# Lesson 23

Comparing Functions

These materials, when encountered before Algebra 1, Unit 7, Lesson 23 support success in that lesson.

## 23.1: Math Talk: Evaluating Functions (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for evaluating the value of functions given an input value. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to evaluate functions to make use of the vertex form of a quadratic.

### 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

Mentally evaluate each of the functions when $$x = 3$$.

$$f(x) = x^2 - 4x + 1$$

$$g(x) = 6x - 2x^2$$

$$h(x) = (x-4)(x-3)$$

$$j(x) = 2(x-1)(x+2)$$

### 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?”

## 23.2: Comparing Functions (15 minutes)

### Activity

In this activity, students compare the values of a function with different $$x$$-coordinates using equations and graphs. In the associated Algebra 1 lesson, students use the vertex form of a quadratic equation to find the maximum or minimum. This work supports students by focusing on the values of the equation and determining which is greater.

### Launch

Display the equation and graph for all to see.

$$f(x) = x^2 - 3x + 5$$

Ask students how they can find the value of $$f(2)$$ using the equation and the graph.

### Student Facing

The notation $$f(2)$$ means the output of function $$f$$ when $$x$$ is 2. For each function, determine whether $$f(2) > f(3)$$, $$f(2) < f(3)$$, or $$f(2) = f(3)$$.

1. $$f(x) = x^2 + 2x+ 3$$
2. $$f(x) = (x-2)(x-3)$$
3. $$f(x) = \text{-}x^2 + 5$$

### Activity Synthesis

The purpose of the discussion is to find methods for comparing function values using equations and graphs. Select students to share their solutions and methods for comparing the values. Ask students,

• “Is it easier to use the graph or equation to determine which value is greater using the equation or the graph?” (For some functions, it was easier to use the graph since one value is clearly higher on the graph, but for other functions, it is difficult to see in the graph, so finding the exact values using the function is easier.)
• “Invent an equation for which $$f(2) > f(3)$$. How did you come up with your function?” (The equation $$f(x) = \text{-}x$$ has $$f(2) > f(3)$$. I wanted an equation that is decreasing so that moving left to right along the graph would have lower values. I thought this equation would be simple and decreasing like I wanted.)

## 23.3: Finding the Vertex (20 minutes)

### Activity

In this activity, students convert functions in factored or standard form to vertex form. In the associated Algebra 1 lesson, students use the vertex form to find the maximum or minimum of the quadratic function. This activity supports students by focusing on the mechanics of changing forms and finding the coordinates of the vertex. Students look for and make use of structure (MP7) when they use an equation to identify the vertex of a graph.

### Student Facing

Write each function in vertex form, then find the coordinates of the vertex.

1. $$y = x^2 - 4x + 7$$
2. $$y = (x-1)(x+3)$$
3. $$y = (x-2)(x+2)$$
4. $$y = x^2 - 2x + 1$$
5. $$y = \text{-}x^2 -2x-6$$
6. $$y = 2x^2 - 12x + 22$$
• “When the coefficient of $$x^2$$ is not _____ , what do you have to do differently?” (The coefficient needs to be factored out first and distributed back in after completing the square.)
• “After completing the square, when one of the terms seems to be missing, like $$y = x^2 - 8$$ or $$y = (x-4)^2$$, what does that mean about the vertex?” (It means it is on one of the axes. In the first example, the vertex is $$(0,\text{-}8)$$ and in the second example, the vertex is $$(4,0)$$. One of the coordinates is zero in these cases, which puts the vertex on an axis.)