21.06 Exponential functions (Extended)
State whether the following graphs show an exponential function:
Consider the graphs of the two exponential functions A and B:
One of the graphs is of y = 3^{x} and the other graph is of y = 6^{x}. Which is the graph of y = 3^{x}?
Consider the graphs of the two exponential functions R and S:
One of the graphs is of y = 3^{x} and the other graph is of y = 5^{x}. Which is the graph of y = 5^{x}?
Which function has its graph below that of the other when x < 0? Explain your answer.
Determine whether following are increasing or decreasing exponential functions:
y = 2^{x}
y = 2^{-x}
Consider the expression y = 4^{x}.
Complete the table of values:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y |
Sketch the graph.
Describe what happens to the y-values as x increases.
Describe what happens to the y-values as x decreases.
Does the curve cross the x-axis?
At what value of y does the graph cross the y-axis?
Use your graph to approximate the solution of the equation 4^x=3.
For each of the following functions:
Find the y-value of the y-intercept of the curve.
Complete the table of values:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y |
Find the horizontal asymptote of the curve.
Sketch the graph.
Consider the expression y = 10^{x}.
Can the value of y ever be negative?
Can the value of y ever be equal to 0?
As the values of x get larger and larger, what value does y approach?
As the values of x get smaller and smaller, what value does y approach?
Find the y-value of the y-intercept of the curve.
How many x-intercepts does the curve have?
Sketch the graph.
Use your graph to approximate the solution of the equation 10^x=2.
Consider the exponentials y = 2^{x} and y = 3^{x}:
Find the horizontal asymptote of each curve:
y = 3^{x}
Complete a table of values for each exponetial:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y |
y = 3^{x}
Sketch the graphs of y = 2^{x} and y = 3^{x} on the same set of axes.
Find the coordinates of the point at which the two graphs intersect.
Consider the two equations y = 5^{ - x } and y = 6^{ - x }. Which is decreasing more rapidly for \\x > 0?
Consider the expression y = 2^{ - x }.
Complete the table of values:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y |
Sketch the graph.
Describe what happens to the y-values as x increases.
Describe what happens to the y-values as x decreases.
Does the curve cross the x-axis?
At what value of y does the graph cross the y-axis?
Use your graph to approximate the solution of the equation 2^{-x}=5.
For each of the following functions:
Find the y-value of the y-intercept of the curve.
Complete the table of values:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y |
Find the horizontal asymptote of the curve.
Sketch the graph.
Consider the expression y = 9^{ - x }.
Can the value of y ever be negative?
Can the value of y ever be equal to 0?
As the values of x get larger and larger, what value does y approach?
As the values of x get smaller and smaller, what value does y approach?
Find the y-value of the y-intercept of the curve.
How many x-intercepts does the curve have?
Sketch the graph.
Use your graph to approximate the solution of the equation 9^{-x}=5.
Consider the exponentials y = 5^{-x} and y = 8^{-x}:
Find the horizontal asymptote of each curve:
y = 8^{-x}
Complete a table of values for each exponetial:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y |
y = 8^{-x}
Sketch the graphs of y = 5^{-x} and y = 8^{-x} on the same set of axes.
Find the coordinates of the point at which the two graphs intersect.
How could we use the graph of y = 7^{x} to draw the graph of y = 7^{ - x }?
Consider the graph of y = 2^{x}:
Describe a transformation of the graph of y = 2^{x} that would obtain y = 2^{ - x }.
Sketch the graph of y = 2^{ - x } on the same set of axes as y = 2^{x}.
Consider the graph of y = 5^{x}:
Describe a transformation of the graph of y = 5^{x} that would obtain y = - 5^{x}.
Sketch the graph of y = - 5^{x} on the same set of axes as y = 5^{x}.
Consider the graph of y = 8^{x}:
Describe a transformation of the graph of y = 8^{x} that would obtain y = - 8^{ - x }.
Sketch the graph of y = - 8^{ - x } on the same set of axes as y = 8^{x}
Consider the graph of y = 6^{ - x }:
Describe a transformation of the graph of y = 6^{ - x } that would obtain y = - 6^{ - x }.
Sketch the graph of y = - 6^{ - x } on the same set of axes as y = 6^{ - x }.
Describe the graph of y = - 6^{ - x } as x increases.
Of the two functions y = 4^{x} and y = \dfrac{4^{x}}{4}, which is increasing more rapidly for x > 0?
Consider the function y = 3 \left(4^{x}\right).
Find the y-value of the y-intercept of the curve.
Can the value of y ever be negative?
As x approaches infinity, what value does y approach?
Sketch the graph of y = 3 \left(4^{x}\right).
Consider the function y = 3 \times 2^{x}.
Find the value of y when x = 0.
Describe a transformation of the graph of y = 3 \times 2^{x} that would obtain \\ y = - 3 \times 2^{x}.
Sketch the graph of y = - 3 \times 2^{x} on the same set of axes as y = 3 \times 2^{x}.
The number of bacteria over time is to be modelled by an exponential function, with x representing time and y representing the number of bacteria.
If the bacteria are increasing, which function should be used?
Consider the original graph y = 2^{x}. The function values of the graph are multiplied by 3 to form a new graph.
Complete the following table by finding the points on the new graph:
| Point on original graph | \left(-1,\dfrac{1}{2}\right) | \left(0,1\right) | \left(1,2\right) | \left(2,4\right) |
|---|---|---|---|---|
| Point on new graph | \left(-1,β¬ \right) | \left(0,β¬\right) | \left(1,β¬\right) | \left(2,β¬\right) |
Find the equation of the new graph.
Sketch the graphs of the original and new function on the same set of axes.
For positive x-values, is the graph of the new function above or below the graph of 2^{x}?
For negative x-values, is the graph of the new function above or below the graph of 2^{x}?
Consider the graph of y = 6^{x}:
Find the y-value of the y-intercept of y = 6^{x}.
Hence, find the y-value of the y-intercept of y = 6^{x} + 3.
Find the horizontal asymptote of y = 6^{x}.
Hence, find the horizontal asymptote of y = 6^{x} + 3.
Consider the graph of y = 2^{x}:
Describe the translation required to shift the graph of y = 2^{x} to obtain the graph of y = 2^{x} - 5.
Sketch the graph of y = 2^{x} - 5 on the same set of axes as y = 2^{x}.
Consider the graph of y = 3^{x}:
Describe the translation required to shift the graph of y = 3^{x} to obtain the graph of y = 3^{x} + 1.
Sketch the graph of y = 3^{x} + 1 on the same set of axes as y = 3^{x}.
Consider the graph of y = 3^{ - x }:
Describe the translation required to shift the graph of y = 3^{ - x } to obtain the graph of y = 3^{ - x } + 5.
Sketch the graph of y = 3^{ - x } + 5 on the same set of axes as y = 3^{ - x }.
Consider the graph of y = 3^{x}:
Describe the translation required to shift the graph of y = 3^{ x } to obtain the graph of y = 3^{ x -2} .
Sketch the graph of y = 3^{ x -2} on the same set of axes as y = 3^{ x }.
Consider the graph of y = 2^{x}:
Describe the translation required to shift the graph of y = 2^{ x } to obtain the graph of y = 2^{ x +3} .
Sketch the graph of y = 2^{ x +3} on the same set of axes as y = 2^{ x }.
Consider the graph of y = 4^{-x}:
Describe the translation required to shift the graph of y = 4^{ -x } to obtain the graph of y = 4^{ -x +1} .
Sketch the graph of y = 4^{ -x +1} on the same set of axes as y = 4^{ - x }.
Consider the graph of y = 6^{-x}:
Describe the translation required to shift the graph of y = 6^{ -x } to obtain the graph of y = 6^{ -x -4} .
Sketch the graph of y = 6^{ -x -4} on the same set of axes as y = 6^{ - x }.
State the new equation when the graph of y = 2^{x} is moved down by 9 units.
State the new equation when the graph of y = 12^{x} is moved left by 6 units.
State the new equation when the graph of y = 3^{-x} is moved right by 2 units and down by 7 units.