3.5 Transformations of Exponential Functions
Chapter
Chapter 3
Section
3.5
Solutions 39 Videos

Complete the second column of the table by describing the transformation associated with the parameter described in the first column.

Q1

Describe the transformation that maps the function y = 5^x onto each function.

y = 5^x +3

Q2a

Describe the transformation that maps the function y = 5^x onto each function.

y = 5^{x - 2}

Q2b

Describe the transformation that maps the function y = 5^x onto each function.

y = 5^{x + 1}

Q2c

Describe the transformation that maps the function y = 5^x onto each function.

y = 5^{x - 4} - 6

Q2d

Sketch a graph. Use the graph of y = 5^x as the base.

y = 5^x + 3

Q3a

Sketch a graph. Use the graph of y = 5^x as the base.

y = 5^{x-2}

Q3b

Sketch a graph. Use the graph of y = 5^x as the base.

y = 5^{x+1}

Q3c

Sketch a graph. Use the graph of y = 5^x as the base.

y = 5^{x-4} - 6

Q3d

Describe one or more transformations that map the function y = 7^x onto each function.

y = (\dfrac{1}{3})7^x

Q4a

Describe one or more transformations that map the function y = 7^x onto each function.

y = 7^{2x}

Q4c

Describe one or more transformations that map the function y = 7^x onto each function.

y = -7^x

Q4d

Describe one or more transformations that map the function y = 7^x onto each function.

y = 7^{-\frac{1}{3}x}

Q4b

Sketch a graph. Use the graph of y = 7^x as the base. Be sure to choose an appropriate scale for your axes.

y = (\dfrac{1}{3})7^x

Q5a

Sketch a graph. Use the graph of y = 7^x as the base. Be sure to choose an appropriate scale for your axes.

y = 7^{2x}

Q5b

Sketch a graph. Use the graph of y = 7^x as the base. Be sure to choose an appropriate scale for your axes.

y = -7^x

Q5c

Sketch a graph. Use the graph of y = 7^x as the base. Be sure to choose an appropriate scale for your axes.

y = 7^{-\frac{1}{3}x}

Q5d

Write an equation for the function that results from each transformation applied to the base y = 11^x.

reflection in the y-axis

Q6a

Write an equation for the function that results from each transformation applied to the base y = 11^x.

stretch vertically by a factor of 4

Q6b

Write an equation for the function that results from each transformation applied to the base y = 11^x.

stretch horizontally by a factor of 3

Q6c

Sketch a graph of y = (-\dfrac{1}{5})5^{x + 4} - 2 by using y = 5^x as the base and applying transformations.

Q8

a) Describe the transformations that must be applied to the graph of  f(x) = 3^x to obtain the transformed function y = -f(4x) - 7. Then write the corresponding equation.

b) State the domain. range. and equation of the horizontal asymptote.

Q9

a) Write the equation of the function that represents f(x) = 2^x after it is reflected in the x-axis. stretched horizontally by a factor of 5, reflected in the y-axis, and then translated down 3 units and right 1 unit.

b) State the domain and range, and the equation of its horizontal asymptote.

Q10

The temperature, in degrees Celsius, of a cooling metal bar is given by the function T = 18 + 100(0.5)^{0.3t}, where t is the time, in minutes.

Sketch a graph of this relation.

Q11a

The temperature, in degrees Celsius, of a cooling metal bar is given by the function T = 18 + 100(0.5)^{0.3t}, where t is the time, in minutes.

What is the asymptote of this function? What does it represent?

Q11b

The temperature, in degrees Celsius, of a cooling metal bar is given by the function T = 18 + 100(0.5)^{0.3t}, where t is the time, in minutes.

How long will it take for the temperature to be within 0.1 ^{\circ}C of the value of the asymptote?

Q11c

a) Rewrite the function y = (\dfrac{1}{16})^x in three different ways, using a different base in each case.

b) State the base function for each equation and describe how the base function is transformed.

Q12

a) Write an equation for a function whose asymptote is y = -5, with a y-intercept of 3.

b) Is the function you produced in part a) the only possible answer? Use transformations to help explain your answer.

Q13

A 250-g sample of one type of radioactive substance has a half-life of 138 days. A 175-g sample of another type of radioactive substance has a half-life of 16 days.

For each substance, write an equation that represents the amount A, in grams, of the radioactive sample that remains after t days.

Q14a

A 250-g sample of one type of radioactive substance has a half-life of 138 days. A 175-g sample of another type of radioactive substance has a half-life of 16 days.

What is the base function for each equation?

Q14b

A 250-g sample of one type of radioactive substance has a half-life of 138 days. A 175-g sample of another type of radioactive substance has a half-life of 16 days.

Describe the transformations that must be applied to the base function to obtain each equation.

Q14c

A 250-g sample of one type of radioactive substance has a half-life of 138 days. A 175-g sample of another type of radioactive substance has a half-life of 16 days.

Use a negative exponent to write an equivalent equation for each of the equations in part a).

i) What is the base function?

ii) Which transformations that must be applied to this base function are the same as those in part c)?

iii) Which new transformation is required?

Q14d

A 250-g sample of one type of radioactive substance has a half-life of 138 days. A 175-g sample of another type of radioactive substance has a half-life of 16 days.

For each substance, determine the mass that remains after 20 weeks.

Q14e

A 250-g sample of one type of radioactive substance has a half-life of 138 days. A 175-g sample of another type of radioactive substance has a half-life of 16 days.

How long does it take for each sample to decay to 10% of its original amount?

Q14f

Refer to question 14. Suppose there is an initial amount of A_0 grams of a radioactive substance that has a half-life of h days.

Write a general equation to represent the amount A, in grams, that remains after t days.

Q15a

Refer to question 14. Suppose there is an initial amount of A_0 grams of a radioactive substance that has a half-life of h days.

Describe the role of A_0 and h in the equation in terms of transformations.

Q15b

Refer to question 14. Suppose there is an initial amount of A_0 grams of a radioactive substance that has a half-life of h days.

Rewrite the equation in part a) so that it includes a reflection in the y-axis.

Q15c

Write an equation for the function that results from each transformation applied to the base y = 11^x.

reflect in the x-axis and compress horizontally by a factor of \dfrac{1}{5}

The graph of y= 4^x is transformed to obtain the graph of y = -3[4^{2(x +1)}] + 5.
a) State the parameters. Describe the corresponding transformations and use these to complete the table. Then use the points to graph y = -3[4^{2(x +1)}] + 5.