Lesson 20

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.

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.

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.

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.

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.

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.)

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.)

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.

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.

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.