Maths-
General
Easy
Question
What is the graph of 
Hint:
A rational function is a function that is the ratio of polynomials. Any function of one variable, x, is called a rational function if, it can be represented as f(x) = 
, where p(x) and q(x) are polynomials such that q(x) ≠ 0.
Rational functions are of the form y = f(x)y = fx , where f(x)fx is a rational expression .
The correct answer is: The vertical asymptote of the rational function is x= 3 and x= -5 . This function has the x -intercept at (1.667,0) and y -intercept at (0,1) . We will find more points on the function and graph the function.
- Find the asymptotes of the rational function, if any.
- Draw the asymptotes as dotted lines.
- Find the x -intercept (s) and y -intercept of the rational function, if any.
- Find the values of y for several different values of x .
- Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x2 + 2x - 15= 0
x2 + 5x -3x - 15= 0
x(x + 5) - 3 (x + 5) = 0
(x – 3) (x + 5) = 0
x – 3 = 0 or x + 5 = 0
x = 3 or x = -5
The vertical asymptote of the rational function is x = 3 and x = -5 .
This function has the x - intercept at (1.667,0) and y -intercept at (0,1) . We will find more points on the function and graph the function.
Related Questions to study
Maths-
Find the equation for a path that passes through the point (-4, 9) and is parallel to 
.
- We have been given an equation that represents y-axis.
- We have to find an equation for a path that passes through the point (-4,9) and is perpendicular to
We have given a line passes through a point (-4, 9) and parallel to a line
Since two parallel lines have same slope.
So, Slope of the line will be
Therefore the equation of the line will be
Find the equation for a path that passes through the point (-4, 9) and is parallel to .
Maths-General
- We have been given an equation that represents y-axis.
- We have to find an equation for a path that passes through the point (-4,9) and is perpendicular to
We have given a line passes through a point (-4, 9) and parallel to a line
Since two parallel lines have same slope.
So, Slope of the line will be
Therefore the equation of the line will be
Maths-
Write explicit formula. 
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = -7
Also, We have a1 = -3
So, The explicit formula will be
.
Write explicit formula.
Maths-General
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = -7
Also, We have a1 = -3
So, The explicit formula will be
.
Maths-
Write explicit formula. 
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = 1
Also, We have a1 = 12
So, The explicit formula will be
Write explicit formula.
Maths-General
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = 1
Also, We have a1 = 12
So, The explicit formula will be
Maths-
Write explicit formula. 
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = -21
Also, We have a1 = 56
So, The explicit formula will be
Write explicit formula.
Maths-General
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = -21
Also, We have a1 = 56
So, The explicit formula will be
Maths-
Graph each function 
Hint :-
Solution:-
1.Find the asymptotes of the rational function, if any.
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x - 1= 0
x = 1
The vertical asymptote of the rational function is x= 1
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x = 1 and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
=3
Solution:-
1.Find the asymptotes of the rational function, if any.
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x - 1= 0
x = 1
The vertical asymptote of the rational function is x= 1
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x = 1 and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
=3Graph each function
Maths-General
Hint :-
Solution:-
1.Find the asymptotes of the rational function, if any.
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x - 1= 0
x = 1
The vertical asymptote of the rational function is x= 1
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x = 1 and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) ==3
Solution:-
1.Find the asymptotes of the rational function, if any.
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x - 1= 0
x = 1
The vertical asymptote of the rational function is x= 1
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x = 1 and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
=3Maths-
Find the equation for a path that passes through the point (6, 6) and is perpendicular to .
- We have been given an equation that represents y-axis.
- We have to find an equation for a path that passes through the point (6, 6) and is perpendicular to
We have given a line passes through a point (6, 6) and perpendicular to a line
Since product of two perpendicular lines is equal to -1.
So,
Therefore the equation of the line will be
Find the equation for a path that passes through the point (6, 6) and is perpendicular to .
Maths-General
- We have been given an equation that represents y-axis.
- We have to find an equation for a path that passes through the point (6, 6) and is perpendicular to
We have given a line passes through a point (6, 6) and perpendicular to a line
Since product of two perpendicular lines is equal to -1.
So,
Therefore the equation of the line will be
Maths-
Write explicit formula. 
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = -2
Also, We have
So, The explicit formula will be
Write explicit formula.
Maths-General
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = -2
Also, We have
So, The explicit formula will be
Maths-
Write explicit formula. 
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = 6
Also, We have a1 = 9
So, The explicit formula will be
Write explicit formula.
Maths-General
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = 6
Also, We have a1 = 9
So, The explicit formula will be
Maths-
Write explicit formula. 
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = 15
Also, We have a1 = 8
So, The explicit formula will be
Write explicit formula.
Maths-General
- We have given
- We have to find the explicit formula of the given sequence.
We know that the recursive formula of an AP is
, where d is common difference.Here we have
So, d = 15
Also, We have a1 = 8
So, The explicit formula will be
Maths-
Graph each function 
1.Find the asymptotes of the rational function, if any.
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x + 8= 0
x = -8
The vertical asymptote of the rational function is x=-8
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x= -8 and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
= 4
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x + 8= 0
x = -8
The vertical asymptote of the rational function is x=-8
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x= -8 and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
= 4Graph each function
Maths-General
1.Find the asymptotes of the rational function, if any.
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x + 8= 0
x = -8
The vertical asymptote of the rational function is x=-8
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x= -8 and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) == 4
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x + 8= 0
x = -8
The vertical asymptote of the rational function is x=-8
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x= -8 and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
= 4Maths-
What is the graph of the function
1.Find the asymptotes of the rational function, if any.
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
3x - 4= 0
3x = 4
x =
The vertical asymptote of the rational function is x=
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x=. and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
3x - 4= 0
3x = 4
x =
The vertical asymptote of the rational function is x=
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x=
. and horizontal asymptote isy = (leading coefficient of numerator) / (leading coefficient of denominator) =
What is the graph of the function
Maths-General
1.Find the asymptotes of the rational function, if any.
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
3x - 4= 0
3x = 4
x =
The vertical asymptote of the rational function is x=
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x=. and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
2.Draw the asymptotes as dotted lines.
3.Find the x -intercept (s) and y -intercept of the rational function, if any.
4.Find the values of y for several different values of x .
5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
3x - 4= 0
3x = 4
x =
The vertical asymptote of the rational function is x=
We will find more points on the function and graph the function.
From the graph we can analyze that the vertical asymptote of the rational function is x=
. and horizontal asymptote isy = (leading coefficient of numerator) / (leading coefficient of denominator) =
Maths-
Write recursive formula and explicit formula. -15,-6,-3,12,21,...
- We have given a sequence -15,-6,-3,12,21,.....
- We have to find the recursive and explicit formula of the given sequence.
We have given a sequence -15,-6,-3,12,21,.....
The given sequence is an AP.
We know that the recursive formula for any AP is
, where d is common difference.Here the common difference is 9
So, The recursive formula is
Step 2 of 2:
The given sequence is an AP.
So, The explicit formula will be
Write recursive formula and explicit formula. -15,-6,-3,12,21,...
Maths-General
- We have given a sequence -15,-6,-3,12,21,.....
- We have to find the recursive and explicit formula of the given sequence.
We have given a sequence -15,-6,-3,12,21,.....
The given sequence is an AP.
We know that the recursive formula for any AP is
, where d is common difference.Here the common difference is 9
So, The recursive formula is
Step 2 of 2:
The given sequence is an AP.
So, The explicit formula will be
Maths-
The equation 2x + 7 represents a north path on a map. Find the equation for a path that passes through the point (6, 3) and is perpendicular to the north path.
- We have been given an equation that represents the north path on a map.
- We have to find an equation for a path that passes through the point (6, 3) and is perpendicular to the north path.
We have given a line passes through a point (6, 3) and perpendicular to a line
Since product of two perpendicular lines is equal to -1.
So,
Therefore the equation of the line will be
The equation 2x + 7 represents a north path on a map. Find the equation for a path that passes through the point (6, 3) and is perpendicular to the north path.
Maths-General
- We have been given an equation that represents the north path on a map.
- We have to find an equation for a path that passes through the point (6, 3) and is perpendicular to the north path.
We have given a line passes through a point (6, 3) and perpendicular to a line
Since product of two perpendicular lines is equal to -1.
So,
Therefore the equation of the line will be
Maths-
Write recursive formula and explicit formula. 62,57,52,47,42,...
- We have given a sequence 62,57,52,47,42,....
- We have to find the recursive and explicit formula of the given sequence.
We have given a sequence 62,57,52,47,42,....
The given sequence is an AP.
We know that the recursive formula for any AP is
, where d is common difference.Here the common difference is -15
So, The recursive formula is
Step 2 of 2:
The given sequence is an AP.
So, The explicit formula will be
Write recursive formula and explicit formula. 62,57,52,47,42,...
Maths-General
- We have given a sequence 62,57,52,47,42,....
- We have to find the recursive and explicit formula of the given sequence.
We have given a sequence 62,57,52,47,42,....
The given sequence is an AP.
We know that the recursive formula for any AP is
, where d is common difference.Here the common difference is -15
So, The recursive formula is
Step 2 of 2:
The given sequence is an AP.
So, The explicit formula will be
Maths-
Write recursive formula and explicit formula. -4, 5,14,23,32,...
Step 1 of 2:
We have given a sequence -4,5,14,23,32,...
The given sequence is an AP.
We know that the recursive formula for any AP is , where d is common difference.
Here the common difference is 9
So, The recursive formula is
We have given a sequence -4,5,14,23,32,...
The given sequence is an AP.
We know that the recursive formula for any AP is
, where d is common difference.Here the common difference is 9
So, The recursive formula is
Step 2 of 2:
The given sequence is an AP.
So, The explicit formula will be
.
Write recursive formula and explicit formula. -4, 5,14,23,32,...
Maths-General
Step 1 of 2:
We have given a sequence -4,5,14,23,32,...
The given sequence is an AP.
We know that the recursive formula for any AP is , where d is common difference.
Here the common difference is 9
So, The recursive formula is
We have given a sequence -4,5,14,23,32,...
The given sequence is an AP.
We know that the recursive formula for any AP is
, where d is common difference.Here the common difference is 9
So, The recursive formula is
Step 2 of 2:
The given sequence is an AP.
So, The explicit formula will be
.
