1.3 Modeling with Linear Functions

Writing Linear Equations Given slope m and y-intercept b Given slope m and point

Example 1 Use slope – intercept form Substitute y= 18350 to find time: The graph shows the distance that Asteroid 2012 DA14 travels in x seconds. Write an equation for the graph and interpret the slope. About how long does it take for the asteroid to travel 18350 miles? y – intercept is 0 Distance (miles) Use slope – intercept form Substitute y= 18350 to find time: Time (s) The asteroid takes about 3823 seconds to travel 18350 miles. The slope is the speed of the asteroid.

Example 2 Two theaters charge for an event an admission fee plus price per person. The table shows costs for different number of students at Oakview Theater. The total cost y (in dollars) for x students at MapleStory Theater is represented by Number of People, x Total Cost, y 2 $25.60 4 $41.20 6 $56.80 7 $64.60 9 $80.20 Which theater charges more per person? How many people must attend for total costs to be equal? Compare slopes (the cost per person) To find number of people for total cost to be equal, set the price equations equal and solve for x. Oakview Theater charges more at $7.80 / person MapleStory Theater only charges $5.80 / person Use point –slope form to find equation for Oakview Theater The cost of the two theaters are the same at 5 people.

Let’s graph functions from example 2: The lines intersect at x = 5. As you can see, the solution from the graphical method is the same as the algebraic approach. Let’s graph functions from example 2: If you use a calculator, this would be the scatter plot:

Steps to Finding Line of Fit Create scatter plot of data Sketch a line that follows the trend of scatter plot Choose two points on the line Write an equation using the points

Example 3 Substitute 33 for x to find height: The table shows femur lengths and heights in centimeters. Does the data have a linear relationship? If so, write an equation for the data and find the height of a person with a femur length of 33 cm. Make a scatter plot and draw line of fit: Femur length, x Height, y 40 170 45 183 32 151 50 195 37 162 41 174 30 141 34 47 185 182 Find m and use point-slope form Height (cm) Femur Length (cm) Substitute 33 for x to find height: Approximate height is 152.5 cm

Scatter Plot on TI-84  

Curve of the best fit using Scatter Plot After you entered the table, go to Y= , PLOT 1, Graph Remember to clear any equations that are already in Y= We want to turn the diagnostic feature ON, so we can see the coefficient of Determination, or This will give us the idea how strong the model is. The closer it to 1, the better the model is. Press 2nd, 0 for Catalog and scroll down for Diagnostic ON, Enter, Enter, Stat, Calc, 4 for LInear Regression, For most TI-84 calculators: Go down to Store RegEQ line. Press Vars and go to Y-Vars. Select Function and then Y1. Press Calculate or press Enter twice. For TI-84 Plus Silver ONLY: LinReg(ax+b) should now be on the screen. Press Vars, Y-Vars, Function, Y1, and then press Enter again. Graph. You will see the graph passing through your scatter plot. To find f(33): VARS, right arrow, ENTER, ENTER, then type (33), ENTER. Answer: height is 150.9 cm (close to 152.5 cm)

Example 4 In February, the store sold 350 books. In August, they sold 2270 books. a)What equation might model the number of books sold each month assuming a linear model? Let x be the month of the year, where January is x=0, and y be the number of books sold in month x. Because January is month 0, February is month 1 and August is month 7. Slope-intercept form: b)Use a sentence to describe what the slope of the equation means in the context of this problem. Plug in known point: 350 books at x=1 The slope is the gain in the number of books sold per month, which is 320 books/month.

Example 5 You are drinking water from a water bottle. The following is a table of the height of the water at various times. Draw & label a graph representing the table and draw a line through the points. Time (sec) Height of water (cm) 5 20 10 16 8 25 4 What does the x-intercept of the graph represent? The x-intercept of the graph represents the time when the height of water reaches 0. In other words, it is the time where there is no more water left in the water bottle. What does the y-intercept of the graph represent? The y-intercept is the height of the water at time t=0. That is, it is the height before drinking any water. Find the slope and describe what the slope represents. The equation of the line is y = -0.8x + 24, using the known point (5, 20). The slope shows that the water’s height decreases by 0.8cm for each second you drink water.