Section 1.4 Linear Functions and Slope

Intro Is there a relationship between literacy and child morality? Check out the data shown and list 3 THINGS YOU NOTICE.  As the percentage of adult females who are literate increases, does the mortality of children under five decrease?  Data presented in a visual form as a set of points is called a scatter plot. A line that best fits the data points is a scatter plot is called a regression line.  By writing the equation of the line, we can obtain a model for the data and make predictions about child mortality based on the percentage of literate adult females in a country. Because data often falls on or near the line, we will use functions to model such data and make predictions. We begin by discussing a line’s steepness.

The Slope of a Line Slope describes steepness of a line. It compares the vertical change (rise) to the horizontal change (run) when moving left to right from one fixed point to another. The slope of the line through the distinct points (x1 , y1) and (x2 , y2) is Find the slope of the line that passes through (-2, 5) and (3, -1).

Find the slope of the line passing through the pair of points. Example 1 Example 2 (5,-2) and (-1,7) (-3, -1) and (-2, 4) Example 3 (-3, 4) and (2, -2)

Study Tip When computing slope, it makes no difference which point you call (x1 , y1) and (x2 , y2). If we let (x1 , y1) = (-2, 4) and (x2 , y2) = (-3, - 1), the slope is… However, what happens if you do the following… (y2 – y1) / (x1 – x2) Why does/doesn’t this make sense?

Special Slopes to Know!! Page 188 Prob.1-10

Point-Slope Form of the Equation of a Line When finding the equation of a line using only slope, m, and one point (x1, y1) is given, we use the following formula Write the point-slope form of the equation of a line that has a slope of 3 that passes through point (-1, 2).

Solving in both forms Write the equation in point slope form of the line with slope 4 that passes through the point (4,-3). Then solve the equation for y x1 y1 (slope intercept form) y-y1 = m(x-x1) y-(-3) = 4(x-4) Substituting the values into the euation y+3 = 4(x-4) This is Point Slope Form. Apply the distributive property for the parentheses. This will give us the slope intercept form. (The equation is solved for y.) -3 -3 y= 4(x-4)-3 y= 4x-16-3 Y=4x-19

Example 4 Write the point slope form of the equation of the line with slope of -4 that passes through (2,5). Then solve for y. Page 188-189 Prob. 11-24

If you are given two points and you need to write an equation in point-slope form, then you can use either point for (x1,y1).

Write the point-slope form of the equation of the line that passes through the point(-1, 2) and (-4, 5). Then solve for y. First: Find the slope Second: Substitute into the point-slope form. Third: Solve for y.

Example 5 Write the point slope form of the equation of the line that passes through (2,5) and (-1,0). Then solve for y. Page 189 Prob. 25-38

The Slope-Intercept form of the Equation of a Line The slope-intercept form of the equation of a NONVERTICAL line with slope, m and y-intercept b is y = mx + b Page 189 Prob. 39 – 48 Just tell the slope and y-intercept

Two forms for Equations of Lines Point Slope Form Slope Intercept Form For a nonvertical line with slope m that passes through (x1,y1) the equation is y-y1 = m(x-x1) For a nonvertical line with slope m and y-intercept b the equation is y=mx + b Example: slope = -3 point on the line(-1,-2) Y-(-2)= -3(x-(-1)) Y+2= -3(x+1) Example: slope =2 y-intercept of 6 Y=2x + 6

Graphing y = mx + b Using Slope- and y-Intercept Plot the point containing the y- intercept on the y-axis. This is the point (0, b). Obtain a second point using the slope, m. Write m as a fraction and use rise over run, starting at the point containing the y-intercept, to plot this point. Use a straightedge to draw a line through the two points. Draw arrow heads at the ends of the line to show that the line continues indefinitely in both directions.

Graph the linear equation y= 2/3x+4 First: Plot the y-intercept of 4 Rise by 2 units Run ( go to the right) by 3 units. Plot the second point (3, 6) Connect the two points with a straight edge or ruler. (3,6) (0,4)

Example

Example Page 189 Prob. 39-48 Graph

Example Graph x=4. Graph y=-2 Page 189 Prob. 49-58

Y intercept slope

Example Find the slope and the y intercept of the line whose equation is 2x+5y-10=0. Page 189 Prob. 59-66

Find x and y intercepts to graph a line 6x-2y=12 X intercept so let y=0 Y intercept so let x=0 6x-2(0)=12 6(0)-2y=12 6x=12 -2y=12 X=2 Y=-6 (2,0) (0,-6)

Example Find the x and y intercepts then graph using those points. X-4y-8=0 Page 189 Prob. 67-72

Summary

Applications

Application Problems The graph gives the median age of the US population in the indicated year. The data is displayed as a scatter plot with two points on the line indicated. Find the equation of the line, in order to make predictions of the US population in the future.

Now we will use the equation to predict the median age of US population in 2010. We will have to plug in 40 for x because the initial date is 1970, thus 2010 – 1970 = 40.

Example The local pizza shop has a special sale on pizzas. Write the slope-intercept equation of the line that describes the price as a function of the diameter of the pizza. If the company decides to make an 18 inch pizza, how much should they charge? Diameter 8 10 12 16 Price 6.40 8.00 9.60 12.80

Graphing Calculator-Linear Regression D $ 8 6.40 10 8.00 12 9.60 16 12.8 More on the next slide.

Graphing Calculator-Linear Regression continued

Exit Ticket Find the equation of the line in slope-intercept form for a line that passes through (0,-4) and has a slope of -2. (a) (b) (c) (d) (a)

Find the equation of the line in slope-intercept form of the line that passes through (-3,-2) and (0,-2). (a) (b) (c) (d) (b)

What is the slope of the line 3x - 7y – 4 = 0. (b) (c) (d) (d)