# Lesson 2

Equations and Graphs

• Let’s explore solutions to equations

### 2.1: The Word List

A group is asked to memorize a list of 20 words, then recall as many as possible later. An equation that models the relationship between the position of the word on the list, $$n$$, and the number of people in the group who remembered the word, $$P$$, is $$P = 0.34n^2 -8.7n +97.3$$.

What do you notice? What do you wonder?

### 2.2: Seeing Solutions

1. A person is hiking from the top of a mountain into a valley. The function $$2,\!000 - 32t$$ represents their elevation in feet above sea level, $$t$$ minutes after they started their hike.
1. What does a solution to the equation $$2000-32t=0$$ mean?
2. Use technology to create a graph of $$y=2,\!000-32t$$. Where do you see the solution to that equation on the graph?
2. A new electronic device originally costs $1,000 but loses$175 worth of value every year.
1. Write a function that represents the worth of the device after $$s$$ years.
2. How many years until the device is worth $0? 3. Use technology to graph the function. Where can you see the solution to your equation on the graph? ### 2.3: Understanding Solutions in Situations 1. The expression $$5.25+0.85x$$ represents the amount a yogurt shop charges for yogurt with $$x$$ ounces of toppings. 1. What does the equation $$5.25+0.85x=7.08$$ mean in this situation? 2. What would a solution to this equation mean? 3. Use technology to graph $$y=5.25+0.85x$$. Where can you see the solution to the equation on the graph? 2. Drinks cost$1.50, sandwiches cost $4.00, and there is a flat delivery fee of$5 for each delivery regardless of the number of orders.
1. Write an expression that represents the amount it costs to have $$x$$ meals including a drink and a sandwich delivered to an office.
2. Write an equation that has a solution representing the number of drink and sandwich orders it would take to cost \$80.
3. Graph $$y=1.5x+4x+5$$. Where can you see the solution to the equation on the graph?
3. The temperature in a deep freezer in a laboratory is -40 degrees Celsius. The freezer breaks, so the temperature starts to rise by 2.5 degrees per hour.
1. Use technology to graph $$y=\text-40+2.5x$$.
2. Explain how to use this graph to find the time (after breaking) when the freezer temperature reaches 0 degrees Celsius.
4. The expression $$400 - 10x^2$$ represents the height in meters of an object above the ground $$x$$ seconds after falling off a 400 meter building.
1. Write an equation that has a solution that would give the time in seconds when the object hit the ground.
2. Use technology to graph $$y=400-10x^2$$ and explain where you can see the solution to your equation on the graph.