6.05 Graphs of logarithmic and exponential functions
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 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 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?
Consider the expression 2^{ - x }.
Evaluate the expression when x = 2.
Evaluate the expression when x = - 2.
What happens to the value of 2^{ - x } as x gets larger?
What happens to the value of 2^{ - x } as x gets smaller?
Consider the graphs of the functions y = 4^{x} and y = 4^{ - x } below. Describe the rate of change for each function.
Consider the two functions y = 4^{x} and y = 5^{x}. Which one increases more rapidly for x > 0?
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 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 points \left(3, n\right), \left(k, 16\right) and \left(m, \dfrac{1}{4}\right) all lie on the curve with equation y = 2^{x}. Find the value of:
n
k
m
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?
Consider the functions y = 2^{-x}, y = 3^{-x} and y = 5^{-x}.
Describe the nature of these functions for large values of 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 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 = 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.
The graph of y = 2^{x} is translated down by 7 units, state its new equation.
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.
Of the two functions y = 2^{x} and y = 3 \times 2^{x}, which is increasing more rapidly for x > 0?
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 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?
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 = 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 }.
Consider the function y = 4^{x} + 3.
Find the y-intercept of the curve.
State the domain of the function.
State the range of the function.
Sketch the graph of y = 4^{x} + 3.
Consider the function y = \left(\dfrac{1}{2}\right)^{x}
Determine 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 }
Hence, 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 set of axes.
Consider the equation y = \left(\dfrac{1}{3}\right)^{x}.
Rewrite the equation in the form y = k^{ - x }.
Describe a trasformation that would obtain the graph of y = \left(\dfrac{1}{3}\right)^{x} from the graph of y =3^{x}.
Graph the functions y = 3^{x} and y = \left(\dfrac{1}{3}\right)^{x} on the same set of axes.
For each of the following functions:
Find the y-intercept of the curve.
State the equation of the horizontal asymptote.
Sketch a graph of the function.
y = 3^{x} + 2.
y = 2^{x} - 2
y = - 3^{x} + 2
y = 3^{ - x }-1
Consider the function y = 2^{x - 2}.
Find the y-intercept of the curve.
Complete table of values for y = 2^{x - 2}.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
State the horizontal asymptote of the curve.
Sketch a graph of the function.
Sketch a graph of each of the following functions:
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).
Consider the function y = \log_{4} x and its given graph:
Complete the following table of values:
| x | \dfrac{1}{16} | \dfrac{1}{4} | 4 | 16 | 256 |
|---|---|---|---|---|---|
| y |
Find the x-intercept.
How many y-intercepts does the function have?
Find the x-value for which \log_{4} x = 1.
Consider the functions graphed below:
Which of these graphs represents a logarithmic function of the form y = \log_{a} \left(x\right)?
Consider the function y = \log_{2} x.
Complete the following table of values:
| x | \dfrac{1}{2} | 1 | 2 | 4 | 16 |
|---|---|---|---|---|---|
| y |
Sketch a graph of the function.
State the equation of the vertical asymptote.
Sketch the graph of y = \log_{5} x.
Consider the function y = \log_{4} x.
Complete the table of values.
| x | \dfrac{1}{1024} | \dfrac{1}{4} | 1 | 4 | 16 | 256 |
|---|---|---|---|---|---|---|
| y |
Is \log_{4} x an increasing or decreasing function?
Describe the behaviour of \log_{4} x as x approaches 0.
State the value of y when x = 0.
Consider the function y = \log_{a} x, where a is a value greater than 1.
For which of the following values of x will \log_{a} x be negative?
x = - 9
x = \dfrac{1}{9}
x = 9
\log_{a} x is never negative
For which of the following values of x will \log_{a} x be positive?
x = 5
x = - 5
x = \dfrac{1}{5}
\log_{a} x will never be positive
Is there a value that \log_{a} x will always be greater than?
Is there a value that \log_{a} x will always be less than?
Consider the given graph of the logarithmic function y = \log_{a} x:
Is \log_{a} x an increasing or decreasing function?
Which is a possible value for a,\dfrac{2}{3} or \dfrac{3}{2} ?
Consider the functions y = \log_{2} x and y = \log_{3} x.
Sketch the two functions on the same set of axes.
Describe how the size of the base relates to the steepness of the graph.
Consider the given graph of f \left( x \right) = \log_{k} x:
Determine the value of the base k.
Hence, state the equation of f \left( x \right).
Consider the functions f\left(x\right) = \log_{2} x and g\left(x\right) = \log_{2} x + 2.
Complete the table of values below:
| x | \dfrac{1}{2} | 1 | 2 | 4 | 8 |
|---|---|---|---|---|---|
| f\left(x\right)=\log_2 x | |||||
| g\left(x\right)=\log_2 x + 2 |
Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.
Describe a transformation that can be used to obtain g \left(x\right) from f \left(x\right).
Determine whether each of the following features of the graph will remain unchanged after the given transformation:
The vertical asymptote.
The general shape of the graph.
The x-intercept.
The range.
Consider the functions f\left(x\right) = \log_{2} \left( - x \right) and g\left(x\right) = \log_{2} \left( - x \right) - 3.
Complete the table of values below:
| x | -8 | -4 | -2 | -1 | -\dfrac{1}{2} |
|---|---|---|---|---|---|
| f\left(x\right)=\log_2 \left( - x \right) | |||||
| g\left(x\right)=\log_2 \left( - x \right) - 3 |
Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.
Describe a transformation that can be used to obtain g \left(x\right) from f \left(x\right).
Determine whether each of the following features of the graph will remain unchanged after the given transformation:
The vertical asymptote.
The general shape of the graph.
The x-intercept.
The domain.
Sketch the graph of the following functions:
y = \log_{3} x translated 4 units down.
y= \log_{2} x + 4.
The graph of y = \log_{6} x is transformed to create the graph of y = \log_{6} x + 4. Describe a tranformation that could achieve this.
For each of the following functions:
State the equation of the function after it has been translated.
Sketch the translated graph.
y = \log_{5} x translated downwards by 2 units.
y = \log_{3} \left( - x \right) translated upwards by 2 units.
Consider the graph of y = \log_{6} x which has a vertical asymptote at x = 0. This graph is transformed to give each of the new functions below. State the equation of the asymptote for each new graph:
Given the graph of y = \log_{8} \left( - x \right), sketch the graph of y = 3 \log_{8} \left( - x \right).
Given the graph of y = \log_{2} x, sketch the graph of the following functions:
Find the equation of the following functions, given it is of the stated form:
y = k \log_{2} x
y = 4 \log_{b} x
y = \log_{4} x + c
The function graphed has an equation of the form y = k \log_{2} x + c and passes through points A\left(4,11\right) and B\left(8,15\right):
Use the given points to form two equations relating c and k.
Hence, find the values of c and k.
State the equation of the function.