Lesson 4

Evaluating Quadratic and Exponential Functions

These materials, when encountered before Algebra 1, Unit 6, Lesson 4 support success in that lesson.

4.1: Math Talk: Exponents (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for evaluating expressions with exponents. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to evaluate more complicated expressions that involve exponents. In this activity, students have an opportunity to notice and make use of structure (MP7) because properties of exponents can be used to rewrite expressions so that they are easier to evaluate.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Evaluate mentally.

\(4^2\)

\(2^4\)

\(2^6\)

\(4^3\)

Student Response

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

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

4.2: Evaluating and Describing Functions (20 minutes)

Activity

This task contains two distinct parts. Depending on what students need and time constraints, you could decide to do either or both.

In the associated Algebra 1 lesson, students will need to evaluate the functions \(6x^2\) and \(3^x\) for different input values. The purpose of the first part is to prepare them to avoid some common mistakes. Analyzing and correcting incorrect work is an opportunity to critique the reasoning of others (MP3).

The purpose of the second part is to revisit ways to tell whether a function is linear, quadratic, or exponential, and to practice the skills of generating a table and sketching a graph for a given function.

Student Facing

  1. Different students are evaluating two expressions, \(3\boldcdot 6^x\) and \(5^x\). Analyze their work, describe any errors made, and then evaluate each expression correctly.

      Noah’s work Mai’s work  corrected work  
    Evaluate \(5^x\) when \(x\) is 6.

    \(5^x\)

    \(5^6\)

    30

    \(5^x\)

    \(5^6\)

    \(6 \boldcdot 6 \boldcdot 6 \boldcdot 6 \boldcdot6\)

    7,776

     
    Evaluate \(3 \boldcdot 6^x\) when \(x\) is 2.

    \(3 \boldcdot 6^x\)

    \(3\boldcdot 6^2\)

    \(3 \boldcdot 12\)

    36

    \(3 \boldcdot 6^x\)

    \(3 \boldcdot 6^2\)

    \(18^2\)

    324

     
  2. Here are three functions. For each function:
    1. Complete the table of values.
    2. Sketch a graph.
    3. Decide whether each function is linear, quadratic, or exponential, and be prepared to explain how you know.

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

    \(x\) -1 0 1 2 3 5
    \(f(x)\)            

     

    horizontal axis, scale 0 to 5, by 1's. vertical axis, scale 0 to 110, by 10's.

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

    \(x\) -1 0 1 2 3 5
    \(g(x)\)            

     

    horizontal axis, scale 0 to 5, by 1's. vertical axis, scale 0 to 110, by 10's.

    \(h(x)=3 \boldcdot 2x\)

    \(x\) -1 0 1 2 3 5
    \(h(x)\)            

     

    horizontal axis, scale 0 to 5, by 1's. vertical axis, scale 0 to 110, by 10's.

Student Response

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

Activity Synthesis

For the first part, invite students to describe the errors they see, and share how to correctly evaluate each expression.

For the second part, display the tables and graphs for all to see. Ask students to share some ways they can tell a function is linear, exponential, or quadratic using the different representations. For example:

  • The graph of the linear function is a straight line.
  • The table shows the exponential function has a growth factor of 2. Each time \(x\) increases by 1, the \(f(x)\) values increase by a factor of 2.
  • The table shows the quadratic function does not have a constant rate of change or a constant growth factor.

4.3: Evaluating Exponential and Quadratic Expressions (20 minutes)

Activity

This practice activity is an opportunity to develop fluency in evaluating expressions that involve exponents using the conventional order, which will be useful in the associated Algebra 1 lesson when they generate tables of values for functions defined by such expressions. Encourage students to evaluate the expressions without using a calculator in order to practice mental math. The structure of a row game gives students an opportunity to construct viable arguments and critique the reasoning of others (MP3).

Launch

Arrange students in groups of 2, assigning one student as partner A and the other as partner B. Explain to students that there are two columns of problems, and that they only do the problems in their column. They should pause after each row to see if their answers match, and if not, work together to resolve any errors.

Student Facing

For each row, you and your partner will each evaluate an expression. You should each get the same answer in each row. If you disagree, work to reach agreement.

row Partner A PartnerB
1 \(4 \boldcdot 2^x\) when \(x\) is 3 \(2 \boldcdot 2^x\) when \(x\) is 4
2 \(19 + x^2\) when \(x\) is 9 \(4 \boldcdot x^2\) when \(x\) is 5
3 \(16 \boldcdot 2^x\) when \(x\) is 0 \(2 \boldcdot 2^x\) when \(x\) is 3
4 \(\frac12 \boldcdot 2^x\) when \(x\) is 4 \(x^2-1\) when \(x\) is 3
5 \(x^2+1\) when \(x\) is 7 \(18+2^x\) when \(x\) is 5
6 \(4+2^x\) when \(x\) is 4 \(\frac15 x^2\) when \(x\) is 10
7 \(0.1 x^2\) when \(x\) is 6 \(0.4 x^2\) when \(x\) is 3
8 \(45 \boldcdot x^2\) when \(x\) is \(\frac13\) \(10 \boldcdot 2^x\) when \(x\) is -1
9 \(x^2\) when \(x\) is -4 \(64x^2\) when \(x\) is \(\frac12\)
10 \(\text-2 x^2\) when \(x\) is 3 \(\text-2 x^2\) when \(x\) is -3

 

Student Response

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

Activity Synthesis

Much of the discussion will happen between partners. These expressions were chosen to highlight errors that people commonly make evaluating expressions. Here are a few specific conventions to draw students’ attention to:

  • When an exponent appears in an expression, evaluate the exponent before multiplication, division, addition, or subtraction. For example, \(4 \boldcdot 2^3\) is \(4 \boldcdot 8\), not \(8^3\).
  • Squaring a negative number results in a positive number, for example, \((\text-3)^2\) is 9. This is because \((\text-3)^2\) means \((\text-3) \boldcdot (\text-3)\), and a negative times a negative is a positive.

Additionally, this may be a good opportunity to revisit mentally evaluating expressions that involve a fraction, for example:

  • \(\frac15 \boldcdot 100 = 20\), because multiplying by \(\frac15\) is the same as dividing by 5.
  • Squaring a fraction just means you are multiplying a fraction by itself, for example \((\frac12)^2 = \frac12 \boldcdot \frac12\), which equals \(\frac14\).