5.06 Transformations of functions
Consider the parabola y = x^{2} - 3.
Complete the table of values:
Sketch the graph of y = x^{2} - 3.
What is the y-intercept of the graph?
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
When adding a constant to the equation y = x^{2}, describe the type of transformation that occurs on its graph.
Consider the graph of y = \dfrac{1}{x}:
What translation is required to shift the graph of y = \dfrac{1}{x} to get the graph of
y = \dfrac{1}{x} + 4.
Hence, sketch y = \dfrac{1}{x} + 4.
Consider the graph of y = 2^{x}:
What translation is required to shift the graph of y = 2^{x} to get the graph of
y = 2^{x} - 5.
Hence, sketch y = 2^{x} - 5.
A graph of y = x^{4} is shown. Sketch the curve after it has undergone a transformation resulting in the function
y = x^{4} - 2.
How should you translate the graph of y = f \left( x \right) to get the graph of y = f \left( x \right) + 4 ?
How should you translate the graph of y=g(x) to get the graph of y=g(x + 6) ?
The functions f \left(x\right) and g \left(x\right) = f \left(x + k\right) have been graphed:
Determine the value of k.
Describe the transformation that occured.
Describe the shift required to transform the graph of y = a^{x} to get the graph of y = a^{\left(x + 5\right)}.
For each of the following equations, determine what the new equation is when their graphs are moved as described:
The graph of y = x^{3} is moved to the right by 10 units.
The graph of y = 2^{x} is moved down by 9 units.
Consider the graph of the hyperbola y = \dfrac{2}{x}:
What would be the new equation if the graph was shifted upwards by 2 units?
What would be the new equation if the graph was shifted to the right by 9 units?
Consider the graph of y = \sqrt{4 - x^{2}}:
What would be the new equation if the graph was translated downwards by 7 units?
What would be the new equation if the graph was translated to the left by 3 units?
If the graph of y = x^{4} is moved to the right by 8 units and up by 6 units, what is its new equation?
Consider the graph of y = x^{3}:
Describe the required translations to shift the graph of y = x^{3} to get the graph of \\y = \left(x + 2\right)^{3} + 4.
Hence, sketch y = \left(x + 2\right)^{3} + 4.
Consider the graph of y = \sqrt{4 - x^{2}}:
Describe the required translations to shift the graph of y = \sqrt{4 - x^{2}} to get the graph of y = \sqrt{4 - x^{2}} + 2.
Hence, sketch y = \sqrt{4 - x^{2}} + 2.
Consider the graph of y = \sqrt{25 - x^{2}}:
Describe the required translations to shift the graph of y = \sqrt{25 - x^{2}} to get the graph of y = \sqrt{25 - \left(x + 4\right)^{2}} - 2.
Hence, sketch y = \sqrt{25 - \left(x + 4\right)^{2}} - 2.
The graph of y = P \left(x\right) is shown. Sketch the graph of y = P\left(x\right) - 20.
Consider the point P \left(8, - 3 \right). Sketch the point that is symmetric to P \left(8, - 3 \right) with respect to:
The x-axis
The y-axis
The origin
State whether the following functions have symmetry. If so, state the line or point of symmetry.
y = x^{2} + 1
y = x + 5
Is the graph shown symmetric with respect to the x-axis, the y-axis, or the origin?
Does the function y = 4 x^{3} have reflective symmetry?
For each of the following functions:
Find f(-x).
State whether the functions is symmetrical along the x-axis, the y-axis or neither.
Consider the point P = \left( - 6 , 1\right). Find the point obtained by:
Reflecting P about the x-axis
Reflecting P about the y-axis
Rotating P by 180 \degree about the origin
Describe the symmetry of the graph of x^{2} + y^{2} = 6.
Suppose f is a function such that f \left( 3 \right) = 2. State a point that lies on the graph of f if:
The graph of y = f \left( x \right) is symmetric with respect to the origin.
The graph of y = f \left( x \right) is symmetric with respect to the y-axis.
The graph of y = f \left( x \right) is symmetric with respect to the line x = 6.
f is an even function.
f is an odd function.
The graph of y = P \left(x\right) is shown. Sketch the graph of y = P \left( - x \right).
Consider the function f \left( x \right) = \sqrt{x}. Write down the new function g \left( x \right) which results from scaling f \left( x \right) vertically by a factor of \dfrac{1}{3}, and scaling horizontally by a factor of \dfrac{1}{2}.
A function f \left(x\right) is transformed into a new function g \left(x\right) = f \left(\dfrac{x}{k}\right). If 0 < k < 1, determine what will be the effect on the graph of f \left(x\right).
Consider the function f \left(x\right) = \sqrt{x}.
Complete the table for f \left(x\right):
Sketch a graph of the function f \left( x \right).
| x | 0 | 1 | 4 | 9 | 16 |
|---|---|---|---|---|---|
| f(x) | |||||
| g(x) |
Complete the table of values for the transformed function g \left(x\right) = f \left( 4 x\right), giving all values in exact form.
Sketch a graph of g \left( x \right).
Describe how g \left(x\right) relates to the graph of f \left(x\right).
Explain why the transformed function h \left(x\right) = f \left( - 4 x \right) can be undefined for the values of x in the table.
The graph of y = P \left(x\right) is shown. Sketch the graph of y = 2 P \left(x\right).
Suppose that the x-intercepts of the graph of y = f \left( x \right) are - 5 and 6.
Find the x-intercepts of the graph of:
Consider the function f \left( x \right) = x^{2} - 5. Using function notation, describe the transformation of f that will result in the function:
Consider the graph of y = x^{3}:
Sketch the curve after it has undergone transformations resulting in the function
y = - 4 \left(x + 4\right)^{3}.
Consider the given graph:
Describe the transformations of the graph of y = x^{3} to the given graph.
Write down the equation of the given graph.
Consider the graphs of f \left(x\right) = \dfrac{3}{x} and g \left(x\right) shown:
Write g \left(x\right) in terms of f(x) using the transformation shown in the graph.
State the equation of g \left(x\right).
Consider the graphs of f \left(x\right) = 2^{x} and g \left(x\right) shown.
Write g \left(x\right) in terms of f(x) using the transformation shown in the graph.
State the equation of g \left(x\right).
For the following base functions, describe the transformations that have occurred:
From y = f \left( x \right) to y = f \left( \dfrac{x}{4} + 3 \right) - 5
From y = g \left( x \right) to y = \dfrac{g \left( x - 3 \right) + 3}{2}
From y = f \left( x \right) to y = 2 f \left( \dfrac{1}{2} x \right) - 3
From y = h \left( x \right) to y = 2 \left(h \left( x \right) + 3\right)
For the following functions, describe the transformations that have occured:
From f(x) = x^{4} to g(x) = - 7 \left(x + 4\right)^{4}
From f(x) = x^{4} to g(x) = - 5 x^{4} + 8
From f(x) = x^{2} to g(x) = - 10 \left(x + 8\right)^{2} - 9
The table below shows values that satisfy the function f \left(x\right) = \left|x\right|.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y=f(x) | 3 | 2 | 1 | 0 | 1 | 2 | 3 |
| g(x)=5f(x) | |||||||
| h(x)=-2f(x) |
Complete the table of values for each transformation of the function f \left(x\right).
Sketch the graph of g \left(x\right).
Describe how to transform the graph of f \left(x\right) into the graph of g \left(x\right).
Sketch the graph of h \left(x\right).
Describe how to transform the graph of f \left(x\right) into the graph of h \left(x\right).
Three functions have been graphed on the number plane. g \left(x\right) and h \left(x\right) are both transformations of f \left(x\right).
State the equation of f \left(x\right).
Write g \left(x\right) in terms of f(x) using the transformation shown in the graph.
State the equation of g \left(x\right).
Write h \left(x\right) in terms of f(x) using the transformation shown in the graph.
State the equation of h \left(x\right).
Some points on the graph of the function y = f \left( x \right) are given in the table below:
| \text{Original point} | g(x) | \text{Corresponding point} |
|---|---|---|
| (9, -12) | g(x)=f(x)-8 | |
| (9, -12) | g(x)=6f(x) | |
| (6, -7) | g(x)=f(x-5) |
Complete the table by finding the corresponding points on the graph of y = g \left( x \right).
Suppose that \left( - 4 , 3\right) is a point on the graph of y = g \left( x \right). Find the corresponding point on the graph of:
y = g \left( x + 7 \right) - 6
y = - 6 g \left( x - 4 \right) + 6
y = g \left( 6 x + 1 \right)
The table below shows coordinates of points on the function y = f(x). By performing the given transformation on them, find the corresponding transformed points.
| \text{Point} | \text{Transformation} | \text{Transformed point} |
|---|---|---|
| (-3, -1) | y=f(x-5)-2 | |
| (0,3) | y=\dfrac{1}{5}f(x-3) | |
| (1, -2) | y=f(3x)-2 | |
| (-3, 5) | y=f\left(\dfrac{x}{2}\right)-5 |