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Change in Tandem

Lesson ~11 min read 8 MCQs

In simple terms: In simple terms, this topic is about telling the story of a relationship between two things by looking at its graph—how fast it's changing, in what direction, and where it's headed next.

Why this matters

Imagine pulling a fresh pizza out of the oven. It's scorching hot, right? You set it on the counter in your 70°F kitchen. For the first few minutes, the temperature drops like a rock. It goes from 350°F to 200°F really fast. But after ten minutes, it's cooling much more slowly. The change from 120°F to 110°F takes longer than the change from 350°F to 340°F did.

The relationship between time and pizza temperature isn't a straight line. It changes. The rate of cooling is not constant. AP Precalculus starts here, with this fundamental idea: how do two quantities change together, or "in tandem"? We'll learn how to describe these changes precisely and turn everyday stories, like a cooling pizza or a runner's race, into graphs that tell us everything we need to know.

Pizza cooling curve: rapid initial drop, then slower cooling over time.

Concept overview

flowchart TD
    A[Verbal Description of a Scenario] --> B{Identify Quantities};
    B --> C[Independent Variable --> x-axis];
    B --> D[Dependent Variable --> y-axis];
    C & D --> E{Analyze Direction};
    E --> F[Uphill --> Increasing];
    E --> G[Downhill --> Decreasing];
    F & G --> H{Analyze Rate of Change};
    H --> I[Speeding Up --> Concave Up];
    H --> J[Slowing Down --> Concave Down];
    I & J --> K[Plot Key Points: Start, End, Zeros];
    K --> L[Sketch the Graph];

Core explanation

Welcome to your first topic in AP Precalculus! We're starting with an idea that's going to be your best friend all year: change in tandem. It sounds fancy, but it's just about how two related things change together.

The Foundation: What is a Function?

Before we can talk about change, let's make sure we're solid on what a function is.

A function is like a rule or a machine. You give it an input, and it gives you back exactly one output.

  • Input
    The value you put in. We call this the independent variable (usually x).
  • Output
    The value you get out. We call this the dependent variable (usually y or f(x)), because its value depends on the input you chose.

Think of a vending machine in your school cafeteria. Your input is pressing button "B4". The output is a bag of chips. For that one input, you get exactly one output. It wouldn't be a very good machine if "B4" sometimes gave you chips and sometimes gave you a soda. That's the key to a function: every input has only one possible output.

The set of all possible inputs is called the domain. The set of all resulting outputs is the range.

Seeing the Change: Increasing vs. Decreasing

Now, let's get things moving. When we graph a function, we're plotting all the (input, output) pairs. We read a graph from left to right, just like a book. As we move to the right (as the input values increase), what are the output values doing?

Visualizing increasing and decreasing intervals on a function graph.
  • A function is increasing if the output values go up as the input values go up. On the graph, you're going uphill.
  • A function is decreasing if the output values go down as the input values go up. On the graph, you're going downhill.

Imagine tracking your phone's battery percentage over time while you're playing a game.

  • Input (Independent Variable)
    Time spent playing (in minutes).
  • Output (Dependent Variable)
    Battery percentage.

As time increases, your battery percentage decreases. So, the function Battery(time) is a decreasing function.

The Next Level: Concavity

This is where things get really interesting. It's not just about if the function is changing, but how it's changing. Is the change speeding up or slowing down? This is called concavity.

Let's use our interactive vase example. We're pouring water into a specially shaped vase at a constant rate.

  • Input (V)
    The volume of water we've added.
  • Output (h)
    The height of the water in the vase.

As we add water (increasing V), the height will always go up (h is always increasing). So, this is an increasing function. But the shape of the graph tells a deeper story.

A graph of water height vs. volume for a non-uniform vase, showing concave down and concave up sections.

Concave Down: The Rate of Change is Decreasing The vase is narrow at the bottom. The first few ounces of water fill it up quickly, so the height shoots up. Then, the vase widens. Now, the same amount of water only raises the height by a little bit.

Even though the height is still increasing, the rate at which it's increasing has slowed down. This is concave down.

  • Analogy
    You're driving and take your foot off the gas. You're still moving forward (increasing distance), but you're slowing down.
  • On the graph
    The curve bends downward, like a frown or a cap.

Concave Up: The Rate of Change is Increasing After the widest point, the vase starts to get narrow again towards the top. Now, as we add water, the height starts to rise more quickly again. The rate of change is increasing. This is concave up.

  • Analogy
    You're pressing the accelerator. You're moving forward, and you're speeding up.
  • On the graph
    The curve bends upward, like a smile or a cup that could hold water.

Putting It All Together: Zeros

One last key feature for now. What happens when the graph crosses the horizontal axis (the x-axis)? This is where the output value is zero. We call the input value that causes this a zero of the function.

For example, if we have a function P(t) that represents a company's profit P in month t, the zeros of the function would be the months where the company broke even (had zero profit). These are often critical points in the story a graph tells.

By combining these ideas—increasing/decreasing, concavity, and zeros—we can look at any graph and tell a detailed story about the relationship it represents.

Concavity: how the rate of change itself is changing.

Worked examples

Example 1

Graphing a Story — The Hot Air Balloon

Problem: Carlos is in a hot air balloon. He starts on the ground. For the first 5 minutes, he rises at a constant, quick pace. For the next 10 minutes, he rises, but more slowly as the air cools. Then, for 5 minutes, he maintains a constant altitude. Finally, he descends back to the ground at a steady rate, taking 10 minutes to land. Sketch a graph of the balloon's altitude versus time.

Solution:

  1. 1
    Identify Variables & Axes
    • The input (independent variable) is time (in minutes). Let's put this on the x-axis.
    • The output (dependent variable) is altitude (in feet). This goes on the y-axis.
    • The graph starts at the origin (0,0) because at time 0, the altitude is 0.
  2. 2
    Translate the Story, Piece by Piece
    • 0-5 minutes
      "rises at a constant, quick pace."
      • "Rises" means the graph is increasing.
      • "Constant pace" means the rate of change is constant. This is a straight line (no concavity).
      • "Quick pace" means it's a steep line. So, from t=0 to t=5, draw a steep, straight line going up.
    • 5-15 minutes
      "rises, but more slowly."
      • "Rises" means the graph is still increasing.
      • "More slowly" means the rate of change is decreasing. The graph is concave down. It's still going up, but it's leveling off. From t=5 to t=15, draw an increasing curve that bends downward.
    • 15-20 minutes
      "maintains a constant altitude."
      • Altitude is not changing. The rate of change is zero. This is a horizontal line.
    • 20-30 minutes
      "descends back to the ground at a steady rate."
      • "Descends" means the graph is decreasing.
      • "Steady rate" means a straight line.
      • It ends on the ground (altitude = 0) at t=30. So, draw a straight line from the point at t=20 down to the point (30, 0).
  3. 3
    The Final Sketch
    Your graph should show these four distinct segments connected, telling the full story of Carlos's balloon ride.
Example 2

Reading a Graph — A Runner's Race

Problem: The graph below shows Priya's velocity during a 100-meter race. Describe her race. When was she accelerating? When was she decelerating?

Priya's velocity during a 100-meter race: acceleration, peak, and deceleration.

(Imagine a graph where the y-axis is Velocity (m/s) and the x-axis is Time (s). The curve starts at (0,0), increases sharply and is concave up for about 2 seconds, then continues to increase but becomes concave down until about 6 seconds, then becomes a nearly horizontal line, and finally dips slightly near the end.)

Solution:

  1. 1
    Analyze the Axes
    The graph shows velocity vs. time. The y-value is her speed. A rising graph means she's speeding up (accelerating). A falling graph means she's slowing down (decelerating).
  2. 2
    Analyze Concavity (The "How")
    • Concave Up means the rate of change is increasing
      Here, the rate of change of velocity is acceleration. So, where the graph is concave up, her acceleration is increasing. She's "accelerating harder."
    • Concave Down means the rate of change is decreasing
      Here, it means her acceleration is decreasing. She's still speeding up (the graph is still rising), but not as effectively.
  3. 3
    Tell the Story
    • Beginning (approx. 0-2s)
      The graph is increasing and concave up. Priya explodes out of the starting blocks. Her velocity is increasing, and her rate of acceleration is also increasing. She's getting faster, faster.
    • Middle (approx. 2-6s)
      The graph is still increasing but is now concave down. She is still getting faster, but the rate at which she's gaining speed is slowing down. This is the main phase of the race where she's approaching her top speed.
    • End (approx. 6-9s)
      The graph is nearly flat. Her velocity is almost constant. She has reached her top speed and is trying to maintain it.
    • Finish Line (approx. 9-10s)
      The graph dips slightly. She is slightly decelerating, likely due to fatigue, as she crosses the finish line.

Try it yourself

Practice Problem 1: A New App Launch

A gaming company, "PixelFun," launches a new mobile game. The number of daily downloads is tracked for the first 30 days.

  • Week 1
    Downloads grow slowly as word gets out.
  • Week 2
    A popular streamer features the game, and downloads explode, growing faster each day.
  • Week 3
    The growth continues, but the rate of increase slows down as the initial hype fades.
  • Week 4
    The number of daily downloads starts to slowly decline as the market becomes saturated.

Your task: Sketch a graph of Daily Downloads versus Time (in days). Label the sections corresponding to the four weeks and describe the intervals where the graph is increasing, decreasing, concave up, and concave down.

Practice Problem 2: Filling a Pool

You are filling a swimming pool with a hose that has a constant flow rate. The pool is shaped like a rectangle, but the bottom is sloped: it's shallow on one end and deep on the other. You start filling from the deep end.

Your task: Sketch a graph of the Water Height versus the Volume of water added. Will the graph be a straight line? If not, will it be concave up or concave down? Explain your reasoning. (Hint: Think about how much surface area the water has to cover as it rises.)

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