5.04 Characteristics of exponential functions
Consider the exponential functions shown in the following tables of values:
| x | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | 1 | 3 | 9 | 27 | 81 |
| x | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | \dfrac{2}{5} | 2 | 10 | 50 | 250 |
| x | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | \dfrac{5}{6} | \dfrac{5}{2} | \dfrac{15}{2} | \dfrac{45}{2} | \dfrac{135}{2} |
| x | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | 4000 | 400 | 40 | 4 | 0.4 |
State whether the following are increasing or decreasing exponential functions:
y = \left(\dfrac{1}{5}\right)^{x}
y = 9 \times 3^{x}
y = 3 \times \left(\dfrac{1}{2}\right)^{x}
y = \dfrac{1}{3} \times 2^{x}
Select the exponential function that represents the following graph.
Select the exponential function that represents the following graph.
Consider the table of values:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 3 | \dfrac{3}{4} | \dfrac{3}{16} | \dfrac{3}{64} | \dfrac{3}{256} |
Select the exponential function that could represent the table.
Consider the table of values:
| x | -4 | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|
| y | 112 | 56 | 28 | 14 | 7 | 3\dfrac{1}{2} | 1\dfrac{3}{4} | \dfrac{7}{8} | \dfrac{7}{16} |
Select the exponential function that could represent the table.
Consider the graph of the equation y = 4^{x}:
State the equation of the horizontal asymptote.
Explain what the horizontal asymptote means for an exponential function.
Do either of the functions y = 9^{x} or y = \left(\dfrac{1}{9}\right)^{ x } have x-intercepts? Explain your answer.
Draw the graphs of the functions y = \left(\dfrac{1}{2}\right)^{x},y = \left(\dfrac{1}{3}\right)^{x} and y = \left(\dfrac{1}{5}\right)^{x}. Then answer the following questions:
State whether the following statements are true for all of the functions:
All of the curves have a maximum value.
All of the curves pass through the point \left(1, 2\right).
All of the curves have the same y-intercept.
None of the curves cross the x-axis.
Describe what happens to the values of y as x gets increasingly larger.
Consider the graph of the functions y = 3^{x} and y = \left(\dfrac{1}{3}\right)^{ x }:
State the coordinates of the point of intersection of the two curves.
Describe what happens to the values of y for each function as x gets increasingly larger.
Consider the table of values for the function y = \left(\dfrac{1}{2}\right)^{ x }.
| x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| y | 32 | 16 | 8 | 4 | 2 | 1 | \dfrac{1}{2} | \dfrac{1}{4} | \dfrac{1}{8} | \dfrac{1}{16} | \dfrac{1}{32} | \dfrac{1}{1024} |
Describe the behavior of the function as x increases.
Determine the y-intercept of the function.
State the domain of the function.
State the range of the function.
Consider the function y = 4 \left(2^{x}\right).
Find the y-value of the y-intercept of the curve.
Can the function values ever be negative?
As x approaches infinity, determine the value that y approaches.
Graph y = 4 \left(2^{x}\right).
List the domain and range for the function.
Consider the function y = 3 \times \left(\dfrac{1}{2}\right)^{ x }.
Find the y-value of the y-intercept of the curve.
Complete the table of values for y = 3 \times \left(\dfrac{1}{2}\right)^{ x } .
| x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
Find the horizontal asymptote of the curve.
Graph y = 3 \times \left(\dfrac{1}{2}\right)^{ x }.
List the domain and range of the function.
Explain how the answers to part (a) for questions 8 and 9 relate to the functions and the general form of an exponential function f(x)=a\left(b\right)^x.
Compare the domain and range for f(x)=4(2)^x and f(x)=3\left(\dfrac{1}{2}\right)^x from questions 8 and 9. What do you notice?
Consider the given graph of y = 5^{x}.
Describe a transformation of the graph of y = 5^{x} that would obtain y = - 5^{x}.
Graph y = 5^{x} and y = - 5^{x} on the same coordinate plane.
Consider the graphs of the two exponential functions R and S:
One of the graphs is of y = 4^{x} and the other graph is of y = 6^{x}.
Identify which is the graph of y = 6^{x}. Explain your answer.
For x < 0, is the graph of y = 6^{x} above or below the graph of y = 4^{x}? Explain your answer.
Consider the function y = \left(\dfrac{1}{2}\right)^{x}.
State whether the following functions are equivalent to y = \left(\dfrac{1}{2}\right)^{x}:
y = \dfrac{1}{2^{x}}
y = 2^{ - x }
y = - 2^{x}
y = - 2^{ - x }
Describe a trasformation that would obtain the graph of y = \left(\dfrac{1}{2}\right)^{x} from the graph of y =2^{x}.
Graph the functions y = 2^{x} and y = \left(\dfrac{1}{2}\right)^{x} on the same coordinate plane.
Consider the equation y = - 10^{x}.
Jenny thinks she has found a set of solutions for the equation as shown in the table:
| x | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y | - \dfrac{1}{100} | - \dfrac{1}{10} | -1 | -10 | -100 | -1000 |
She notices that all the y values are negative and concludes that for any value of x, y must always be negative. Is she correct? Explain your answer.
Graph y = - 10^{x}.
Find the values of x for which y = 0.
Consider the original graph y = 3^{x}. The function values of the graph are multiplied by 2 to form a new graph.
For each point on the original graph, find the point on the new graph.
| Point on original graph | \left(-1, \dfrac{1}{3}\right) | \left(0, 1\right) | \left(1, 3\right) | \left(2, 9\right) |
|---|---|---|---|---|
| Point on new graph | \left(-1, ⬚ \right) | \left(0, ⬚ \right) | \left(1, ⬚ \right) | \left(2, ⬚ \right) |
State the equation of the new graph.
Graph the functions y = 3^{x} and y = 2 \times 3^{x} on the same coordinate plane.
For negative x-values, is the graph of y = 2 \times 3^{x} above or below y = 3^{x}?
For positive x-values, is the graph of y = 2 \times 3^{x} above or below 3^{x}?