Pre-Class Work

Work through the problems at the following links in Kahn Academy, record your work and bring it to the next class for discussion.

Relating linear contexts to graph features

Graphing linear relationships-word problems

Comparing linear rates-word problems

Many real world scenarios involve quantities that increase or decrease at a constant rate and hence may be modeled by linear functions.  In this section we focus on applications of linear functions and work through several modeling problems.

4.2 Examples

Example 4.2.1

Franco plants a dozen corn seedlings, each 6 inches tall. With plenty of water and sunlight, they will grow approximately 2 inches per day.

a) Complete the table of values for the height, h, of the seedlings after t

b) What is the (average) rate of change of the height of the plants from 0 to 5 days? What about from 5 – 10 days? What about 10 – 20 days?

c) Write an equation for the height h of the seedlings in terms of the number t of days since they were planted.

d) Graph the equation with t on the horizontal axis and h on the vertical axis.

Example 4.2.1 illustrates a linear function.  Recall…

A linear function is a function in which the rate of change/slope is constant.

The rate of change/slope is the amount that the output variable changes per unit change in the input variable.

In example 4.2.1 we view the input as time t in days and the output as height h in inches.  Here, the slope is 2 because the height changes by 2 inches per day.

Recall the different forms of linear functions (with input variable x and output variable y)

Slope-intercept form: [latex]y = mx + b[/latex] where m is the slope and (0,b) is the y-intercept.

Standard or general form: [latex]𝑎𝑥 + 𝑏𝑦 = 𝑐[/latex] where a, b and c are constants

Point-Slope form: [latex]𝑦 − 𝑘 = m(𝑥 − ℎ)[/latex] where 𝑚 is the slope and [latex](ℎ, 𝑘)[/latex] is a point on the line

In general, for any function, the y-intercept is the point on the graph where x = 0, that is, where the graph intersects the y-axis and the x-intercept is the point on the graph where y = 0, that is, where the graph crosses the x-axis.

Example 4.2.2

Consider a line that goes through the points (7, 5) and (-7, -1)

a) Find an equation for this line in each of the three forms given above.

b) Find the x and y intercepts of the line and graph it below, showing the intercepts.

Example 4.2.3

Suppose Keng has a job raking leaves where he gets paid $5.00 an hour for raking and an additional $3.00 for hauling the leaves to the curb for pick up.

a) How much will he get paid if he rakes for 2 hours and hauls the leaves? Explain your answer.

b) How long must he rake to earn $19.00 including the hauling fee? Explain your answer.

c) Represent the relationship between hours worked (x) and money earned (y), including hauling charge as an equation, a table and as a graph.

d) What is the slope and what does it tell us about Keng’s earnings? Be specific and include units in your explanation.

e) What is the y-intercept and what does it tell us about Keng’s earnings? Be specific and include units in your explanation.

Example 4.2.4

Delbert must increase his daily potassium intake by 1800 mg. He decides to eat a combination of figs and bananas. There are 9 mg of potassium per gram of fig, and 4 mg of potassium per gram of banana.

a) Write an expression that represents the amount of potassium in x grams of fig.

b) Write an expression that represents the amount of potassium in y grams of banana.

c) Write an equation, in standard form, that relates the number of grams of fig (x) and the number of grams of banana (y) Delbert needs to consume in order to get 1800 mg of potassium.

d) Find the x and y intercepts of this equation and sketch the graph.

e) What do each of the intercepts tell us about Delbert’s diet?

x-intercept:

y-intercept:

Example 4.2.5

According to one study, 27.5% of high school seniors reported vaping nicotine in 2018 and approximately 39.5% of high school seniors reported vaping nicotine in 2021.

a) Use this data to write a linear function with y as the percent of teens who vaped nicotine x years after 2010.

b) Use your function to predict the percent of high school seniors who vaped nicotine in 2022.

c) Find the x-intercept of the graph of your equation (you may approximate to the nearest tenth).  Explain what this point tells us about the percent of high school seniors vaping.

d) What is the slope of your function? Explain what this number tells us about the percent of high school seniors vaping.

e) Sketch the function.  Be sure to label the axes with the corresponding values they represent in terms of the application.