4.02 Transformations of exponential graphs
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-value of the y-intercept.
Determine the horizontal asymptote.
Sketch the graph of the function.
For each of the following exponential functions:
Find the y-value of 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-value of 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-value of 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.
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 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.
Sketch the graph of y = 3^{ - x } + 2.
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 }
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).
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 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)