Evaluating Graphs and Equations Using Function Notation

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SWBAT evaluate f(x) and x using given information and the equation or graph of the function.

Big Idea

To be able to read and comprehend function notation to know the difference between when to find x and when to find y.

Warm Up

10 minutes

I plan for today's Warm Up to take about 10 minutes for the students to complete and to review with the class.  I chose this Warm Up to reinforce my students' fluency with the vocabulary of independent variable, dependent variable, and function. I use the student's prior knowledge of these concepts to introduce how all of these words relate to function notation f(x). I demonstrate reviewing the Warm Up in the video below. 

Guided Practice

15 minutes

At the beginning of today's Guided Practice I provide a quadratic function and a t-table for students to complete. I ask students to evaluate the function for a set of input values. My primary objective of this lesson is for students to understand when to find x and when to find f(x) when evaluating functions.

I give only the first input value, -2. I expect most of my students will complete the table with the input values of -1. 0, 1, and 2.  This will include the vertex and increasing values on each side of the vertex. I have not yet taught my students how to find the vertex of a quadratic function. Building on the previous lesson, Evaluating Functions using Function Notation, in this practice students will learn about transformations of functions. My students know enough about the parent function f(x) = x^2, to begin to understand this function in relation to its parent function. 

The second part of the Guided Practice gives students practice evaluating the graph of a function using function notation.  I give students about 5 minutes to complete Part 2. Then, I will randomly call on students to provide the answer to different problems.  

By the end of the Guided Practice, my students should understand when to find x or f(x) from a graph using function notation.  This understanding should come from the structure of the problem (MP7) and repeated reasoning (MP8) provided from the Warm Up and the Guided Practice.

Independent Practice

15 minutes

Today's Independent Practice provides more practice evaluating functions. I include problems that focus on the following skills:

  • Evaluating a function for x and f(x) using an equation
  • Graphing a function using a t-table
  • Evaluating a function for x and f(x) using a graph

The Independent Practice should take my students about 15 minutes to complete. On some occasions, I have my students begin with Number 6. I think that it is a good problem for students to discuss in pairs and provide feedback to each other before working independently (MP3).  For Problem 6, I like to have students work the problem independently for a few minutes, and then compare to their table partner.

Teacher's Note: Problem 6 is a multi-step problem.  Students have to write the equation for the situation of the bathtub draining, and use the equation or the graph to answer questions about G(s) and s.  With G(s) defined as number of gallons of water remaining in the tub, and s as the number of seconds since the plug has been pulled.  While students are working, I walk around the room to monitor progress.  As I am monitoring progress, I am looking for students that have used different methods, students with common incorrect responses, and correct responses to share out with the class.

I have provided three student responses of this problem to show the progress of students as an indication of how students may struggle with this work.

  • Student 1 and Student2 have defined the independent variable as x to represent seconds since the plug had been pulled.  They also both define y as the dependent variable to represent the number of gallons of water remaining in the tub.  Student 1 scales the x axis by tens and the y axis by fives. Student 2 scales the x axis by tens and the y axis by twos.  
  • Student 3 defined the x axis as gallons of water and the y axis as number of seconds, which is incorrect for this context. 

In the above examples, all three students have difficulty stating the correct slope because they did not apply their scale of the graph correctly.  For example, Student 1 moved down one and two to the right, so the student stated the slope as 1/2. However, using the correct scale of the graph, the slope should have been down 5 and 20 to the right which is -1/4.

Student 3 had some correct answers even though the independent and dependent variable were switched.  The students evaluated the y-intercept, the number of gallons left after 120 seconds, and how many seconds have passed to have five gallons of water remaining. However, the students obtained these answers by reading the graph.  So when reading the graph for how many gallons of water was left in the tub after 30 seconds, the students had to estimate if they used the graph because it was in between two whole numbers.  For a precise answer, the student would need to substitute 30 in for x seconds, and evaluate the equation for y gallons of water. 

Exit Slip

5 minutes

I use the Exit slip as a quick formative assessment to check for each student's understanding of evaluating a function from a graph using function notation.  I give the students the Exit Slip about 10 minutes before the end of class.  Students are to complete the Exit Slips on their own without assistance from their table partner or me.  This way I can individually assess the objective which is when and how to find x and f(x) from a graph using function notation.

Some students may not have completed the Independent Practice before I give out the Exit Slip.  I instruct all students to work on the Exit Slip with 10 minutes left in the class, and to hand the Exit Slip in before they leave.  If students have not completed the Independent Practice, then I assign it as homework.  If students do not finish the Exit Slip, I still take it up for a formative assessment if I think that the student has had adequate time to work on it.