Lesson 7
Expressing Transformations of Functions Algebraically
7.1: Describing Translations (5 minutes)
Warm-up
The purpose of this warm-up is for students to compare different ways to describe a transformation of a function \(g\). They complete missing entries in a table where a series of transformations are described using words, function notation, and an expression in terms of \(x\). Several of the transformations purposefully relate to one another in order for students to make connections, such as how a horizontal translation ends up "inside" the function since it affects the input while a vertical is "outside" since it affects the output.
Student Facing
Let \(g(x)=\sqrt{x}\). Complete the table. Be prepared to explain your reasoning.
| words (the graph of \(y=g(x)\) is...) | function notation | expression |
|---|---|---|
| translated left 5 units | \(g(x+5)\) | |
| translated left 5 units and down 3 units | \(\sqrt{x+5}-3\) | |
| \(g(\text-x)\) | \(\sqrt{\text-x}\) | |
| translated left 5 units, then down 3 units, then reflected across the \(y\)-axis |
Student Response
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Activity Synthesis
Invite students to explain how they filled in the missing entries in the table. Here are some questions for discussion:
- “How did you decide which values go 'inside' the square root?” (Translating right 5 is something that affects the input of a function, so the +5 goes with the \(x\) inside of the square root.)
- “How did you decide which values go 'outside' the square root?” (Translating down 3 units is something that affects the output of a function, so the -3 goes outside of the square root.)
An important takeaway here is for students to separate transformations that affect inputs and transformations that affect outputs when writing equations to describe a transformation. Writing equations in terms of \(x\) for transformations is the focus of the following activity.
7.2: Translating Vertex Form (15 minutes)
Activity
In this activity, students apply the ideas from the warm-up to write a transformed quadratic function in vertex form. Monitor for students who reason via transformations and for those who use their prior knowledge of vertex form to identify the vertex of \(g\). Also monitor for students who wrote different but correct equations for \(h\) and \(k\). For example, if someone used a coefficient other than 1.
Launch
Supports accessibility for: Conceptual processing; Language
Student Facing
Let \(f\) be the function given by \(f(x) = x^2\).
- Write an equation for the function \(g\) whose graph is the graph of \(f\) translated 3 units left and up 5 units.
- What is the vertex of the graph of \(g\)? Explain how you know.
- Write an equation for a quadratic function \(h\) whose graph has a vertex at \((1.5, 2.6)\).
- Write an equation for a quadratic function \(k\) whose graph opens downward and has a vertex at \((3.2, \text-4.7)\).
Student Response
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Activity Synthesis
Begin the discussion by inviting previously identified students to share their explanations for how they found the vertex of the graph of \(g\). Next, invite the previously identified students to share their equations for \(h\) and \(k\), recording them for all to see. If no student wrote an equation with a coefficient, there is no need to mention them. Students will study the effect of coefficients in the following lessons.
7.3: An Even Better Fit (15 minutes)
Activity
Students revisit a data set from the beginning of the unit now that they have developed more skills to transform functions. In order to create a function to fit the data well, they need to translate the given graph vertically and horizontally. Students then write an equation in terms of input \(h\) and interpret the equation in the given context.
Launch
Remind students of the task at the beginning of this unit where they chose the best function to fit the data of a soda water cooling. Now that they have more tools to adjust functions, their job is to write a function to get an even better fit.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
In an earlier lesson, we looked at the temperature \(T\), in degrees Fahrenheit, of a bottle of soda water left outside for \(h\) hours. Let’s model this data with a function. This time, we will start with the function \(f(h) = 33(0.6)^{h}\). This graph has a shape that fits the data well.
- Describe a translation of this graph that fits the data.
- Write an equation defining a function \(g\) that models the data.
- What does your function tell you about the temperature outside?
Student Response
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Launch
Remind students of the task at the beginning of this unit where they chose the best function to fit the data of a soda water cooling. Now that they have more tools to transform functions, their job is to write a function to get an even better fit.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
In an earlier lesson, we looked at the temperature \(T\), in degrees Fahrenheit, of a bottle of soda water left outside for \(h\) hours. Let’s model this data with a function. This time, we will start with the function \(f(h) = 33(0.6)^{h}\). This graph has a shape that fits the data well.
- Describe a translation of this graph that fits the data.
- Write an equation defining a function \(g\) that models the data.
- What does your function tell you about the temperature outside?
Student Response
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Student Facing
Are you ready for more?
Han tried the following steps to model the soda water temperature. First he shifts the given graph left by one hour, then he applies a vertical shift.
- What vertical shift does Han need to apply to model the 45 degree Fahrenheit temperature in the refrigerator?
- How does Han’s model compare to yours?
Student Response
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Anticipated Misconceptions
If students have trouble seeing the amount of translation required for the function to match the data, suggest that they copy the function onto tracing paper and shift the paper directly over the graph with the data.
Activity Synthesis
| \(h\) (hours) | \(T\) (°F) |
|---|---|
| 0.03 | 69 |
| 0.12 | 67.8 |
| 0.22 | 67.4 |
| 0.3 | 66.3 |
| 0.93 | 59.9 |
| 1.02 | 59.1 |
| 1.28 | 57.5 |
| 1.55 | 57.3 |
| 2.17 | 55 |
| 2.77 | 52.3 |
| 5.7 | 47.1 |
Here is the data set from the graph. Invite students to share their equations and why they think it is the best fit. Since there is no single correct answer, attend to students’ explanations and ensure the reasons given are correct. Display the results of the equations students suggest for all to see using graphing technology alongside the data set, and invite the class to suggest further adjustments.
Conclude the discussion by asking, "What does your function tell you about the temperature outside?" After a brief quiet think time, select 2–3 students to share their reasoning. If not mentioned by students, remind the class of the term “horizontal asymptote” and ask them to interpret the asymptote in context. (The constant term in the equation is the asymptote, the graph approaches this value, so the temperature of the soda water is approaching this value, which must be the temperature outside.)
Supports accessibility for: Language; Social-emotional skills
Lesson Synthesis
Lesson Synthesis
Say, “So far we have learned how to translate and reflect functions both horizontally and vertically. For each function displayed, be prepared to describe how its graph is a sequence of translations and reflections of the graph of a simpler function.”
Display one equation at a time. Give students quiet think time for each problem and then ask them to give a signal when they have an answer. Invite students to share their responses, recording them for all to see, before displaying the next problem.
- \(f(x)=e^{\text-x}+5\)
- \(g(x)=\text-(x-3)^2\)
- \(h(x)=(\text-x+2)^3-6\)
Solutions:
- \(F(x)=e^x\), reflect over the \(y\)-axis, translate up 5 units.
- \(G(x)=x^2\), translate right 3 units, reflect vertically over the \(x\)-axis.
- \(H(x)=x^3\), translate left 2 units, translate down 6 units, reflect over the \(y\)-axis.
If not brought up by students, point out that the order of the transformations matters for function \(h\). For example, translate left 2 units, then translate down 6 units, and reflect over the \(y\)-axis is not the same as reflect over the \(y\)-axis, translate left 2 units, and then translate down 6 units. (Though it is the same as reflect over the \(y\)-axis, translate right 2 units, and then translate down 6 units since \((\text-x+2)^3-6=(\text-(x-2))^3-6\).
7.4: Cool-down - A Translated Equation (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
You can use the equation of a function to write an equation for its transformation. For example, let \(f(x) = x^2\). Take the graph of \(f\), reflect it across the \(x\)-axis, translate it up 10 units, and translate it left 3 units. What is an equation for this new function? The new function \(g\) is related to \(f\) by \(g(x) =\text-f(x+3)+10\), since
Which means \(g(x) =\text-(x+3)^2 + 10\).
Sometimes you can recognize from the expression for a function that it is the transformation of a simpler function. For example, consider:
\(\displaystyle H(t) = 10 - (1.2)^{t+5}\)
One way to obtain the expression for \(H\) from \(1.2^t\) is:
- adding 5 to the input to get \((1.2)^{t+5}\)
- multiplying the output by -1 to get \(\text-(1.2)^{t+5}\)
- adding 10 to the output to get \(10 - (1.2)^{t+5}\)
So the graph of \(H\) is obtained from the graph of \(f(t) = 1.2^t\) by translating left 5 units, reflecting across the \(x\)-axis, and translating up 10 units. Consider the point \((0,1)\) on the graph of \(f\). After translating, reflecting, and translating again, it becomes the point \((\text-5,9)\) on the graph of \(H\).
