Maths-
General
Easy
Question
What is the graph of the function
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 .
- If both the polynomials have the same degree, divide the coefficients of the leading terms. This is your asymptote.
- If the degree of the numerator is less than the denominator, then the asymptote is located at y = 0 (which is the x-axis).
- If the degree of the numerator is greater than the denominator, then there is no horizontal asymptote.
The correct answer is: From the graph we can analyze that the vertical asymptote of the rational function is x= 4/3. and horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator) = 2/3
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) = 
Related Questions to study
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
.
Maths-
What are the vertical and horizontal asymptotes of the graph of 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 .
3x2 -12= 0
3x2 = 12
x2 = 4
x = -2 or x = 2
The vertical asymptote of the rational function is x = −2 and x = 2
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= -2 and x = 2. 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 .
3x2 -12= 0
3x2 = 12
x2 = 4
x = -2 or x = 2
The vertical asymptote of the rational function is x = −2 and x = 2
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= -2 and x = 2. and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
What are the vertical and horizontal asymptotes of the graph of 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 .
3x2 -12= 0
3x2 = 12
x2 = 4
x = -2 or x = 2
The vertical asymptote of the rational function is x = −2 and x = 2
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= -2 and x = 2. 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 .
3x2 -12= 0
3x2 = 12
x2 = 4
x = -2 or x = 2
The vertical asymptote of the rational function is x = −2 and x = 2
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= -2 and x = 2. and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
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 parallel 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 parallel to a line y = 2x + 7
Since two parallel lines have same slope.
So, Slope of the line will be 2
Therefore the equation of the line will be
y - 3 = 2(x - 6)
y = 2x - 9
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 parallel 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 parallel to a line y = 2x + 7
Since two parallel lines have same slope.
So, Slope of the line will be 2
Therefore the equation of the line will be
y - 3 = 2(x - 6)
y = 2x - 9
Maths-
Write recursive formula and explicit formula. 12,19,26,33,40,...
- We have given a sequence 12,19,26,33,40,....
- We have to find the recursive and explicit formula of the given sequence.
We have given a sequence 12,19,26,33,40,....
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 7
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. 12,19,26,33,40,...
Maths-General
- We have given a sequence 12,19,26,33,40,....
- We have to find the recursive and explicit formula of the given sequence.
We have given a sequence 12,19,26,33,40,....
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 7
So, The recursive formula is
Step 2 of 2:
The given sequence is an AP.
So, The explicit formula will be
.
Maths-
What are the vertical and horizontal asymptotes of the graph of each 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.
x2 - 2x - 8 = 0
x2 + 2x - 4x - 8 = 0
x(x + 2)- 4(x + 2) = 0
(x + 2)(x - 4)=0
x= -2 or x= 4
The vertical asymptote of the rational function is x =−2 and x = 4
This function has x -intercept at (-2.386,0) and y -intercept at (0,1.125) . 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= -3 and x = -4. and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
=2What are the vertical and horizontal asymptotes of the graph of each function?
Maths-General
- 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.
x2 - 2x - 8 = 0
x2 + 2x - 4x - 8 = 0
x(x + 2)- 4(x + 2) = 0
(x + 2)(x - 4)=0
x= -2 or x= 4
The vertical asymptote of the rational function is x =−2 and x = 4
This function has x -intercept at (-2.386,0) and y -intercept at (0,1.125) . 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= -3 and x = -4. and horizontal asymptote is
y = (leading coefficient of numerator) / (leading coefficient of denominator) =
=2Maths-
Are graphs of the equations parallel, perpendicular or neither?
- We have been given two equations in the question for which we have to tell are graphs of the equations parallel, perpendicular or neither.
We have given two equations
Slope of both lines are -5, -5 respectively
Since slope are equal then both are parallel.
Are graphs of the equations parallel, perpendicular or neither?
Maths-General
- We have been given two equations in the question for which we have to tell are graphs of the equations parallel, perpendicular or neither.
We have given two equations
Slope of both lines are -5, -5 respectively
Since slope are equal then both are parallel.
Maths-
Tell whether the given sequence is an arithmetic sequence. 93,86,79,72,66,...
- We have given a sequence 93,86,79,72,66,....
- We have to find weather the given sequence is AP or not.
We have given sequence 93,86,79,72,66,....
The difference in first two terms is
Now the difference in next two terms is
Then, The difference between next two terms will be
Since the difference is not constant
The given sequence is not an arithmetic sequence.
Tell whether the given sequence is an arithmetic sequence. 93,86,79,72,66,...
Maths-General
- We have given a sequence 93,86,79,72,66,....
- We have to find weather the given sequence is AP or not.
We have given sequence 93,86,79,72,66,....
The difference in first two terms is
Now the difference in next two terms is
Then, The difference between next two terms will be
Since the difference is not constant
The given sequence is not an arithmetic sequence.
Maths-
Are graphs of the equations parallel, perpendicular or neither?
- We have been given two equations in the question for which we have to tell are graphs of the equations parallel, perpendicular or neither.
We have given two equations
Slope of both lines are 2,
respectively
Since slope are not equal then both are not parallel.
Product of both slope is 
So, both are not perpendicular also.
So,
Both are nor parallel neither perpendicular.
Are graphs of the equations parallel, perpendicular or neither?
Maths-General
- We have been given two equations in the question for which we have to tell are graphs of the equations parallel, perpendicular or neither.
We have given two equations
Slope of both lines are 2, respectively
Since slope are not equal then both are not parallel.
Product of both slope is
So, both are not perpendicular also.
So,
Both are nor parallel neither perpendicular.
Maths-
Tell whether the given sequence is an arithmetic sequence. 37,3,31,29,26,23,...
- We have given a sequence 37,34,31,29,26,23,...
- We have to find weather the given sequence is AP or not.
We have given sequence 37,34,31,29,26,23,...
The difference in first two terms is
Now the difference in next two terms is
Then, The difference between next two terms will be
Since the difference is constant
The given sequence is an arithmetic sequence.
Tell whether the given sequence is an arithmetic sequence. 37,3,31,29,26,23,...
Maths-General
- We have given a sequence 37,34,31,29,26,23,...
- We have to find weather the given sequence is AP or not.
We have given sequence 37,34,31,29,26,23,...
The difference in first two terms is
Now the difference in next two terms is
Then, The difference between next two terms will be
Since the difference is constant
The given sequence is an arithmetic sequence.
Maths-
What are the horizontal asymptotes for the graph
- 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.
x2 + 7x + 12 = 0
x2 + 3x + 4x + 12 = 0
x(x + 3) + 4(x + 3) = 0
(x + 3) (x + 4) = 0
x = -3 or x = -4
The vertical asymptote of the rational function is x =−3 and x = -4
This function has x -intercept at (4,0) and y -intercept at (0,7) . 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 = -3 and x = -4.
What are the horizontal asymptotes for the graph
Maths-General
- 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.
x2 + 7x + 12 = 0
x2 + 3x + 4x + 12 = 0
x(x + 3) + 4(x + 3) = 0
(x + 3) (x + 4) = 0
x = -3 or x = -4
The vertical asymptote of the rational function is x =−3 and x = -4
This function has x -intercept at (4,0) and y -intercept at (0,7) . 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 = -3 and x = -4.
Maths-
Tell whether the given sequence is an arithmetic sequence. 3,6,9,15,18,...
- We have given a sequence 3,6,9,15,18,...
- We have to find weather the given sequence is AP or not.
We have given sequence 3,6,9,15,18,...
The difference in first two terms is
Now the difference in next two terms is
Then, The difference between next two terms will be
Since the difference is not constant
The given sequence is not an arithmetic sequence.
Tell whether the given sequence is an arithmetic sequence. 3,6,9,15,18,...
Maths-General
- We have given a sequence 3,6,9,15,18,...
- We have to find weather the given sequence is AP or not.
We have given sequence 3,6,9,15,18,...
The difference in first two terms is
Now the difference in next two terms is
Then, The difference between next two terms will be
Since the difference is not constant
The given sequence is not an arithmetic sequence.
Maths-
Tell whether the given sequence is an arithmetic sequence. 3,6,9,12,15,18,...
- We have given a sequence 3,6,9,12,15,18,...
- We have to find weather the given sequence is AP or not.
We have given sequence 3,6,9,12,15,18,...
The difference in first two terms is
Now the difference in next two terms is
Then, The difference between next two terms will be
Since the difference is constant
The given sequence is an arithmetic sequence.
Tell whether the given sequence is an arithmetic sequence. 3,6,9,12,15,18,...
Maths-General
- We have given a sequence 3,6,9,12,15,18,...
- We have to find weather the given sequence is AP or not.
We have given sequence 3,6,9,12,15,18,...
The difference in first two terms is
Now the difference in next two terms is
Then, The difference between next two terms will be
Since the difference is constant
The given sequence is an arithmetic sequence.
