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Precalculus MATH F156X Summer 2026

Dr. Latrice Bowman

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Section BF-4 Transformations and Combinations of Functions

Worksheet BF-4 Quick Notes

For this outcome we will be looking at creating new functions by shifting, reflecting, stretching, or compressing a graph. We will also be combining two or more functions using algebraic operations and the evaluation of functions to create new functions. Specifically, for this outcome, students should be able to:
  • Given a transformed function, determine the basic function that was transformed.
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  • Given two functions, identify the transformations that will change one function into the other.
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  • Apply transformations in the correct order to move points on a graph.
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  • Given a transformed graph and that of a basic function, determine the equation of the transformed graph.
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  • Given two or more functions, be able to evaluate:
    • The sum or difference function.
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    • The sum or difference function evaluated at a point.
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    • The product or quotient function.
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    • The product or quotient function evaluated at a point.
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    • The composition function.
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    • The composition function evaluated at a point.
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    • The domain of any combination.
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Transformations of Functions
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There are several transformations that can be applied to a function to create a new function. These include shifting the graph up or down, left or right, reflecting the graph across an axis, and stretching or compressing the graph.
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Translations will shift a graph up or down or left or right. Reflections reflect a graph over one of the axes or the origin. Some transformations can shrink or stretch a graph either vertically or horizontally. In each of the following let \(c \gt 0\text{.}\)
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Note: The order of transformations is important. When applying multiple transformations, the order in which they are applied can affect the final result. Generally, it is recommended to apply transformations in the following order: reflections, stretches/compressions, and then translations (applying to the horizontal axis first, then the vertical axis).
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Note: When applying transformations to a graph, it is often helpful to start with key points on the graph (such as intercepts, vertices, or other easily identifiable points) and apply the transformations to those points to understand how the graph changes.
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  • \(y = f(x) + c\) Shifts the graph up c units
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  • \(y = f(x) - c\) Shifts the graph down c units
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  • \(y = f(x + c)\) Shifts the graph left c units
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  • \(y = f(x - c)\) Shifts the graph right c units
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  • \(y = -f(x)\) Reflects the graph about the x-axis
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  • \(y = f(-x)\) Reflects the graph about the y-axis
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  • \(y = cf(x), c \gt 1\) Stretches the graph vertically by a factor of c
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  • \(y = cf(x), c \lt 1\) Shrinks the graph vertically by a factor of c
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Combinations of Functions
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Given two functions \(f(x)\) and \(g(x)\text{,}\) we can create new functions by combining them algebraically:
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  • Sum: \((f + g)(x) = f(x) + g(x)\)
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  • Difference: \((f - g)(x) = f(x) - g(x)\)
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  • Product: \((fg)(x) = f(x) \cdot g(x)\)
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  • Quotient: \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\text{,}\) where \(g(x) \ne 0\)
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The domain of the sum, difference, and product functions is the intersection of the domains of \(f\) and \(g\text{.}\) The domain of the quotient function is the intersection of the domains of \(f\) and \(g\text{,}\) excluding any values where \(g(x) = 0\text{.}\)
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Compositions of Functions
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Given two functions \(f(x)\) and \(g(x)\text{,}\) we can create a new function by composing them:
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\((f \circ g)(x) = f(g(x))\)
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This means that we first evaluate \(g(x)\text{,}\) and then evaluate \(f\) at the result of \(g(x)\text{.}\)
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The domain of the composition function \(f \circ g\) is the set of all values of \(x\) such that \(g(x)\) is in the domain of \(f\text{.}\)
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Section BF-4 Videos

Transformations
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Combinations of Functions
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Compositions of Functions
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Section BF-4 Rubric

Worksheet Worksheet

Before attempting the BF-4 outcome, you should review the following rubric. This shows the criteria you will be graded on and the expectations to earn each mastery level.
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Table 13. BF4 Rubric
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Criteria M P R N
Simplifying Algebraic Expressions All answers are fully simplified. Exact answers are given, not approximations. Radical and exponential expressions are used appropriately and are simplified in the correct manner No algebraic errors OR some expressions are not simplified. There are some algebraic errors and some expressions were not simplified. There are some algebraic errors and some expressions were not simplified.
Basics of Graphing Graph is labeled with units for each axis. The graph includes all of the important features (intercepts, asymptotes, deleted points, etc).  The graph is appropriately drawn to accommodate all features. Graph is labeled with units for each axis and includes most of the important features or the graph has all of the important features but is not labeled. Graph is not labeled and has some of the important features. No graph was given or the given graph is not labeled and does not have the correct features of the function.
Problem Structure and Mathematical Communication Problems have a clear beginning, middle, and end. Work progresses clearly from one step to the next. The work provided uses appropriate methods and notation and provides a clear solution. Problems have clear beginnings and ends. Work progresses from one step to the next. The work provided uses algebraic methods, correct notation, and provides a solution. Parts of the mathematical structure are missing, or steps in the progress of the solution are missing. There are errors in the mathematical notation. The work provided uses algebraic methods and provides a solution. There is not a clear structure to the solution, notation is used incorrectly, and it is unclear what methods are being used.
Identify Transformations Correctly stated horizontal shifts, vertical shifts, horizontal and vertical stretches or compressions, and reflections; all transformations have both direction and length. Stated horizontal shifts, vertical shifts, horizontal and vertical stretches or compressions, and reflections; all transformations have both direction and length. minor errors in at most one of these. Errors in horizontal shifts, vertical shifts, horizontal and vertical stretches or compressions, or reflections; missing either direction or length on transformations. No transformations given.
Transform a point or a graph of a function Showed through a series of transformations how final point or final graph was reached, correct point or graph was found. Showed through some transformations how final point or final graph was reached. Missing or made an error in at most one transformation. Provided a final point or final graph but with minimal steps shown how the answer was reached. No transformed point or transformed graph given.
Combinations Applied the correct operations and rewrote to get the correct value or expression.  Function is correctly labeled. Applied operations and rewrote to get a value or an expression. Incorrectly applied the operations OR did not show understanding of function notation OR had multiple errors in finding the value or expression. No combinations were given.
Compositions Applied the composition in the correct order and rewrote to get the correct value or expression.  Function is correctly labeled. Applied the composition and rewrote to get a value or  an expression. Incorrectly applied the composition OR errors in rewriting the function OR did not show understanding of function notation OR multiple errors in obtaining the value or expression. No composition was given.
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Worksheet BF-4 Sample Outcomes

This outcome covers transformations and combinations of functions. You will be asked to identify basic functions, and the transformations applied to them. You will be asked to use transformations in the correct order to find new points or graphs. Given two functions, you will be asked to evaluate the combinations or the combination at a point. Given two functions, you should be able to evaluate their composition or the composition at a point. You should be able to write out complete steps to solve these types of problems by hand, without the use of a calculator.
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For the quiz on this outcome you would be given 30 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:
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1.

Given the function \(f(x)=\frac{2}{x+4}-1\text{.}\)
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(b)
State the transformations applied to the basic function.
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Hint.
Look at the function \(f(x)=\frac{2}{x+4}-1\) and compare it to the basic function \(y=\frac{1}{x}\text{.}\) Think about what the numerical coefficients and constants represent, as well as any negative signs.
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Solution.
The transformations applied are: a horizontal shift left by 4 units, a vertical stretch by a factor of 2, and a vertical shift down by 1 unit.
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(c)
Determine where the point \((2,\frac{1}{2})\) would be translated to on \(f\) under the transformations.
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Hint.
Use the order of transformations to move the point: horizontal shift, vertical stretch, vertical shift.
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Answer.
\((-2,0)\)
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Solution.
First we will apply the horizontal shift left by 4 units, which moves the point from \((2,\frac{1}{2})\) to \((-2,\frac{1}{2})\text{.}\) Next, we will apply the vertical stretch by a factor of 2, which multiplies the y-coordinate by 2, resulting in \((-2,1)\text{.}\) Finally, we will apply the vertical shift down by 1 unit, which subtracts 1 from the y-coordinate, resulting in \((-2,0)\text{.}\)
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2.

Given the function \(f(x)\) shown below, find the graph of the transformed function \(g(x)\text{.}\)
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Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
(a)
If \(g(x)=3-2f(x+1)\text{,}\) state the transformations applied to \(f(x)\text{.}\)
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Hint.
Look at the numbers and minus signs in the function for g. Think about what these represent in terms of transformations.
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Solution.
The transformations applied are: a horizontal shift left by 1 unit, a vertical stretch by a factor of 2, a reflection about the x-axis, and a vertical shift up by 3 units.
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(b)
Sketch the graph of \(g(x)\text{.}\)
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Hint.
Use the order of transformations to move the graph: horizontal shift, vertical stretch, reflection, and vertical shift.
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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Solution.
First we will apply the horizontal shift left by 1 unit which subtracts 1 from all of the x-coordinates. Next, we will apply the vertical stretch by a factor of 2, which multiplies all of the y-coordinates by 2. Then we will apply the reflection about the x-axis, which multiplies all of the y-coordinates by -1. Finally, we will apply the vertical shift up by 3 units, which adds 3 to all of the y-coordinates.
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3.

Given \(f(x)=\frac{x}{x+1}\) and \(g(x)=3x^2+3\text{,}\) find and simplify the following:
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(a)
\((f+g)(-4)\)
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Hint.
Recall that \((f+g)(x)=f(x)+g(x)\)
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Answer.
\((f+g)(-4)= \frac{157}{3}\)
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Solution.
\((f+g)(-4) \amp = f(-4) + g(-4)\\ \amp = \frac{-4}{-4+1} + 3(-4)^2 + 3\\ \amp = \frac{4}{3} + 48 + 3\\ \amp = \frac{157}{3}\end{aligned}\)
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(b)
\((f-g)(x)\)
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Hint.
Recall that \((f-g)(x)=f(x)-g(x)\)
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Answer.
\((f-g)(x)= \frac{-3x^3-3x^2+4x+3}{x+1}\)
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Solution.
\((f-g)(x) \amp = f(x)-g(x)\\ \amp= \frac{x}{x+1}-3x^2+3\\ \amp= \frac{x-3x^2(x+1)+3(x+1)}{x+1}\\ \amp= \frac{-3x^3-3x^2+4x+3}{x+1}\end{aligned}\)
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(c)
\(\left(\frac{f}{g}\right)(3)\)
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Hint.
Recall that \(\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}\)
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Answer.
\(\left(\frac{f}{g}\right)(3)= \frac{1}{40}\)
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Solution.
\(\left(\frac{f}{g}\right)(3) \amp = \frac{f(3)}{g(3)}\\ \amp = \frac{\frac{3}{3+1}}{3(3)^2+3}\\ \amp = \frac{\frac{3}{4}}{27+3}\\ \amp = \frac{\frac{3}{4}}{30}\\ \amp = \frac{3}{4} \cdot \frac{1}{30}\\ \amp = \frac{1}{40}\end{aligned}\)
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(d)
\((f\circ g)(x)\)
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Hint.
Recall that \((f\circ g)(x)=f(g(x))\)
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Answer.
\((f\circ g)(x) =\frac{3x^2+3}{3x^2+4}\end{aligned}\)
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Solution.
\((f\circ g)(x) \amp = f(g(x))\\ \amp= f(3x^2+3)\\ \amp= \frac{3x^2+3}{3x^2+3+1}\\ \amp= \frac{3x^2+3}{3x^2+4}\end{aligned}\)
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