Consider the functions y = 2^{x}, y = 3^{x} and y = 5^{x}.
Determine whether each of the following statements are true:
None of the curves cross the x-axis.
They all have the same y-intercept.
All of the curves pass through the point \left(1, 2\right).
All of the curves have a maximum value.
State the y-intercept of each curve.
Consider the function y = - \left(2^{x}\right).
Complete the table of values:
| x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| y |
Can the value of y ever be zero or positive? Explain your answer.
Is y = - \left(2^{x}\right) an increasing or decreasing function?
Describe the behaviour of the function as x increases.
State the domain.
State the range.
Find the missing coordinate in each ordered pair so that the pair is a solution of y = - 3^{x}:
\left(5, ⬚\right)
\left(⬚, - \dfrac{1}{27} \right)
\left( - 1 , ⬚\right)
\left(⬚, - 81 \right)
Determine the y-intercept of all exponential functions of the form:
y = a^{x}
y = a^{ - x }
y = - a^{x}
y = - a^{-x}
Consider the function y = - 2.5 \times 4^{x}.
Is y = - 2.5 \times 4^{x} an increasing or decreasing function?
As x approaches -\infty, what value does y approach?
As x approaches \infty, y what value does y approach?
State the y-intercept of the graph.
Determine whether following are increasing or decreasing exponential functions:
y = 10\times\left(\dfrac{3}{5}\right)^{x}
y = 9 \times 3^{x}
y = 3 \times \left(0.5\right)^{x}
y = 0.2 \times 2^{x}
y = 2.2 \times 1.05^{x}
y = 50\times\left(\dfrac{7}{4}\right)^{x}
y = 500 \times 0.75^{x}
y = 80 \times 2^{-x}
y = -5 \times 2^{x}
y = -0.5 \times 0.8^{x}
y = -2\times\left(\dfrac{1}{4}\right)^{x}
y = -4 \times 3^{-x}
Which of the following graphs of exponential functions rises most steeply?
y = 2 \times \left(2.1\right)^{x}
y = 2 \times \left(2.2\right)^{x}
y = 2 \times \left(1.7\right)^{x}
y = 2 \times \left(1.2\right)^{x}
Of the two functions y = 2^{x} and y = 3 \times 2^{x}, which is increasing more rapidly for x > 0?
Consider the functions y = 2^{x} and y = 2^{x} - 2.
Find the y-intercept of y = 2^{x}.
Hence, determine the y-intercept of y = 2^{x} - 2.
State the horizontal asymptote of y = 2^{x}.
Hence, determine the horizontal asymptote of y = 2^{x} - 2.
For each of the following exponential functions:
Find the y-intercept.
Determine the horizontal asymptote.
Sketch the graph of the function.
For each of the following exponential functions:
Find the y-intercept.
State the domain.
State the range.
Sketch the graph of the function.
For each of the following exponential functions:
Find the y-intercept.
Create a table of values for -3 \leq x \leq 3.
Determine the horizontal asymptote.
Sketch the graph of the function.
Is the function increasing or decreasing?
Consider the function y = 8^{ - x } + 6.
What value is 8^{ - x } always greater than?
Hence, what value is 8^{ - x } + 6 always greater than?
How many x-intercepts does y = 8^{ - x } + 6 have?
State the equation of the asymptote of the curve y = 8^{ - x } + 6.
State the domain.
State the range.
Consider the function y = 2 \left(3^{x}\right).
Find the y-intercept.
Can the function value ever be negative? Explain your answer.
State the domain.
State the range.
As x approaches positive infinity, what value does y approach?
Sketch the graph of the function.
Consider the function y = - 10^{x}.
Complete the following table:
| x | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y |
For any value of x, is y always negative for this equation? Explain your answer.
Sketch the graph of the function.
For each of the following functions describe transformations that can be used to obtain the graph of f \left( x \right) from the graph of y=2^x:
Consider a graph of y = 6^{ 3 x}.
A horizontal dilation by what factor takes the graph of y=6^{x} to y=6^{ 3 x}?
Determine whether the following functions are equivalent to y = 6^{ 3 x}:
Sketch the graph of y = 6^{ 3 x}.
Consider a graph of y = 9^{\frac{x}{2}}.
A horizontal dilation by what factor takes the graph of y=9^{x} to y=9^{\frac{x}{2}}?
Determine whether the following functions are equivalent to y =9^{\frac{x}{2}}:
Sketch the graph of y=9^{\frac{x}{2}}.
Consider the graph of y = 2^{x+5}.
Using index laws, rewrite 2^{x + 5} as A \times 2^{x}.
Hence, for the graph of y=2^x, a horizontal translation left by 5 units is equivalent to a vertical dilation by what factor?
Sketch the graph of y = 2^{x + 5}.
Consider the graph of y = 5^{x-1}.
Using index laws, rewrite 5^{x - 1} as A \times 5^{x}.
Hence, for the graph of y=5^x, a horizontal translation right by 1 unit is equivalent to a vertical dilation by what factor?
Sketch the graph of y = 5^{x - 1}.
Consider the graph of y = 3^{ 2 x + 3}.
Determine whether the following functions are equivalent to y = 3^{ 2 x + 3}:
State the domain.
State the range.
Consider the given graph of y = - 3^{x}:
State the asymptote of y = - 3^{x}.
Hence, find the asymptote of y = 2 - 3^{x}.
How many x-intercepts would \\ y = 2 - 3^{x} have?
State the domain of y = 2 - 3^{x}.
State the range of y = 2 - 3^{x}.
Consider the given graph of y = 4^{x}:
Describe a transformation of the graph y = 4^{x} that would obtain y = 4^{ - x }.
Sketch the graph of y = 4^{ - x }.
Consider the given 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}.
Consider the given graph of y = 3^{x}:
Describe a transformation of the graph of y = 3^{x} that would obtain y = 3^{x} - 4.
Sketch the graph of y = 3^{x} - 4.
Consider the given graph of y = 3^{x}:
Describe a transformation of the graph of y = 3^{x} that would obtain y = 3^{x - 2}.
Skecth the graph of y = 3^{x - 2}.
Consider the given graph of y = 5^{x}:
Describe a transformation of the graph of y = 5^{x} that would obtain y = 5^{x - 3}.
Sketch the graph of y = 5^{x - 3}.
Consider the given graph of y = 3^{ - x }:
Describe a transformation of the graph of y = 3^{ - x } that would obtain \\ y = 3^{ - x } + 2.
Sketch the graph of y = 3^{ - x } + 2.
Consider the given graph of y = 4^{ - x }:
Describe a transformation of the graph of y = 4^{ - x } that would obtain y = 4^{ - \left( x + 3 \right)}.
Sketch the graph of y = 4^{ - \left( x + 3 \right)}.
Determine whether each of the following exponential functions has a range of y < 5:
y = -4 \left(2^{ - x }\right) + 5
y = 5 + 2^{ - x }
y = - 4 \left(2^{x-5}\right)
y = - 2^{x} + 5
y = - 2^{x} - 5
y = 2^{x} - 5
Find the equation of the new function that results from the following transformations:
The graph of y = 2^{x} is translated down by 7 units.
The graph of y = 11^{x} is moved to the left by 10 units.
For each of the following, write the equation of the new function that results from y = 7^{x} after the following translations:
Consider the function y = 4^{x}. Find the equation of the new function that results from the following transformations:
The function is first reflected about the x-axis.
This new function is then multiplied by - 2.
Consider the function y = 6^{x}.
Find the equation of the new function that results from the following transformations:
The function is dilated by a factor of 5 vertically.
This new function is then translated 3 units up.
Find the horizontal asymptote of the new function.
Find the y-intercept of the new function.
Hence, sketch the graph of the new function.
Consider the function y = 5^{x}.
Find the equation of the new function that results from y = 5^{x} being translated 3 units to the right and 5 units downwards.
Find the horizontal asymptote of the new function.
Find the y-intercept of the new function.
Find the x-intercept of the new function.
Hence, sketch the graph of the new function.
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 | Point on new graph |
|---|---|
| \left(-1,\dfrac{1}{3}\right) | (-1,⬚) |
| (0,1) | (0,⬚) |
| (1,3) | (1,⬚) |
| (2,9) | (2,⬚) |
State the equation of the new graph.
Graph the original and new graph on the same set of axes.
Describe the postion of new graph in relation to the original graph.
The graph of the exponential function P, given by y = 4^{ - x } is shown on the right:
Sketch the graph of Q given by the equation y = 8 x + 12 on the same set of axes as P.
How many times do P and Q intersect?
Which function has the higher function value at the y-intercept?
Which function has the higher function value at x = 1?
The graph of the exponential function P, given by y = - 4^{x} is shown on the right:
Sketch the graph of Q given by the equation y = - \dfrac{4 \left(x - 3\right) \left(x - 7\right)}{21} on the same set of axes as P.
How many times do P and Q intersect?
Which function has the higher function value at the y-intercept?
Describe the behaviour of functions P and Q on the domain 0 < x < 5.
Consider the graph of exponential function P given by y = 2^{x} + 2 and a table of values for parabola Q given by \\ y = - 3 \left(x + 5\right) \left(x + 2\right) + 2.
Sketch the graphs of Q and P on the same set of axes.
Describe the behaviour of function P and function Q as x approaches infinity.
Using the graph, determine how many solutions are there to the equation \\2^{x} + 2 + 3 \left(x + 5\right) \left(x + 2\right) - 2 = 0.
Parabola Q
| x | -5 | -4 | -3 | -2 | -1 |
|---|---|---|---|---|---|
| y | 2 | 8 | 8 | 2 | -10 |
Function P
Consider the exponential functions P given by y = 9 \left(3^{ - x }\right) and Q given by y = 5 \left(3^{ - x }\right).
Are the exponential functions P and Q increasing or decreasing?
Sketch the two functions on the same set of axes.
As x tends to \infty, what value does each function approach?
Describe a transformation of the graph y=3^{-x} that would obtain:
Consider the given graphs of the two exponential functions P and Q:
State whether the following pairs of equations could be the equations of the graphs P and Q:
P: \, y = 2^{x} \\ Q: \, y = 2^{ - x }
P: \, y = \left(3.5\right)^{x} \\ Q: \, y = 6^{ - x }
P: \, y = 2^{x} \\ Q: \, y = 5^{ - x }
P: \, y = 5^{x} \\ Q: \, y = 2^{ - x }
Consider the given graphs of f \left(x\right) = 3^{x} and g \left(x\right):
Describe a transformation that can be used to obtain g \left(x\right) from f \left(x\right).
State the equation of g \left(x\right).
The graph of f \left(x\right) = 9^{x} and another exponential function, g \left(x\right) is shown:
g(x) is increasing at exactly the same rate as f \left(x\right), but has a different y-intercept. Write down the equation of function g \left(x\right).
For each of the following graphs of exponential functions in the form y = a^{x}, state the equation of the function:
For each of the following graphs of exponential functions in the form y = a^{x} + k:
State the equation for the horizontal asymptote.
State the equation for the exponential function.
For each of the following graphs of exponential functions in the form y = a^{x - h}:
Find the value of h for the function.
State the equation of the function.
For each of the following graphs of exponential functions in the form y = a^{x - h} + k:
Find the value of k for the function.
Find the value of h for the function.
State the equation of the function.
For each of the given pair of points, find the equation of the exponential function of the form y =A\times a^{x} that passes through the points:
P\left(0,3\right) and Q\left(1,6\right)
P\left(1,30\right) and Q\left(2,90\right)
P\left(1,20\right) and Q\left(2,5\right)
P\left(1,30\right) and Q\left(2,10\right)
P\left(2,20\right) and Q\left(3,40\right)
P\left(2,-18\right) and Q\left(5,-486\right)