equation that is not a function:
The graph below is the graph of a piece-wise function in some interval. Identify, in interval notation, the decreasing interval.
The graph below is the graph of a piece-wise function in some interval. Identify, in interval notation, the decreasing interval.
Tap to see back →
As is clear from the graph, in the interval between 
(
included) to 
, the 
is constant at 
and then from 
(
not included) to 
(
not included), the 
is a decreasing function.
As is clear from the graph, in the interval between
Without graphing, determine the relationship between the following two lines. Select the most appropriate answer.
Without graphing, determine the relationship between the following two lines. Select the most appropriate answer.
Tap to see back →
Perpendicular lines have slopes that are negative reciprocals. This is the case with these two lines. Although these lines interesect, this is not the most appropriate answer since it does not account for the fact that they are perpendicular.
Perpendicular lines have slopes that are negative reciprocals. This is the case with these two lines. Although these lines interesect, this is not the most appropriate answer since it does not account for the fact that they are perpendicular.
Which is a vertical asymptote of the graph of the function
?
(a) 
(b) 
Which is a vertical asymptote of the graph of the function
(a)
(b)
Tap to see back →
The vertical asymptote(s) of the graph of a rational function such as 
can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the numerator and denominator.
The numerator is a perfect square trinomial and can be factored as such:
The denominator can be factored as the difference of squares:
Rewrite
as
The expression can be reduced by cancelling 
in both halves:
Set the denominator equal to 0 and solve:
The only asymptote is therefore the line of the equation 
.
The vertical asymptote(s) of the graph of a rational function such as
First, factor the numerator and denominator.
The numerator is a perfect square trinomial and can be factored as such:
The denominator can be factored as the difference of squares:
Rewrite
as
The expression can be reduced by cancelling
Set the denominator equal to 0 and solve:
The only asymptote is therefore the line of the equation
Define a function 
.
Is this function even, odd, or neither?
Define a function
Is this function even, odd, or neither?
Tap to see back →
To identify a function 
as even odd, or neither, determine 
by replacing 
with 
, then simplifying. If 
, the function is even; if 
is odd.
,
so
By the Power of a Product Property,
,
so 
is an odd function
To identify a function
so
By the Power of a Product Property,
so
Define a function 
.
Is this function even, odd, or neither?
Define a function
Is this function even, odd, or neither?
Tap to see back →
To identify a function 
as even odd, or neither, determine 
by replacing 
with 
, then simplifying. If 
, the function is even; if 
is odd.
so
By the Power of a Product Property,
, so 
is not an even function.
,
, so 
is not an odd function.
To identify a function
so
By the Power of a Product Property,
Define a function 
.
Is this function even, odd, or neither?
Define a function
Is this function even, odd, or neither?
Tap to see back →
To identify a function 
as even, odd, or neither, determine 
by replacing 
with 
, then simplifying. If 
, the function is even; if 
is odd.
Since 
, 
is an odd function.
To identify a function
Since
Define a function 
.
Is this function even, odd, or neither?
Define a function
Is this function even, odd, or neither?
Tap to see back →
To identify a function as even, odd, or neither, determine by replacing with , then simplifying. If , the function is even; if is odd.
, so is an even function.
To identify a function
Which is a vertical asymptote of the graph of the function 
?
(a) 
(b) 
Which is a vertical asymptote of the graph of the function
(a)
(b)
Tap to see back →
The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced. The expression is in simplest form, so set the denominator equal to 0 and solve for 
:
The graph of 
has the line of the equation 
as its only vertical asymptote.
The vertical asymptote(s) of the graph of a rational function such as
The graph of
Which of the following is true of the relation graphed above?
Which of the following is true of the relation graphed above?
Tap to see back →
The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as seen below:
Also, it is seen to be symmetrical about the 
-axis. This proves the function even.
The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as seen below:
Also, it is seen to be symmetrical about the
Which of the following is true of the relation graphed above?
Which of the following is true of the relation graphed above?
Tap to see back →
The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:
Also, it can be seen to be symmetrical about the origin. Consequently, for each in the domain, 
- the function is odd.
The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:
Also, it can be seen to be symmetrical about the origin. Consequently, for each
Which of the following is a vertical asymptote of the graph of the function 
?
(a) 
(b) 
Which of the following is a vertical asymptote of the graph of the function
(a)
(b)
Tap to see back →
The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the numerator. It is a quadratic trinomial with lead term 
, so look to "reverse-FOIL" it as
by finding two integers with sum 
and product 30. By trial and error, these integers can be found to be 
and 
, so
Therefore, 
can be rewritten as
.
Cancelling 
, this can be seen to be essentially a polynomial function:
,
which does not have a vertical asymptote.
The vertical asymptote(s) of the graph of a rational function such as
First, factor the numerator. It is a quadratic trinomial with lead term
by finding two integers with sum
Therefore,
Cancelling
which does not have a vertical asymptote.
True or false: The graph of 
has as a horizontal asymptote the line of the equation 
.
True or false: The graph of
Tap to see back →
is a rational function whose numerator is a polynomial with degree greater than that of the polynomial in its denominator (2 and 1, respectively). The graph of such a function does not have a horizontal asymptote.
True or false: The graph of 
has as a horizontal asymptote the graph of the equation 
.
True or false: The graph of
Tap to see back →
is a rational function whose numerator and denominator have the same degree (1). As such, it has as a horizontal asymptote the line of the equation 
, where 
is the quotient of the coefficients of the highest-degree terms of its numerator and denominator. Consequently, the horizontal asymptote of
is
,
or
.
is
or
True or false: The graph of 
has as a horizontal asymptote the graph of the equation 
.
True or false: The graph of
Tap to see back →
is a rational function whose numerator and denominator are polynomials of the same degree (1). As such, it has as a horizontal asymptote the line of the equation 
, where 
is the quotient of the coefficients of the highest-degree terms of its numerator and denominator. Consequently, the horizontal asymptote of
is
or
.
is
or