10.7: Linear, Exponential, and Quadratic Models
So far in this text you have learned how to graph three very important types of equations.
- Linear equations in slope-intercept form \begin{align*}y=mx+b\end{align*}
- Exponential equations of the form \begin{align*}y=a(b)^x\end{align*}
- Quadratic equations in standard form \begin{align*}y=ax^2+bx+c\end{align*}
In real-world applications, the function that describes some physical situation is not given. Finding the function is an important part of solving problems. For example, scientific data such as observations of planetary motion are often collected as a set of measurements given in a table. One job for the scientist is to figure out which function best fits the data. In this lesson, you will learn some methods that are used to identify which function describes the relationship between the dependent and independent variables in a problem.
Using Differences to Determine the Model
By finding the differences between the dependent values, we can determine the degree of the model for the data.
- If the first difference is the same value, the model will be linear.
- If the second difference is the same value, the model will be quadratic.
- If the number of times the difference has been taken exceeds five, the model may be exponential or some other special equation.
Example: The first difference is the same value (3). This data can be modeled using a linear regression line.
The equation to represent this data is \begin{align*}y=3x+2\end{align*}
When we look at the difference of the \begin{align*}y-\end{align*}values, we must make sure that we examine entries for which the \begin{align*}x-\end{align*}values increase by the same amount.
For example, examine the values in the following table.
At first glance, this function might not look linear because the difference in the \begin{align*}y-\end{align*}values is not always the same.
However, we see that the difference in \begin{align*}y-\end{align*}values is 5 when we increase the \begin{align*}x-\end{align*}values by 1, and it is 10 when we increase the \begin{align*}x-\end{align*}values by 2. This means that the difference in \begin{align*}y-\end{align*}values is always 5 when we increase the \begin{align*}x-\end{align*}values by 1. Therefore, the function is linear.
The equation is modeled by \begin{align*}y=5x+5\end{align*}.
An example of a quadratic model would have the following look when taking the second difference.
Using Ratios to Determine the Model
Finding the difference involves subtracting the dependent values leading to a degree of the model. By taking the ratio of the values, one can obtain whether the model is exponential.
If the ratio of dependent values is the same, then the data is modeled by an exponential equation, as in the example below.
Determine the Model Using a Graphing Calculator
To enter data into your graphing calculator, find the [STAT] button. Choose [EDIT].
- [L1] represents your independent variable, your \begin{align*}x\end{align*}.
- [L2] represents your dependent variable, your \begin{align*}y\end{align*}.
Enter the data into the appropriate list. Using the first set of data to illustrate yields:
You already know this data is best modeled by a linear regression line. Using the [CALCULATE] menu of your calculator, find the linear regression line, linreg.
Look at the screen above. This is where you can find the quadratic regression line [QUADREG], the cubic regression line [CUBICREG], and the exponential regression line, [EXPREG].
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK-12 Basic Algebra: Linear, Quadratic, and Exponential Models (8:15)
- The second set of differences have the same value. What can be concluded?
- Suppose you find the difference five different times and still don't come to a common value. What can you safely assume?
- Why would you test the ratio of differences?
- If you had a cubic (\begin{align*}3^{rd}\end{align*}-degree) function, what could you conclude about the differences?
Determine whether the data can be modeled by a linear equation, a quadratic equation, or neither.
- \begin{align*}& x && -4 && -3 && -2 && -1 && 0 && 1\\ & y && 10 && 7 && 4 && 1 && -2 && -5\end{align*}
- \begin{align*}& x && -2 && -1 && 0 && 1 && 2 && 3\\ & y && 4 && 3 && 2 && 3 && 6 && 11\end{align*}
- \begin{align*}& x && 0 && 1 && 2 && 3 && 4 && 5\\ & y && 50 && 75 && 100 && 125 && 150 && 175\end{align*}
- \begin{align*}& x && -10 && -5 && 0 && 5 && 10 && 15\\ & y && 10 && 2.5 && 0 && 2.5 && 10 && 22.5\end{align*}
- \begin{align*}& x && 1 && 2 && 3 && 4 && 5 && 6\\ & y && 4 && 6 && 6 && 4 && 0 && -6\end{align*}
- \begin{align*}& x && -3 && -2 && -1 && 0 && 1 && 2\\ & y && -27 && -8 && -1 && 0 && 1 && 8\end{align*}
Can the following data be modeled with an exponential function?
- \begin{align*}& x && 0 && 1 && 2 && 3 && 4 && 5\\ & y && 200 && 300 && 1800 && 8300 && 25,800 && 62,700\end{align*}
- \begin{align*}& x && 0 && 1 && 2 && 3 && 4 && 5\\ & y && 120 && 180 && 270 && 405 && 607.5 && 911.25\end{align*}
- \begin{align*}& x && 0 && 1 && 2 && 3 && 4 && 5\\ & y && 4000 && 2400 && 1440 && 864 && 518.4 && 311.04\end{align*}
Determine whether the data is best represented by a quadratic, linear, or exponential function. Find the function that best models the data.
- \begin{align*}& x && 0 && 1 && 2 && 3 && 4\\ & y && 400 && 500 && 625 && 781.25 && 976.5625\end{align*}
- \begin{align*}& x && -9 && -7 && -5 && -3 && -1 && 1\\ & y && -3 && -2 && -1 && 0 && 1 && 2\end{align*}
- \begin{align*}& x && -3 && -2 && -1 && 0 && 1 && 2 && 3\\ & y && 14 && 4 && -2 && -4 && -2 && 4 && 14\end{align*}
- As a ball bounces up and down, the maximum height it reaches continually decreases. The table below shows the height of the bounce with regard to time.
- Using a graphing calculator, create a scatter plot of this data.
- Find the quadratic function of best fit.
- Draw the quadratic function of best fit on top of the scatter plot.
- Find the maximum height the ball reaches.
- Predict how high the ball is at 2.5 seconds.
Time (seconds) | Height (inches) |
---|---|
2 | 2 |
2.2 | 16 |
2.4 | 24 |
2.6 | 33 |
2.8 | 38 |
3.0 | 42 |
3.2 | 36 |
3.4 | 30 |
3.6 | 28 |
3.8 | 14 |
4.0 | 6 |
- A chemist has a 250-gram sample of a radioactive material. She records the amount remaining in the sample every day for a week and obtains the following data.
- Draw a scatter plot of the data.
- Which function best suits the data: exponential, linear, or quadratic?
- Find the function of best fit and draw it through the scatter plot.
- Predict the amount of material present after 10 days.
Day | Weight(grams) |
---|---|
0 | 250 |
1 | 208 |
2 | 158 |
3 | 130 |
4 | 102 |
5 | 80 |
6 | 65 |
7 | 50 |
- The following table show the pregnancy rate (per 1000) for U.S. women aged 15 – 19 (source: US Census Bureau). Make a scatter plot with the rate as the dependent variable and the number of years since 1990 as the independent variable. Find which model fits the data best. Use this model to predict the rate of teen pregnancy in the year 2010.
Year | Rate of Pregnancy (per 1000) |
---|---|
1990 | 116.9 |
1991 | 115.3 |
1992 | 111.0 |
1993 | 108.0 |
1994 | 104.6 |
1995 | 99.6 |
1996 | 95.6 |
1997 | 91.4 |
1998 | 88.7 |
1999 | 85.7 |
2000 | 83.6 |
2001 | 79.5 |
2002 | 75.4 |
Mixed Review
- Cam bought a bag containing 16 cups of flour. He needs \begin{align*}2 \frac{1}{2}\end{align*} cups for each loaf of bread. Write this as an equation in slope-intercept form. When will Cam run out of flour?
- A basketball is shot from an initial height of 7 feet with an velocity of 10 ft/sec.
- Write an equation to model this situation.
- What is the maximum height the ball reaches?
- What is the \begin{align*}y-\end{align*}intercept? What does it mean?
- When will the ball hit the ground?
- Using the discriminant, determine whether the ball will reach 11 feet. If so, how many times?
- Graph \begin{align*}y=|x-2|+3\end{align*}. Identify the domain and range of the graph.
- Solve \begin{align*}6 \ge -5(c+4)+10\end{align*}.
- Is this relation a function? \begin{align*}\{(-6,5),(-5,-3),(-2,-1),(0,-3),(2,5)\}\end{align*}. If so, identify its domain and range.
- Name and describe five problem-solving strategies you have learned so far in this chapter.