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Learn how to look at a set of data and decide which mathematical model fits best for making predictions.
Would you rather have $1,000,000 today, or a single penny that doubles in value every day for a month? Your choice could be the difference between a nice car and enough money to buy a sports team!
A linear function is the simplest model. It represents a situation where something changes by the exact same amount every time. In a table of values, we look at the first differences—the subtraction of consecutive -values. If these differences are constant, the model is linear. The equation takes the form , where is that constant rate of change. Think of it like walking at a steady pace: every second, you cover the exact same distance.
Determine if the following data is linear: 1. Data: 2. Calculate first differences: , , . 3. Since the first difference is a constant , this is a linear function with the equation .
Quick Check
If the first differences of a table are all -5, what type of function are you looking at?
Answer
A linear function.
Sometimes the rate of change isn't constant—it's accelerating or decelerating. A quadratic function, written as , shows up when the second differences are constant. To find these, you first calculate the differences between -values, and then you calculate the differences between those results. This model is perfect for things like gravity or the area of a growing square. On a graph, it forms a U-shaped curve called a parabola.
Check this data for a quadratic pattern: 1. Data: 2. First differences: , , . 3. Second differences: , . 4. Because the second difference is a constant , this is a quadratic function.
Quick Check
If the first differences of a data set are 3, 5, 7, and 9, what is the constant second difference?
Answer
The constant second difference is 2.
An exponential function, , doesn't have constant differences. Instead, it has a common ratio. This means you multiply the previous -value by the same number to get the next one. While linear and quadratic functions eventually grow quite large, exponential functions are the 'speed kings' of math. As gets larger, an exponential function will eventually surpass any linear or quadratic function, no matter how big the other functions start.
Compare (Quadratic) and (Exponential): 1. At : while . The quadratic is winning! 2. At : while . The exponential has taken the lead. 3. At : while . The exponential function is now vastly larger.
Which model fits a table where the -values are ?
If a function's second differences are all , what is the function type?
True or False: A quadratic function will eventually grow faster than an exponential function if the quadratic's starting value is much higher.
Review Tomorrow
Tomorrow morning, try to explain to a friend how to tell the difference between a constant difference and a common ratio.
Practice Activity
Find a table of data from a science experiment or sports stats and calculate the first and second differences to see which model fits best.