Learning objectives
When determining whether a linear or exponential model is a better fit for a given scenario, consider:
Decide whether you think the function is linear then drag the slider to investigate further. Try again by clicking "Try a new function."
Exponential functions can be difficult to identify from a graph or table when the amount of change is small relative to the size of the numbers.
A quadratic function is a polynomial function of degree 2. A quadratic function can be written in the form f(x)=ax^2+bx+c where a, b, and c are real numbers.
From the graph of a quadratic function, called a parabola, we can identify key features including domain and range, x- and y-intercepts, increasing and decreasing intervals, positive and negative intervals, average rate of change, and end behavior. The parabola also has the following two features that help us identify it, and that we can use when drawing the graph:
Determine whether an exponential or linear model would better model the data. Justify your choice.
A real estate agent earns 3\% of the value of every house sold.
The average median house price of a home, y, sold in the U.S. from 2019 to the beginning of 2022 is shown in the graph where x represents the number of years since 2019.
The cost of college tuition in the United States has increased by 1200\% since 1980. Consider the average annual tuition and fees presented in the table:
Year | Public university | Private university |
---|---|---|
1980 | \$1\,856 | \$10\,227 |
1990 | \$2\,750 | \$16\,590 |
2000 | \$3\,706 | \$21\,698 |
2010 | \$5\,814 | \$25\,250 |
2020 | \$9\,403 | \$34\,059 |
Create a linear model to represent the average annual tuition cost for each type of university.
Create an exponential model to represent the average annual tuition cost for each type of university.
Determine if a linear model or an exponential model would be better to predict tuition for the next decade. Explain the differences between the models and what they tell us about the context.
Consider the quadratic function: f(x)=x^2-2x+1
Graph the function.
State the axis of symmetry.
Linear functions can help model relations with a near-constant of change, and exponential functions can help model relations with an increasing or decreasing rate of change.
Consider the table below:
x | y=3x | y=3x^2 | y=3^x |
---|---|---|---|
1 | 3 | 3 | 3 |
2 | 6 | 12 | 9 |
3 | 9 | 27 | 27 |
5 | 15 | 125 | 243 |
The way a function is represented can affect the characteristics we are able to identify for the function. Different representations can highlight or hide certain characteristics. Remember that key features of functions include:
One way to compare functions is to look at growth rates as the x-values increase over regular intervals. In order to compare the growth rates of quadratics with those of exponential or linear functions, we will examine only the increasing interval of a quadratic function.
When the leading coefficient of the quadratic equation is positive, the parabola opens upward. In this case, we know y increases at an increasing rate as x approaches infinity.
Since a linear function increases at a constant rate and the quadratic function increases at an increasing rate as x increases, eventually the quadratic function will increase faster than the linear function.
Next, we need to examine how an exponential growth function compares to the increasing portion of the quadratic function, since both functions increase at an increasing rate. Consider a situation where we compare the increasing interval of the quadratic function g(x) with a positive leading coefficient, to an exponential growth function h(x), as shown in the graph.
Notice starting at x=0, g(x) is greater than h(x) and is increasing at a greater rate. But, as x continues to increase, the quadratic function g(x) is increasing at a slower rate than the exponential function, and eventually the exponential function will overtake the quadratic function.
An exponential growth function will always exceed a linear or quadratic growth function as values of x become larger.
Consider the functions shown below. Assume that the domain of f is all real numbers.
Function 1:
x | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|
f\left(x\right) | -3.75 | -2 | -0.25 | 1.5 | 3.25 | 5 | 6.75 |
Function 2:
Determine which function has a higher y-intercept.
Find the average rate of change for each function over the following intervals:
Using part (b), determine which function will be greater as x approaches positive infinity.
Consider functions representing three options to earn money one of the following ways:
Note: Option 3 starts with \$2 on day one and doubles each day after this.
Compare the average rate of change of each function over the intervals 2 \leq x \leq 3 and 4 \leq x \leq 5.
Find the equation that represents each option, where x is the number of days that have passed.
Find the value of each option at 8 days, 12 days, and 14 days.
Determine which option will be greater for larger and larger values of x.
It is important to be able to compare the key features of functions whether they are represented in similar or different ways:
Use the linear and exponential models to fit the data on the graph.
Sometimes we need to consider fitting something other than a line of best fit or regression line, which both refer to a linear regression model, to model data. A fitted function could include another type of function, such as an exponential function.
We already learned about the correlation coefficient, r, a statistic that describes both the strength and direction of a linear association. But, we also need a measure that determines how well our fitted function can actually predict an outcome.
This value is known as the coefficient of determination, or the value R^2, and is a measure of the proportion of the variation in the dependent variable that is predicted by the independent variable. Since the coefficient of determination represents a proportion it will only ever return a value between 0 and 1.
In the case where the fitted function is a linear model with one independent variable, R^2 is equal to our correlation coefficient squared, r^2. This only holds true for linear models with one independent variable and is not the case when the fitted function is from any other function family, e.g. exponential, quadratic, etc.
Bivariate data can be modeled with a fitted function also called a regression model. Depending on the goodness of fit, measured with the coefficient of determination (R^2), a regression function may pass exactly through all of the points, some of the points, or none of the points.
\text{ Exponential regression } R^2=0.763
R^2 for the exponential regression shown means that 76.3\% of the variation in the dependent variable is explained by the variation in the independent variable. The closer R^2 is to 1, the more that the variation in the dependent variable is explained by the variation in the independent variable.
A teacher recorded the number of days since a student last studied for an exam and their score out of a possible 80 points on the exam.
Number days since studying | 3 | 2 | 6 | 4 | 4 | 1 | 6 | 3 | 4 | 2 |
---|---|---|---|---|---|---|---|---|---|---|
Exam score | 64 | 59 | 42 | 57 | 58 | 72 | 33 | 63 | 55 | 62 |
Describe the association between the number of days since student and the exam score.
Calculate the line of best fit and correlation coefficient. Interpret the correlation coefficient.
Interpret the meaning of the slope and y-intercept of the line of best fit in context of the data.
The population P of fish in a small lake over t years is shown in the table below:
Years (t) | Fish Population (P) |
---|---|
0 | 1000 |
0.5 | 550 |
1 | 500 |
1.5 | 425 |
1.75 | 350 |
2 | 290 |
2.25 | 210 |
2.5 | 160 |
3.75 | 100 |
Determine whether a linear or exponential model best fits the relationship between the years, t, and the population of fish P.
Calculate the regression model for the data and use it to predict the population of fish in the lake after 5 years.
Interpret the coefficient of determination for the regression model.
Linear and exponential data can be fitted to a regression model. We can analyze the closeness of the fit using the coefficient of determination.
Consider the scatter plot of data relating the number of guests at a restaurant and the cost of the meal and the residual plot of the data:
From a scatter plot and a line of fit, we can further analyze an association between two variables by examining the residuals of the model.
By taking the residuals of each point in the data set and plotting them at their corresponding x-values, we form a residual plot for the data.
The residual plot is constructed using the same x-axis scale and x-coordinates from the original scatter plot, and plotting the residual values as the y-coordinates.
A residual plot can be used to decide if a straight line is an appropriate model for the data. And, it identifies the strength of the relationship by showing how much the model over-predicts (negative residual) and under-predicts (positive residual) the actual data. Looking for unusually large residuals can help us identify outliers in the data set.
Two key features will help provide evidence about whether or not a linear model is appropriate, and indicate the strength of the relationship:
Pattern - if the linear model fitted is appropriate, then points on the residual plot should be randomly scattered about the x-axis without a noticeable pattern.
Size of residuals - residuals that are small in size relative to the data being predicted indicate a stronger association. Large residuals would indicate the model significantly under- or over-predicts the actual data.
Following are some example scatter plots with the line of best fit and residuals, and their corresponding residual plots.
Scatter plot and residual plot of weak positive linear association:
From the scatterplot we can see the association is positive. The residual plot has no obvious pattern, suggesting a linear model is appropriate. The residuals are relatively large indicating a weak relationship.
Scatter plot and residual plot of strong negative linear association with an outlier:
From the scatterplot we can see the association is negative. Other than the outlier, the residuals are relatively small, indicating a strong relationship. The outlier in the scatterplot stands out in the residual plot. Its inclusion leads to most of the data points being over-predicted by the best fit line.
Scatter plot and residual plot of non-linear association:
The residual plot displays a clear pattern, indicating that a linear model is not appropriate for this data set.
The scatter plot shows the relationship between the electricity usage of a household and the cost of their monthly utility bill.
The equation of the line of best fit is y=0.255x-81.49
The residual plot of the data is shown:
Interpret the strength and linear association of the data using the line of best fit and residual plot.
Find and interpret the residual for the point \left(930, 150\right).
Consider the following data set and scatterplot with line of fit.
x | 10 | 11 | 13 | 18 | 19 | 21 | 23 | 25 | 28 | 29 | 31 |
---|---|---|---|---|---|---|---|---|---|---|---|
y | 12 | 13 | 9 | 8 | 7 | 7 | 4 | 2 | 3 | -1 | -2 |
Create a residual plot for the data.
Determine if a linear model is an appropriate choice for the data.
A residual plot shows the strength of the correlation between two variables. The closer the data points on a residual plot are to the x-axis, the stronger the correlation between the data. A model is considered strong when the residuals are small relative to the value being predicted. Calculate the residuals for a residual plot using the formula:\text{residual}=\text{actual}-\text{predicted}
In general, a residual plot with points randomly dispersed about the x-axis indicates that the model is appropriate for the data.