Lesson 20

Changes over Equal Intervals

20.1: Writing Equivalent Expressions (5 minutes)

Warm-up

This warm-up allows students to practice writing equivalent expressions. It prepares students to use this skill as they work with complex expressions arising from linear and exponential functions in this lesson.

Student Facing

For each given expression, write an equivalent expression with as few terms as possible.

  1. $7p -3+ 2(p +1)$
  2. $[4(n+1)+10] - 4(n+1)$
  3. $9^5 \boldcdot 9^2\boldcdot 9^x$
  4. $\frac {2^{4n}}{2^n}$

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share how they manipulated the expressions. If not already mentioned in students' explanations, bring up how arithmetic properties of operations (distributive, commutative) and properties of exponents come into play in writing equivalent expressions.

20.2: Outputs of A Linear Function (10 minutes)

Activity

In grade 8, students learned that a linear function has a constant rate of change. Each time the input increases by 1, the output changes by a particular amount, and this amount is called the rate of change of the function. In this activity, students revisit and extend their understanding of this structure. If \(f\) is a linear function, then whenever the input \(x\) increases by 2, the output always changes by the same amount. The same is true if 2 is replaced by any non-zero quantity. When the input \(x\) changes by the same quantity, the output \(f(x)\) also changes by the same quantity.

This idea is connected to the fact that a line representing the linear function has a well-defined slope. Look for students who draw slope triangles in order to calculate the differences in \(x\) and \(y\) and invite them to share during the discussion.

Launch

Tell students that they will now revisit a linear function in order to describe its behavior more generally and clearly. Later, they will compare and contrast this behavior with that of an exponential function.

Arrange students in groups of 2. Give partners a moment to discuss the first question. Consider asking partners to use different pairs of consecutive values when checking how \(f(x)\) changes when \(x\) increases by 1 (for example, one partner uses 1 and 2 for the values of \(x\) and the other partner uses 19 and 20).

Briefly discuss students’ responses and reasoning on the first question before they proceed to the last two questions. If no students mentioned evaluating the function at two consecutive values, ask them about it.

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Consider pausing after most students have completed the first question for a brief, small-group or whole-class discussion before moving on.
Supports accessibility for: Memory; Organization

Student Facing

Here is a graph of \(y =f(x)\) where \(f(x) = 2x + 5\).

 

Graph of an increasing linear function, x y plane, origin O.

 

  1. How do the values of \(f\) change whenever \(x\) increases by 1, for instance, when it increases from 1 to 2, or from 19 to 20? Be prepared to explain or show how you know.
  2. Here is an expression we can use to find the difference in the values of \(f\) when the input changes from \(x\) to \(x+1\).

    \(\displaystyle  [2(x+1) +5] - [2x+5]\)

    Does this expression have the same value as what you found in the previous questions? Show your reasoning.

    1. How do the values of \(f\) change whenever \(x\) increases by 4? Explain or show how you know.
    2. Write an expression that shows the change in the values of \(f\) when the input value changes from \(x\) to \(x+4\).
    3. Show or explain how that expression has a value of 8.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students seem to have trouble making sense of how the expression \([2(x+1) +5] - [2x+5]\) represents the change in \(f(x)\) when \(x\) increases by 1, or how the expression has a value of 2, consider asking them to reason about the expression as if \(x\) and \(x+1\) were numerical expressions (say, 3 and \(3+1\)).

Students may have trouble with the first part of the last question (about how the values of \(f\) change whenever \(x\) increases by 4). Ask them to try several pairs of values (e.g., 1 and 5, 3 and 7, etc.) to see if there is a pattern. Then, prompt them to refer to how an increase of 1 was expressed algebraically and rearranged in an earlier question.

Watch for students who overlook the fact that the change in the value of \(x\) must be 1 if the change in the value of \(y\) is the same as the slope.

Activity Synthesis

Invite students to share their responses to the last two questions. Relate the algebraic work done in this activity to what students have learned about slope and similar triangles in grade 8. Discuss questions such as:

  • Can you find an example where the output of \(f\) does not increase by 2 when \(x\) goes up by 1? (No.) Why not? (The graph of \(f\), a line, has a particular slope, an amount by which the output value changes when \(x\) changes by 1. So any two points on the line whose \(x\)-values are 1 unit apart will have \(y\)-values that differ by the value of the slope.)
  • Can you find an example where the input \(x\) increases by 4, but the output \(f(x)\) does not increase by 8? (No.) Why not? (If increasing the input by 1 always changes the output by 2, then increasing by 1 four times always changes the output by 4 times the slope, or \(4 \boldcdot 2\). Consider showing a graph.)
  • How does the expression \([2(x+4)+5]- [2x + 5]\) show that when \(x\) changes by 4, the output changes by 8? (The \(2(x+4)+5\) is the output of \(f\) for \(x+4\). The \(2x+5\) is the output of \(f\) for \(x\). Subtracting the two gives us: \(2x + 8 + 5 - 2x - 5\), which is 8.)
Graph.
 

Emphasize that in any linear function, when \(x\) increases by an equal amount, the output also changes by an equal amount.

To further highlight that this observation is true for any linear function, and if time permits, consider showing a diagram of an abstract case of a linear function with slope \(s\).

When the input changes by \(a\), the output changes by \(a \boldcdot s\). (The triangles drawn are similar triangles, so the ratios of their corresponding sides are equivalent.)

Graph.
 
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Present an incorrect answer and explanation for the question “How do the values of \(f\) change whenever \(x\) increases by 4? Explain or show how you know.” Consider using this ambiguous response, “The values of \(f\) change by 6, because I used two points on the graph and counted the spaces.” Ask students to identify the error, critique the reasoning, and write an improved explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe the process for finding the amount of change in the graph. This helps students evaluate, and improve on, the written mathematical arguments of others, as they clarify their understanding of discrete graphs.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

20.3: Outputs of An Exponential Function (20 minutes)

Activity

This activity complements the previous one. Students first examine what happens to the values of an exponential function \(f\) when the input is increased by 1. If \(f\) is an exponential function defined by \(f(x) = a \boldcdot b^x\), then its value changes by a factor of \(b\). Similarly, if the input is increased by 7, then the function value increases by a factor of \(b^7\). More generally, an exponential function always changes by equal factors over equal intervals.

Students may have little trouble recognizing how the output values change when dealing with numerical input values, but may find it challenging to generalize their observations. Focus the discussion on this transition and on using properties of exponents to extend their observations toward generalized cases.

Student Facing

Here is a table that shows some input and output values of an exponential function \(g\). The equation \(g(x) = 3^x\) defines the function.

\(x\) \(g(x)\)
3 27
4 81
5 243
6 729
7 2,187
8 6,561
   
   
\(x\)  
\(x+1\)  
  1. How does \(g(x)\) change every time \(x\) increases by 1? Show or explain your reasoning.
  2. Choose two new input values that are consecutive whole numbers and find their output values. Record them in the table. How do the output values change for those two input values?
  3. Complete the table with the output when the input is \(x\) and when it is \(x+1\).
  4. Look at the change in output values as the \(x\) increases by 1. Does it still agree with your findings earlier? Show your reasoning.

    Pause here for a class discussion. Then, work with your group on the next few questions.

  5. Choose two \(x\)-values where one is 3 more than the other (for example, 1 and 4). How do the output values of \(g\) change as \(x\) increases by 3? (Each group member should choose a different pair of numbers and study the outputs.)
  6. Complete this table with the output when the input is \(x\) and when it is \(x+3\). Look at the change in output values as \(x\) increases by 3. Does it agree with your group's findings in the previous question? Show your reasoning.

    \(x\) \(g(x)\)
    \(x\)  
    \(x+3\)  

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

For integer inputs, we can think of multiplication as repeated addition and exponentiation as repeated multiplication:

\(\displaystyle 3\cdot 5=3+3+3+3+3\qquad\qquad\text{ and }\qquad\qquad 3^5=3\cdot 3\cdot 3\cdot 3\cdot 3\)

We could continue this process with a new operation called tetration. It uses the symbol \(\uparrow\uparrow\), and is defined as repeated exponentiation:

\(\displaystyle 3\uparrow\uparrow5 = 3^{3^{3^{3^3}}}.\)

Compute \(2\uparrow\uparrow 3\) and \(3\uparrow\uparrow 2\). If \(f(x)=3\uparrow\uparrow x\), what is the relationship between \(f(x)\) and \(f(x+1)\)?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

After the previous activity on linear relationships, students may initially look for something similar here. They should notice that the differences in consecutive rows of the table are not the same. Encourage them to look for other patterns.

Activity Synthesis

Invite students to share what they noticed about values of \(g\) for consecutive whole numbers. If not mentioned by students, point out the pattern that as the \(x\)-value increases by 1, the output changes by a factor of 3. For example: \(g(5)\) is 3 times \(g(4)\)\(g(8)\) is 3 times \(g(7)\), and so on.

Discuss questions such as:

  • What do we have when dividing the outputs for two consecutive \(x\)-values, such as \(\frac{g(5)}{g(4)}\) or \(\frac{g(8)}{g(7)}\)? (a quotient of 3)
  • How are the two outputs \(g(x+1)\) and \(g(x)\) related, then? (\(g(x+1)\) is \(g(x)\) times 3, or when we divide \(g(x+1)\) by \(g(x)\), the quotient is 3.)
  • How can we use the rules of exponents to show that this is always the case? (\(g(x) = 3^x\), so \(g(x+1) = 3^{x+1}\). When dividing two powers with the same bases, we can subtract the exponents. Dividing the two, we have \(\frac{3^{x+1}}{3^x}\), which is 3.)
  • Does this reasoning work if the function has a different initial value, say \(h\) given by \(h(x) = 5 \boldcdot 3^x\)? (Yes,\(\frac{5\boldcdot 3^{x+1}}{5 \boldcdot 3^x}\) still equals 3.)
  • What happens to the output of \(g\) when \(x\) increases by 2? (\(g(x)\) increases by a factor of 3 twice, or \(3^2\).) What about when it increases by 5? (\(g(x)\) increases by a factor of 3 five times.)
  • What happens to the output when \(x\) increases by \(a\)? (\(g(x)\) increases by a factor of 3 \(a\) times, or \(3^a\). The rule of exponents can help us here as well. If we divide \(g(x+a)\) by \(g(x)\), we have: \(\frac{3^{x+a}}{3^x} = 3^a\).)

Consider using a graph of \(g\) to help students visualize growth of \(g\) over equal intervals of \(x\). The successive quotients of the outputs are always 3.

Graph.
 

Lesson Synthesis

Lesson Synthesis

Previously, we saw that an exponential function eventually overtakes a linear function. We can understand better why this is so by examining how these functions change when we change their input. Present these equations and tables for functions \(f\) and \(g\):

\(f(x) = 3x + 2\)

\(x\) 0 1 2 3 4 5 10 15
\(f(x)\) 2 5 8 11 14 17 32 47

\(g(x) = 2 \boldcdot 3^x\)

\(x\) 0 1 2 3 4 5 10 15
\(g(x)\) 2 6 18 54 162 486 118,098 28,697,814

Ask students:

  • How does the function \(f\) change when \(x\) grows by 1? By 5? By 10?
  • How does the function \(g\) change when \(x\) grows by 1? By 5? By 10?

Make sure students see that any time \(x\) increases by 2, \(f\) grows by \(3 \boldcdot 2\) or 6, and \(g\) grows by a factor \(3^2\) or 9. Any time \(x\) increases by 5, the value of \(f\) grows by \(3 \boldcdot 5\) or 15, but the value of \(g\) grows by a factor of \(3^5\) or 243.

20.4: Cool-down - Increasing Input by One (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Linear and exponential functions each behave in a particular way every time their input value increases by the same amount.

Take the linear function \(f\) defined by \(f(x) = 5x + 3\). The graph of this function has a slope of 5. That means that each time \(x\) increases by 1, \(f(x)\) increases by 5. For example, the points \((7,38)\) and \((8,43)\) are both on the graph. When \(x\) increases by 1 (from 7 to 8), \(y\) increases by 5 (because \(43-38=5\)). We can show algebraically that this is always true, regardless of what value \(x\) takes.

Graph of an increasing linear function, x y plane, origin O.

The value of \(f\) when \(x\) increases by 1, or \(f(x+1)\), is \(5(x+1) +3\). Subtracting \(f(x+1)\) and \(f(x)\), we have:

\(\displaystyle \begin {align} f(x+1) - f(x)=5(x+1) + 3 - (5x+3)\\ =5x + 5 +3 -5x-3\\ =5\end{align}\)

This tells us that whenever \(x\) increases by 1, the difference in the output is always 5. In the lesson, we also saw that when \(x\) increases by an amount other than 1, the output always increases by the same amount if the function is linear.

Now let's look at an exponential function \(g\) defined by \(g(x) = 2^x\). If we graph \(g\), we see that each time \(x\) increases by 1, the value \(g(x)\) doubles. We can show algebraically that this is always true, regardless of what value \(x\) takes.

Graph of an increasing exponential function, x y plane, origin O. 

The value of \(g\) when \(x\) increases by 1, or \(g(x+1)\), is \(2^{x+1}\). Dividing \(g(x+1)\) by \(g(x)\), we have:

\(\displaystyle \begin {align} \frac{g(x+1)}{g(x)}=\frac{2^{(x+1)}}{2^x}\\ =2^{x+1-x}\\ =2^1\\ =2\\ \end{align}\)

This means that, whenever \(x\) increases by 1, the value of \(g\) always increases by a multiple of 2. In the lesson, we also saw that when \(x\) increases by an amount other than 1, the output always increases by the same factor if the function is exponential.

A linear function always increases (or decreases) by the same amount over equal intervals. An exponential function increases (or decreases) by equal factors over equal intervals.