# Lesson 24

Quadratic Situations

These materials, when encountered before Algebra 1, Unit 7, Lesson 24 support success in that lesson.

## 24.1: Growing Plants (10 minutes)

### Warm-up

In this warm-up, students match linear equations to their graphs, then find points on each line and interpret them in the situation. This work prepares students for later in the lesson when they interpret values in similar situations.

### Student Facing

Plant A’s height over time is represented by \(y=\frac{1}{2}x+4\). Plant B’s height is \(y=\frac{1}{3}x+3\) for which \(x\) represents the number of weeks since the plants were found, and \(y\) represents the height in inches.

- Which graph goes with which equation? How do you know?
- What is a pair of values that works for Plant A but not B? What does it represent?
- What is a pair of values that works for Plant B but not A? What does it represent?
- What is a pair of values that works for both plants? What does it represent?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to help students connect the situation to the equations and graphs. Ask students,

- “Which plant has a greater value for the slope? What does the value mean in terms of plant growth?” (Plant A has a greater slope since \(\frac{1}{2} > \frac{1}{3}\). This means that every 2 weeks, Plant A grows 1 inch.)
- “What do the vertical intercepts mean for these plants?” (They represent the height of the plants when they were found.)
- “What do the horizontal intercepts mean for these plants?” (They represent the time when the plants just broke through the ground.)
- “Do negative \(x\)- and \(y\)-values make sense in this situation?” (Yes, to a point. The plants were growing before they were found, so some negative \(x\)-values can make sense. The seeds for these plants were also underground at some point, so the plant could be considered to be below ground level, which would give meaning to negative \(y\)-values.)

## 24.2: Diego’s Plant (15 minutes)

### Activity

In this activity, students interpret a quadratic formula in a situation. In the associated Algebra 1 lesson, students solve quadratic equations in situations. This activity supports students by helping them connect the equation to the situation. Students use appropriate tools strategically (MP5) when they use graphing technology to examine a function.

### Launch

Allow students to use graphing technology to examine the function if they choose.

### Student Facing

- The height, in centimeters, of Diego’s plant is represented by the equation \(p(t) = \text{-}0.5(t-10)^2+58\) where \(t\) represents the number of weeks since Diego has started nurturing the plant. Determine if each statement is true or false. Explain your reasoning.
- Diego’s plant shrinks each week.
- Diego’s plant is 8 cm tall when he starts to nurture it.
- Diego’s plant grows to be 58 cm tall.
- The plant shrinks 4 weeks after Diego begins to nurture it.

- Write your own true statement about Diego’s plant.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to connect the equation to the situation. Select students to share their solutions and statements. After each statement, ask another student if they agree that the statement is true and how they can know it is true based on the equation or graph. Ask students,

- “After a while, Diego’s plant shrinks to be 0 cm tall. If this represents the time when the plant dies, what is an equation that could be solved to find the number of weeks after Diego begins nurturing the plant that the plant dies?” (\(p(t) = 0\) or \(\text{-}0.5(t-10)^2 + 58 = 0\))
- “What form is the equation given in? What information is easy to get from that form?” (The equation is given in vertex form. It is easy to get the vertex and not too difficult to find the \(y\)-intercept in this form by substituting 0 for \(t\).)

## 24.3: Making the Grades (15 minutes)

### Activity

In this activity, students solve a system of linear equations in a given context. In the associated Algebra 1 lesson, students solve a system of equations involving a quadratic equation. This activity supports students by reminding them of methods used to solve systems of equations.

### Launch

Remind students that an intersection point for graphs can be found by solving a system of equations. Ask students what methods they remember for solving a system of equations. If none of these methods are mentioned, consider telling students about them:

- Examining the graph
- Substitution method
- Elimination method

### Student Facing

Jada’s quiz grade after \(h\) hours of studying is given by the equation \(Q(h) = 10h + 70\). Her test grade after \(h\) hours of studying is given by the equation \(T(h) = 6h + 76\).

Here’s a graph of both functions:

- Which graph represents Jada’s quiz grade after \(h\) hours of studying?
- What do the \(y\)-intercepts of the lines mean in this situation?
- Find the coordinates of the \(y\)-intercepts.
- The 2 lines intersect at a point. What does that point represent in this situation?
- Find the coordinates of the intersection point. Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to recall methods for solving systems of equations and the interpretation of the solution. Select students to share their solutions. After students share their solution for the coordinate of the intersection point, ask if any other students solved it using a different method. Ask students,

- “If Jada’s test grade was given by a quadratic function, like \(T(h) = \text{-}(h-4)^2 + 100\), would the intersection have the same meaning? Could the intersection be found using a method like graphing or substitution?” (Yes, the intersection has the same meaning, but there may be more than 1 intersection point or there may be no intersection points. Examining a graph or using a substitution method could be used to find any intersection points.)
- “What are some values of intersection points that do not make sense in this situation?” (If \(h < 0\) or if the function values are much greater than 100, the values do not make much sense. Negative \(h\) values would represent negative study time which is not reasonable. Similarly, intersections with function values greater than 100 would represent scores that are not reasonable unless the teacher is giving a lot of extra credit on both the quiz and test.)