Introduction

Function modeling is like being a detective with math skills. Which model is best for modeling constant rates of change? It’s questions like these that are important for an AP® Precalculus student learning 1.13 Function Model Selection. It helps you understand the patterns and relationships between pieces of data. Why is this important? Well, knowing which function model to choose can make a complex situation much easier to understand and predict. This guide will walk through several types of function models, showing how each can be used to represent real-world scenarios effectively.

Understanding the Constant Rate of Change

The constant rate of change is a term used in math to describe a situation where something changes consistently over time. Think of driving a car at a steady speed. This concept is closely tied to linear functions.

• Definition: The constant rate of change is the rate at which a quantity increases or decreases. It’s the same at every interval.
• Connection to Linear Functions: In a linear function, graphing the data points forms a straight line, showing a constant rate of change.

Example: Calculating the Constant Rate of Change

Consider a situation where a car travels 60 miles in 2 hours and 90 miles in 3 hours. The rate of change of the distance is calculated as:

\text{Rate of change} = \frac{\text{Change in distance}}{\text{Change in time}}

Calculate: = \frac{90 - 60}{3 - 2} = \frac{30}{1} = 30 \text{ miles per hour}

Therefore, the car travels at a constant speed of 30 miles per hour.

Modeling with Linear Functions: Appropriate Contexts

Linear functions represent situations where the change between any two points is consistent.

• Characteristics:
  • Graph is a straight line.
  • Represented by y = mx + b, where m is the slope and b is the y-intercept.
• When to Use Linear Functions: Use when modeling scenarios with a constant rate of change, like speed or cost per item.

Example: Creating a Linear Function that Models the Data Table

Imagine the weekly cost C of renting a car as dependent on the number of days d rented. The cost data is:

Days (d)	Cost (C)
2	$100
4	$200

Calculate the linear function:

Find the slope (m):

m = \dfrac{\text{Change in cost}}{\text{Change in days}} = \dfrac{200 - 100}{4 - 2} = 50

The equation is: C = 50d

So, the cost changes by $50 per day.

Modeling with Quadratic Functions: Symmetry

Quadratic functions are key when things change direction or curve. Sporting events, such as a ball thrown in the air, often use quadratic functions to model their paths.

• Understanding Quadratic Functions:
  • Characterized by the equation y = ax^2 + bx + c.
  • The graph forms a parabola.
• Situations to Apply: Use in scenarios like projectile motion or profit maximization.

Example: Determining a Quadratic Function from Real-World Data

If a ball is thrown upwards, its height (h) over time (t) might be modeled as:

h(t) = -16t^2 + 32t + 5

Here, the coefficients describe the influence of gravity and initial velocity.

To find height at t = 1 second:

h(1) = -16(1)^2 + 32(1) + 5 = -16 + 32 + 5 = 21

The ball reaches 21 feet at 1 second.

Modeling with Polynomial Functions

For more complex curves, polynomial functions of a higher degree come into play.

• Explanation:
  • A polynomial function of degree n has at most n – 1 turning points.
  • For example, p(x) = ax^3 + bx^2 + cx + d is a cubic polynomial.
• Usage: They’re necessary when data has multiple peaks or changes directions multiple times.

Real-World Context Example: Using a Polynomial to Model Roller Coaster Motion

Context: Polynomial functions are often used to model the motion of a roller coaster along a track. The height of the roller coaster at different points along the track can be described using a polynomial with multiple maxima and minima, corresponding to the peaks and valleys of the ride.

Problem: A roller coaster’s height, h(x), in meters, as a function of horizontal distance, x, in meters, is modeled by a cubic polynomial. The coaster starts at ground level, rises to a peak, dips into a valley, and ends at a higher platform. The following data points describe the coaster’s height at key distances:

• h(0) = 0 (start at ground level)
• h(50) = 80 (first peak)
• h(100) = 20 (valley)
• h(150) = 60 (higher platform)

Find a cubic polynomial h(x) that models this data.

Solution: Assume the cubic polynomial is: h(x) = ax^3 + bx^2 + cx + d.

Substitute the given points into the polynomial to form equations:

• For h(0) = 0: d = 0.
• For h(50) = 80: a(50)^3 + b(50)^2 + c(50) + 0 = 80, or 125000a + 2500b + 50c = 80.
• For h(100) = 20: a(100)^3 + b(100)^2 + c(100) + 0 = 20, or 1000000a + 10000b + 100c = 20.
• For h(150) = 60: a(150)^3 + b(150)^2 + c(150) + 0 = 60, or 3375000a + 22500b + 150c = 60.

Simplify and solve the system of equations:

• 125000a + 2500b + 50c = 80
• 1000000a + 10000b + 100c = 20
• 3375000a + 22500b + 150c = 60

Divide each equation by powers of 10 for simplicity:

• 125a + 2.5b + 0.5c = 0.8
• 1000a + 10b + c = 0.2
• 3375a + 22.5b + 1.5c = 0.6

After solving the system using substitution or matrices, we find the following values: a=0.00032, b=-0.076, c=4.6

So, the polynomial is: h(x)=0.00032x^3-0.076x^2 +4.6x

Piecewise Functions: Addressing Variability

Piecewise functions tackle situations with different rules at different intervals. A classic example is tax brackets, where the rate changes at certain income levels.

• Definition: They are composed of multiple sub-functions, each with its domain.
• Utility: Situations with abrupt changes or different conditions.

Example: Constructing a Piecewise Function from a Scenario

Traffic fines increase based on speed over the limit. The fine F(x) is $100 for speeds over 50 but $150 for speeds over 70.

Express F(x) as:

F(x) = \begin{cases}100, & \text{if } 50 < x \leq 70 \\ 150, & \text{if } x > 70\end{cases}

This piecewise-defined function is useful because this scenario demonstrates different characteristics over different intervals.

Quick Reference Chart: Vocabulary and Definitions

Term	Definition
Constant Rate of Change	Rate at which a quantity increases; remains the same over intervals.
Linear Function	A function with a constant rate of change, modeled as y = mx + b.
Quadratic Function	A curved function represented by y = ax^2 + bx + c.
Polynomial Function	A function with multiple terms, such as p(x) = ax^n + \ldots + c.
Piecewise Function	Functions with different expressions over specific portions of their domain.

Conclusion

The ability to select the correct function model is vital for understanding and predicting real-world phenomena. Whether calculating constant rates or modeling complex data with polynomial functions, practice helps in mastering these skills.

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