Lesson 4

Comparing Quadratic and Exponential Functions

4.1: From Least to Greatest (5 minutes)

Warm-up

In this warm-up, students compare the values of exponential expressions by making use of their structure (MP7). The reasoning here prepares them to think about exponential growth later in the lesson.

Students should recognize that \(9^2 < 10^2\) and \(2^9 < 2^{10}\). Deciding whether \(10^2\) or \(2^9\) is greater requires some estimation or further reasoning using properties of exponents.

For example, some students may recognize that \(2^4 = 16\) and \(2^8 = 2^4 \boldcdot 2^4 = \left(2^4\right)^2\), so \(2^8 = 16^2\), which is 256. Because \(2^9\) is greater than \(2^8\), it follows that \(2^9\) is greater than 256 and therefore greater than \(10^2\).

As students discuss their thinking, listen for strategies that involve using properties of exponents or thinking about the structure of the expressions.

Launch

Arrange students in groups of 2. Give students a moment of quiet think time and then time to share their thinking with a partner.

Students should not use a calculator to evaluate the expressions in order to encourage them to rely on the structure of the expressions.

Student Facing

List these quantities in order, from least to greatest, without evaluating each expression. Be prepared to explain your reasoning.

A. \(2^{10}\)

B. \(10^2\)

C. \(2^9\)

D. \(9^2\)

 

Student Response

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Activity Synthesis

Select students to share their responses and reasoning. Highlight explanations that show that the expressions can be compared by analyzing their structure (as in the example in the Activity Narrative), and that it is not necessary to know their exact values to put the expressions in order.

4.2: Which One Grows Faster? (15 minutes)

Activity

This activity prompts students to contrast quantities that grow exponentially and quadratically by writing equations and creating tables of values. Before students begin working, they are asked to make an estimate the number of squares in each pattern at Step 5 and Step 10. Making a reasonable estimate and comparing a computed value to one’s estimate is often an important aspect of making sense of problems (MP1). Later from the tables, students notice that the output of the exponential function eventually outgrows that of the quadratic function. In the next activity, they will think further about whether this is always the case.

If students opt to use spreadsheet or graphing technology, they practice choosing appropriate tools strategically (MP5).

Launch

Display the image of the patterns for all to see and ask students to read the description of how the patterns grow. Ask students to predict which pattern will have more small squares in Step 5.

Then, ask students to predict which pattern will have more small squares in Step 10. Poll the class to collect their predictions. Display the number of students who think pattern A will have more small squares and the number who think pattern B will have more small squares.

Arrange students in groups of 2. If time is limited, ask partners to each complete the questions one pattern and then work together to compare the patterns and make observations.

Some students may choose to use a spreadsheet tool to study the pattern, and subsequently to use graphing technology to plot the data. Others may wish to use a calculator to compute growth factors. Provide access to devices that can run a spreadsheet tool, graphing technology, or a scientific calculator.

Student Facing

  • In Pattern A, the length and width of the rectangle grow by one small square from each step to the next.
  • In Pattern B, the number of small squares doubles from each step to the next.
  • In each pattern, the number of small squares is a function of the step number, \(n\).

Pattern A

Four steps of a growing pattern. Step 0 is blank. Step 1: One square.  Step 2: Four squares in a two by two square. Step 3: Nine squares in a three by three square.

Pattern B

Four steps of a growing pattern. Step 0: One square. Step 1: Two squares, one atop the other.  Step 2: Four squares in a two by two square. Step 3: Eight squares in a two by four rectangle.
  1. Write an equation to represent the number of small squares at Step \(n\) in Pattern A.
  2. Is the function linear, quadratic, or exponential?
  3. Complete the table:
    \(n\), step number \(f(n)\), number of small squares
    0  
    1  
    2  
    3  
    4  
    5  
    6  
    7  
    8  
  1. Write an equation to represent the number of small squares at Step \(n\) in Pattern B.
  2. Is the function linear, quadratic, or exponential?
  3. Complete the table:
    \(n\), step number \(g(n)\), number of small squares
    0  
    1  
    2  
    3  
    4  
    5  
    6  
    7  
    8  

How would the two patterns compare if they continue to grow? Make 1–2 observations.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Some students may write the equation for pattern B as \(g(n)=2n\). Point out that pattern B is doubling the number of small squares. Step 3 would have 4 small squares. Prompt students to test their equation when \(n=3\) to see if it gives the correct output. \(g(3)=2(3)=6\) not 4, so a linear function does not work. Since pattern B is doubling, the function is exponential not linear. A linear pattern such as \(2n\) would add 2 small squares at each step rather than double the number of small squares.

Activity Synthesis

Select students to share their equations and to display their tables for all to see. Invite others to share their observations about the values in the tables.

To help students understand why the value of the exponential function outgrows that of the quadratic function, consider showing tables that contrast the output values of \(f\) and \(g\) and amending each with a third column that shows their growth factors as \(n\) goes up by 1.

\(n\), step number \(f(n)\), number of squares growth factor (to 2 places)
0 0  
1 1 undefined
2 4 \(\frac41=4\)
3 9 \(\frac94=2.25\)
4 16 \(\frac{16}{9}=1.78\)
5 25 \(\frac{25}{16}=1.56\)
6 36 \(\frac{36}{25}=1.44\)
7 49 \(\frac{49}{36}=1.36\)
8 64 \(\frac{64}{49}=1.31\)
\(n\), step number \(g(n)\), number of squares growth factor
0 1  
1 2 \(\frac21=2\)
2 4 \(\frac42=2\)
3 8 \(\frac84=2\)
4 16 \(\frac{16}{8}=2\)
5 32 \(\frac{32}{16}=2\)
6 64 \(\frac{64}{32}=2\)
7 128 \(\frac{128}{64}=2\)
8 256 \(\frac{256}{128}=2\)

Highlight the fact that a fundamental feature of an exponential function is that it changes by equal factors over equal intervals. In this exponential function, the output increases by a factor of 2 at each step.

In the quadratic function, we can see that the output changes by a factor of 4, then \(2\frac14\), then \(1\frac79\), and so on. Even though it started out growing faster than the exponential function, the growth factor of the quadratic function decreases at each step and falls below 2 after a couple of steps. In the meantime, the growth factor of the exponential function stays at 2.

A graph plotting the outputs of f and g at whole-number inputs.

Also consider showing the graphs representing the two functions to help students visualize the data in the tables. This graph shows plots the outputs of \(f\) and \(g\) at whole-number inputs.

Discuss how the graphs representing both quadratic and exponential functions curve upward. The two are very close together for small values of \(x\). As \(x\) continues to grow, however, the values of \(g\) become much greater than those of \(f\) and continue to increase more quickly.

Representing, Conversing: MLR7 Compare and Connect. As students share their equations and tables with the class, call students’ attention to the ways the dimensions are represented within the context of each equation. Take a close look at the equation to distinguish how each pattern is represented. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness; Support sense-making
Representation: Internalize Comprehension. Demonstrate and encourage students to use color-coding and annotations to highlight connections between representations. Use one color to highlight connections between the growth factor and the graph of the quadratic function, and another color for the exponential function. 
Supports accessibility for: Visual-spatial processing

4.3: Comparing Two More Functions (15 minutes)

Activity

Students continue to compare quadratic and exponential functions in this activity. This time, they decide how to compare the functions.

Monitor for different strategies students may use to compare the functions. Identify students who:

  • Create one or more tables of values and compare the values of \(p(x)\) and \(q(x)\) at increasingly large values of \(x\).
  • Graph the functions defined by \(p(x) = 6x^2\) and \(q(x) = 3^x\) and compare the graphs.
  • Create one or more tables of values, calculate the growth factors at equal intervals of input, and then compare the growth factors.

If students choose graphing or spreadsheet tools strategically to perform comparisons, they practice MP5.

Launch

Ask students to observe the equations representing the two functions and determine which function is exponential and which is quadratic. Invite students to share how they know. Make sure students recognize that \(p\) is quadratic and \(q\) is exponential before they begin the activity.

Provide access to graphing technology and spreadsheet tools. This may be a good opportunity for students to experiment with the graphing window. If the horizontal dimension is very small (for example, \(0<x<5\)) or the vertical dimension is very large (for example, \(0<y<3,\!000\)), the two graphs will be hard to distinguish. As needed, remind students to think about adjusting the graphing window to make the graphs more informative. 

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to adjust the graphing window to experiment with the horizontal and vertical dimensions. 
Supports accessibility for: Organization; Conceptual processing; Attention

Student Facing

Here are two functions: \(p(x) = 6x^2\) and \(q(x) = 3^x\).

Investigate the output of \(p\) and \(q\) for different values of \(x\). For large enough values of \(x\), one function will have a greater value than the other. Which function will have a greater value as \(x\) increases?

Support your answer with tables, graphs, or other representations.

Student Response

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Student Facing

Are you ready for more?

  1. Jada says that some exponential functions grow more slowly than the quadratic function as \(x\) increases. Do you agree with Jada? Explain your reasoning.
  2. Could you have an exponential function \(g(x)=b^x\) so that \(g(x) < f(x)\) for all values of \(x\)?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Select students to present their strategies in the sequence listed in the Activity Narrative. If no students chose to graph the functions, consider displaying the graphs for all to see.  By presenting strategies that involved calculation of growth factors last, students will deepen their understanding of why exponential values will eventually overtake the quadratic values.

Again, one way to make sense of why exponential functions eventually grow faster than quadratic functions is by thinking of the growth factors. The output of an exponential function always increases by the same factor when the input increases by 1 (for example, 3 for the exponential function \(q\) studied here). The output of a quadratic function, on the other hand, increases by smaller and smaller amounts when the input increases by 1. So even though a quadratic function may take larger values than an exponential function for many inputs, the values of the exponential function will eventually overtake the quadratic.

Conversing: MLR2 Collect and Display. As students share their analysis with the class, listen for and collect the language students use to identify and describe how the output of the exponential function eventually outgrows that of the quadratic function. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Ask students to reflect on how they analyzed the behaviors of quadratic and exponential functions. Discuss questions such as:

  • “What are some ways for comparing a quadratic growth and an exponential growth?” (By comparing their values in a table, by graphing the equations representing them, or by comparing the growth factors as the input increases.)
  • “When we compared \(n^2\) and \(2^n\), we saw the value of \(2^n\) become greater than \(n^2\) at \(n=5\). When we compared \(6x^2\) and \(3^x\), we saw \(3^x\) overtaking \(6x^2\) by the time \(x\) reaches 5. If we compare, say, \(1,\!000x^2\) and \(2^x\), will the exponential still overtake the quadratic? If so, at what \(x\) value do you think it would happen? If not, why not?” (Yes. It’d probably happen when \(x\) is between 15 and 20.)

4.4: Cool-down - Comparing $5x^2$ and $2^x$ (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

We have seen that the graphs of quadratic functions can curve upward. Graphs of exponential functions, with base larger than 1, also curve upward. To compare the two, let’s look at the quadratic expression \(3n^2\) and the exponential expression \(2^n\).

A table of values shows that \(3n^2\) is initially greater than \(2^n\) but \(2^n\) eventually becomes greater.

\(n\) \(3n^2\) \(2^n\)
1 3 2
2 12 4
3 27 8
4 48 16
5 75 32
6 108 64
7 147 128
8 192 256

We also saw an explanation for why exponential growth eventually overtakes quadratic growth.

  • When \(n\) increases by 1, the exponential expression \(2^n\) always increases by a factor of 2.
  • The quadratic expression \(3n^2\) increases by different factors, depending on \(n\), but these factors get smaller. For example, when \(n\) increases from 2 to 3, the factor is \(\frac{27}{12}\) or 2.25. When \(n\) increases from 6 to 7, the factor is \(\frac {147}{108}\) or about 1.36. As \(n\) increases to larger and larger values, \(3n^2\) grows by a factor that gets closer and closer to 1.

A quantity that always doubles will eventually overtake a quantity growing by this smaller factor at each step.