y = a sin[ b ( x - d) ] + c or y = a cos [ b ( x - d) ] + c
given its graph.
Find a Sinusoidal Function for Each of the Graphs Below
Question 1
Solution to Question 1
The scaling along the y-axis is one unit for one large division and therefore the maximum value of y: ymax = 1 and the minimum value of y: ymin = - 7.
The scaling along the x axis is ? for one large division and ?/5 for one small division.
Points A and B mark the start and the end of one period P which is equal to 5?. These points are useful because they are maximum points with clear coordinates.
Since A and B are maximum points, it is easier to write an equation for the graph as y = a cos[ b(x - d) ] + c assuming that originally it is a cos(x), which starts with a maximum at x = 0, that is transformed by vertical and horizontal shifting (translation) and vertical and horizontal stretching/shringking.
Let us calculate a, and c.
|a| = (ymax - ymin) = (1 - (-7)) / 2 = 4. Which gives two possible values for a: a = 4 or a = - 4
The graph between A and B has no reflection when compared to the period of cos(x) between 0 and 2? and we may therefore take a = 4.
c = (ymax + ymin) = (1 + (-7)) / 2 = - 3
Period: P = 2? / |b|= 5?
solve the above for |b| to obtain: |b| = 2/5.
Again here we have two possible values for b: b = 2/5 and b = -2/5. We take b = 2/5 to make our calculations for d easier.
We now write the function for the graph as follows:
y = 4 cos[ (2/5)(x - d) ] - 3
d indicates the shift. The shift is determined by comparing the graphs of y = 4 cos[ (2/5)(x) ] - 3 ( note d = 0 ) and the given graph. We note that the shift (x coordinate of point A) d = - ?/5 from the graph (one small division the the left). Hence the equation of the graph is:
We now check that the function found corresponds to the given graph by checking few points.
Point A: x = -?/5; evaluate y at this value of x.
y( - ?/5)= 4 cos[ (2/5)( - ?/5 + ?/5) ] - 3 = 4 cos[ (2/5)( 0 ) ] - 3 = 1 which corresponds to the value on the graph.
Point B: x = 4? + 4 ?/5 = 24?/5 ( 4 small division after 4?)
y( 24?/5 )= 4 cos[ (2/5)( 24?/5 + ?/5) ] - 3 = 4 cos[ (2/5)( 25?/5) ] - 3 = 4 cos[ (2/5)( 5?) ] - 3 = 4 cos[ (2?) ] - 3 = 1
which corresponds to the value on the graph.
Question 2
Solution to Question 2
Maximum value of y: ymax = 0.2 and the minimum value of y: ymin = - 1.4 (one large division along the y axis is equal to 1 unit. A small division is 1/5 = 0.2).
The scaling along the x axis is ? for one large division and ?/5 for one small division.
Points A and B mark the start and the end of one period P which is equal to 4?.
The coordinates of points A and B are: A(?/2 , 0.2 ) , B(9?/2 , 0.2).
The graph between A and B may be assumed that of a cos(x) that has been transformed. Hence a possible equation for the given graph is: y = a cos[ b(x - d) ] + c.
Let us calculate a, and c.
|a| = (ymax - ymin) = (0.2 - (-1.4)) / 2 = 0.8. Which gives two possible values for a: a = 0.8 or a = - 0.8
The period between A and B has no reflection when compared to the period of cos(x) between 0 and 2? and we may therefore take a = 0.8.
c = (ymax + ymin) = (0.2 + (-1.4)) / 2 = - 0.6
Period: P = 2? / |b|= 4?
solve for |b| to obtain: |b| = 1/2. Again here we have two possible values for b: b = 1/2 and b = - 1/2. We take b = 1/2 to make our calculations for d easier.
We now write the function for the graph as follows:
y = 0.8 cos[ (1/2)(x - d) ] - 0.6
d indicates the shift. The shift is determined by comparing the graphs of y = 0.8 cos[ (1/2)(x) ] - 0.6 ( note d = 0 ) and the given graph. We note that the shift (x coordinate of point A) d = ?/2 from the graph (one half a large division the the right). Hence the equation of the graph is:
y = 0.8 cos[ (1/2)(x - ?/2) ] - 0.6
Check Answer Found
We now check that the function found corresponds to the given graph by checking few point.
Point A: x = ?/2; evaluate y at this value of x.
y(?/2)= 0.8 cos[ (1/2)(?/2 - ?/2) ] - 0.6 = 0.8 cos (0) - 0.6 = 0.2 , which corresponds to the value on the graph.
Point B: x = 4? + ?/2 = 9?/2 ( half a large division after 4?)
y( 9?/2 )= 0.8 cos[ (1/2)(9?/2 - ?/2) ] - 0.6 = 0.8 cos (2?) - 0.6 = 0.2
which is equal to the value on the graph.
Question 3
Solution to Question 3
Maximum value of y: ymax = 0 and the minimum value of y: ymin = - 2 (one large division along the y axis is equal to 1 unit).
The scaling along the x axis is 1 unit for one large division and 1/5 = 0.2 for one small division.
Points A and B mark the start and the end of one period P which is calculated as follows: P = 2.6 - 0.6 = 2.
The coordinates of points A and B are: A(0.6 , 0 ) , B(2.6 , 0).
The graph between A and B may be assumed that of a sin(x) that has been transformed. Hence a possible equation for the given graph is: y = a sin[ b(x - d) ] + c.
Let us calculate a, and c.
|a| = (ymax - ymin) = (0 - (-2)) / 2 = 1. Which gives two possible values for a: a = 1 or a = - 1
The period between A and B has no reflection when compared to the period of sin(x) between 0 and 2? and we may therefore take a = 1.
c = (ymax + ymin) = (0 + (-2)) / 2 = - 1
Period: P = 2? / |b|= 2;
solve for |b| to obtain: |b| = ?. Again here we have two possible values for b: b = ? and b = - ?. We take b = ? to make our calculations for d simpler.
We now write the function for the graph as follows:
y = sin[ ?(x - d) ] - 1
d indicates the shift. The shift is determined by comparing the graphs of y = sin[ ?(x) ] - 1 ( note d = 0 ) and the given graph. We note the shift (x coordinate of A) d = 0.6 from the graph; shift to the right. Hence the equation of the graph is:
We now check that the function found corresponds to the given graph by checking few point.
Point A: x = 0.6; evaluate y at this value of x.
y( 0.6)= sin[ ?(0.6 - 3/5) ] - 1 = sin[ ?(0) ] - 1 = -1 , which corresponds to the value on the graph.
Point B: x = 1.6
y( 1.6)= sin[ ?(1.6 - 3/5) ] - 1 = sin(?) - 1 = -1 , which corresponds to the value on the graph.
Checking values at A and B is not enough because they would give same values if the function - sin[ ?x - 3?/5) ] - 1 was used. We need to check a maximum or a minimum beside A and B. The first maximum point after point A is at x = 1 + (1/2)0.2 = 1.1
y( 1.1)= sin[ ?(1.1 - 3/5) ] - 1 = sin (0.5 ?) - 1 = 0 , which corresponds to the value on the graph.
Question 4
Solution to Question 4
Maximum value of y: ymax = -1 and the minimum value of y: ymin = - 3 (one large division along the y axis is equal to 1 unit. A small division is 1/5 = 0.2). The scaling along the x axis is ?/5 for one large division and ?/25 for one small division.
Points A and B mark the start and the end of one period P which is equal to 8?/5 - 3?/5 = ?.
The coordinates of points A and B are: A(3?/5 , - 1) , B(8?/5 , - 1).
The period between A and B may be considered as that of a cos(x) that has been transformed. Hence a possible equation for the given graph is: y = a cos[ b(x - d) ] + c.
Let us calculate a, and c.
|a| = (ymax - ymin) = (-1 - (-3)) / 2 = 1. Which gives two possible values for a: a = 1 or a = - 1 .
The period between A and B has no reflection when compared to the period of cos(x) between 0 and 2? and we may therefore take a = 1.
c = (ymax + ymin) = (-1 + (-3)) / 2 = - 2
Period: P = 2? / |b|= ?
solve for |b| to obtain: |b| = 2. Two possible values for b: b = 2 and b = - 2. We take b = 2 to make our calculations for d easier.
We now write the function for the graph as follows:
y = cos[ 2(x - d) ] - 2
The x coordinate of point A indicates the shift d which is determined by comparing the graphs of y = cos[ 2(x) ] - 2 ( note d = 0 ) and the given graph. We note that d = 3?/5 from the graph. Hence the equation of the graph is:
y = cos[ 2(x - 3?/5) ] - 2
Check Answer Found
We now check that the function found corresponds to the given graph by checking few points.
Point A: x = 3?/5 evaluate y at this value of x.
y( 3?/5 )= cos[ 2(3?/5 - 3?/5) ] - 2 = cos(0) - 2 = - 1 , which corresponds to the value on the graph.
Point B: x = 8?/5
y( 8?/5 )= cos[ 2(8?/5 - 3?/5) ] - 2 = cos (2?) - 2 = - 1
which is equal to the value on the graph.
