Consider the equation y = 5 \times 4^{x}.
Find the value of y when:
x = 0
x = 2
x = - 1
Consider the function f \left( x \right) = 2 + 3^{x}.
Evaluate f \left( 4 \right).
Evaluate f \left( - 3 \right).
Consider the function f \left( x \right) = 3^{x} + 3^{ - x }.
Evaluate f \left( 3 \right).
Evaluate f \left( - 3 \right).
Does f \left( 3 \right) = f \left( - 3 \right)?
Would an exponential function generate the values shown in the following tables?
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| f(x) | 5 | 25 | 125 | 625 | 3125 | 15\,625 |
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| f(x) | 4 | 9 | 11.5 | 15 | 13.5 | 11 |
What type of function would generate values as shown in the table?
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| f(x) | 2 | 3 | 4.8 | 7.6 | 11.8 | 18 |
approximately exponential
exactly exponential
not exponential
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 graph of the following functions y = 3^{x} and y = 3^{ - x }:
State the coordinates of the point of intersection of the two curves.
Describe the behaviour of both these functions for large values of x.
Consider the expression 3^{x}.
Evaluate the expression when x = - 4.
Evaluate the expression when x = 0.
Evaluate the expression when x = 4.
What happens to the value of 3^{x} as x gets larger?
What happens to the value of 3^{x} as x gets smaller?
For each of the following functions:
Complete the following table of values:
| x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| y |
State whether the function is an increasing or decreasing function.
Describe the rate of change of the function.
State the y-intercept of the curve.
Consider the functions y = 2^{x}, y = 3^{x} and y = 5^{x}.
Determine whether each of the following statements is 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?
A linear function and exponential function have been graphed on the following number plane:
Over a 1 unit interval of x, by what constant amount does the linear function grow?
Over a 1 unit interval of x, by what constant ratio does the exponential function grow?
Would it be correct to state that the linear function always produces greater values than the exponential function? Explain your answer.
As x approaches infinity, which function increases more rapidly?
Consider the graph of the equation y = 4^{x}:
Is each y-value of the function positive or negative?
State the value of y the graph approaches but does not reach.
State the equation and name of the horizontal line, which y = 4^{x} gets closer and closer to but never intersects.
Do either of the functions y = 9^{x} or y = 9^{ - x } have x-intercepts? Explain your answer.
Consider the function y = 6^{x}.
Can the value of y ever be negative?
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?
Can the value of y ever be equal to 0?
Find the y-intercept of the curve.
How many x-intercepts does the curve have?
Sketch the graph of y = 6^{x}.
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}.
Which is the graph of y = 6^{x}?
For x < 0, is the graph of y = 6^{x} above or below the graph of y = 4^{x}. Explain your answer.
Consider the graphs of the functions y = 4^{x} and y = 4^{ - x } below. Describe the rate of change for each function.
State whether the following are increasing or decreasing exponential functions:
Which of the following graphs of exponential functions rises most steeply?
Consider the function y = 4 \left(2^{x}\right).
Find the y-intercept of the curve.
Is the function value ever negative?
As x approaches infinity, what value does y approach?
Sketch the graph of y = 4 \left(2^{x}\right).
Consider the function y = 3^{ - x }.
Complete the table of values:
| x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| y |
Is y = 3^{ - x } an increasing function or a decreasing function?
Describe the rate of decrease of the function.
Find the y-intercept of the curve.
Sketch the graph of y = 3^{ - x }.
Consider the function y = - 3^{x}.
Complete the table of values:
| x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| y |
Can the function values ever be positive?
Can the function value ever be 0?
Is y = - \left(3^{x}\right) an increasing function or a decreasing function?
Describe the rate of decrease of the function.
Find the y-intercept of the curve.
Find the horizontal asymptote of the curve y = - 3^{x}.
Sketch the graph of y = 3^{ - x }.
Consider the function y = 8^{ - x }.
Can the value of y ever be negative?
As the value of x gets larger and larger, what value does y approach?
As the value of x gets smaller and smaller, what value does y approach?
Can the value of y ever be equal to 0?
Find the y-intercept of the curve.
How many x-intercepts does the curve have?
Sketch the graph of y = 8^{ - x }.
Given the function y = - 10^{x}, what is the largest possible function value? Explain your answer.
Consider the function y = - 5^{x}.
State the equation of the asymptote of y = - 5^{x}.
Hence, determine the equation of the asymptote of y = 2 - 5^{x}.
How many x-intercepts would the graph of y = 2 - 5^{x} have?
State the range of the following functions:
Consider the function y = - 6^{x}.
Find the y-intercept of the curve y = - 6^{x}.
Find the horizontal asymptote of the curve y = - 6^{x}.
Is the function y = - 6^{x}, an increasing or decreasing function?
Sketch the graph of y = - 6^{x}.
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.
Sketch the curve of y = - 10^{x}.
Find the values of x for which y = 0.
State whether the following statements are true or false for the graph of y = - 2.5 \times 4^{x}:
As x approaches -\infty, y approaches 0.
The graph is decreasing.
As x approaches \infty, y approaches 0.
The graph has a y-intercept of \left(0, - 4 \right).
The graph has a y-intercept of \left(0, - 2.5 \right).
The graph of y = 2^{x} is translated down by 7 units, state its new equation.
For each of the following exponential functions:
Consider the function y = 3^{ - x } + 2.
What value is 3^{ - x } always greater than?
Hence, what value is 3^{ - x } + 2 always greater than?
Hence how many x-intercepts does y = 3^{ - x } + 2 have?
State the equation of the asymptote of the curve y = 3^{ - x } + 2.
Consider the function y = 9^{x} + 5.
What value is 9^{x} always greater than?
Hence, what value is 9^{x} + 5 always greater than?
How many x-intercepts does y = 9^{x} + 5 have?
State the equation of the asymptote of the curve y = 9^{x} + 5.
Consider the linear function f \left(x\right) = 5 x + 2 and the exponential function g \left(x\right) = 5 \left(3\right)^{x}.
Find the value of f\left(5\right) - f\left(4\right).
Find the value of \dfrac{g \left( 5 \right)}{g \left( 4 \right)}.
Simplify f \left(k + 1\right) - f \left(k\right).
What does the result of part (c) demonstrate?
Simplify \dfrac{g \left(k + 1\right)}{g \left(k\right)}.
What does the result of part (e) demonstrate?
Consider the function y = - 3^{ - x }.
Find the y-intercept of the curve y = - 3^{ - x }.
Find the horizontal asymptote of the curve y = - 3^{ - x }.
Hence sketch the curve y = - 3^{ - x }.
Is the function y = - 3^{ - x }, an increasing or decreasing function?
Consider the function y = 2 - 4^{ - x }.
Find the y-intercept of the curve.
Is this an increasing or decreasing function?
As x approaches infinity, what value does y approach?
Sketch the graph of y = 2 - 4^{ - x }.
The graph of f \left(x\right) = 9^{x} and another exponential function, g \left(x\right) is shown:
g(x) 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).
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.
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 graph of y = 3^{ - x }:
What transformation must be done to obtain the graph of y = 3^{ - x } + 2 from y = 3^{ - x }?
Hence sketch the graph of y = 3^{ - x } + 2 on a number plane.
Write the equation of the resulting function if the graph of y = 7^{x} is translated three units to the right and then five units downward.
Describe how the graph of y = 3^{4 - x} could be obtained from the graph of y = 3^{x}.
Describe how the graph of y = 3 \times 2^{x} - 1 could be obtained from the graph of y = 2^{x}.
Consider the function y = 5^{x - 3}.
Describe how the graph of y = 5^{x - 3} could be obtained from the graph of y = 5^{x}?
Given the graph of y = 5^{x}, sketch the graph of y = 5^{x - 3}.
Consider the graph of y = 4^{ - x }:
Describe how to shift the graph of \\ y = 4^{ - x } to get the graph of y = 4^{ - \left( x + 3 \right)}.
Hence, sketch the graph of y = 4^{ - \left( x + 3 \right)}.
Consider the graph of the function y = 3^{x}.
Beginning with the above function, what equation do we obtain if we:
Horizontally dilate the graph by a factor of 2 from the x-axis and then translate the function by 5 units to the right?
Translate the graph by 5 units to the right and then horizontally dilate the graph by a factor of 2 from the x-axis?
Did the order of the transformations affect the final equation in part (a)?
Consider the equation y = 5^{x}.
The above function is translated 3 units to the right and 5 units downwards. What is the equation of the new function?
What is the horizontal asymptote of the new function?
What is the y-intercept of the new function?
What is the x-intercept of the new function?
Sketch the graph of the new function.
Consider the equation y = 6^{x}.
The above function is dilated by a factor of 5 vertically, and then translated by 3 units upwards. Find the equation of the new function.
What is the horizontal asymptote of the new function?
What is the y-intercept of the new function?
Sketch the graph of the new function.
Consider the function y = 2^{x}. By translating this function horizontally 5 units to the left we get the function y = 2^{x + 5}.
Using index laws, rewrite 2^{x + 5} as A \times 2^{x}, where A is a positive constant.
Hence, 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}. By translating this function horizontally 4 units to the right we get the function y = 5^{x - 4}.
Using index laws, rewrite 5^{x - 4} as A \times 5^{x}, where A is a positive constant.
Hence, a horizontal translation right by 4 units is equivalent to a vertical dilation of what factor?
Sketch the graph of y = 5^{x - 4}.
Consider the function y = 3^{ 2 x + 3}.
Which function below is equivalent?
y=8 \times 3^{x}
y=27 \times 9^{x}
y=9 \times 6^{x}
y=\dfrac{1}{2} \times 9^{x}
What is the domain of y = 3^{ 2 x + 3}?
What is the range of y = 3^{ 2 x + 3}?
Find the equation corresponding to the graph of y = 7^{x} after having been translated 5 units upward, 3 units to the left then reflected across the x-axis.
For each of the following exponential functions:
The function y = 4^{x} is dilated by a factor of 4 vertically, and then translated 3 unit downwards.
Find the equation of the new function.
What is the equation horizontal asymptote of the new function?
What is the value of the y-intercept of the new function?
Hence, sketch the graph of the new function.
The function y = 2^{x} is translated 4 units to the left and 4 units up.
What is the equation of the new function?
What is the equation of the horizontal asymptote of the new function?
What is the value of the y-intercept of the new function?
Does the new function have an x-intercept?
Hence, sketch the graph of the new function.
The function y = 4^{x} is translated 3 units to the right and is then dilated graphically by a factor of 2 horizontally from the y-axis.
What is the equation of the new function?
What is the equation of the horizontal asymptote of the new function?
What is the value of the y-intercept of the new function?
Hence, sketch the graph of the new function.
The function y = 4^{ - x } is dilated graphically by a factor of 2 horizontally from the y-axis and then translated 1 unit downwards.
What is the equation of the new function?
What is the equation of the horizontal asymptote of the new function?
What is the value of the y-intercept of the new function?
Hence, sketch the graph of the new function.