Section 1.4 Linear Functions and Slope

Intro

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 (x 1, y 1 ) and (x 2, y2)y2) is Find the slope of the line that passes through (-2, 5) and (3, -1).

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

Study Tip When computing slope, it makes no difference which point you call (x 1, y 1 ) and (x 2, y 2 ). If we let (x 1, y 1 ) = (-2, 4) and (x 2, y 2 ) = (-3, -1) the slope is… However, what happens if you do the following… Why does this make sense?

Special Slopes to Know!! Page 199 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 (x 1, y 1 ) 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 y1y1 x1x1 y - (-3) = 4(x - 4) Substituting the values into the equation 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.) y + 3 = 4x-16 y - y 1 = m(x - x 1 ) (slope intercept form) y = 4x - 19 m

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 199 Prob

If you are given two points and you need to write an equation in point-slope form, then you can use either point for (x 1, y 1 ).

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. Point- Slope Slope- intercept

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 199 Prob Point- Slope Slope- intercept

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 (0, b) is y = mx + b Page 189 Prob. 39 – 48 Just tell the slope and y-intercept The y-intercept is (0, -4) The y-intercept is (0, 2) The slope is 2 The slope is 1/2

Point Slope FormSlope Intercept Form For a nonvertical line with slope m that passes through (x 1, y 1 ) the equation is y - y 1 = m(x - x 1 ) 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 is (0, 6) y = 2x + 6 Two forms for Equations of Lines

1.Plot the point containing the y-intercept on the y-axis. This is the point (0, b). 2.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. 3.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. Graphing y = mx + b Using Slope- and y-Intercept

Graph the linear equation y = 2/3 x + 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. (0,4) (3,6)

Example

Page 199 Prob Graph

Example Graph x = 4 Graph y = -2 Page 199 Prob

y-intercept slope y-intercept slope

Example Find the slope and the y intercept of the line whose equation is 2x + 5y – 10 = 0. Page 199 Prob Y intercept slope

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

Example Find the x and y intercepts then graph using those points. x - 4y – 8 = 0 Page 199 Prob

Summary

Application Problems

Now we will use the equation to predict the median age of US population in We will have to plug in 50 for x because the initial date is 1970, thus 2020 – 1970 = 50.

Make sure you Grab your Calculator and Log In!

Example Diameter Price 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?

Graphing Calculator-Linear Regression Open a “New Document” Select 4: Lists & Spreadsheets. Type Information like pictured. Now add a new page, make it 5: Data & Statistics. Add variables to the bottom and side of graph. Now choose b and option 4: Analyze then select 6: Regression, then 1: Show Linear Use the equation to predict the amount an 18 inch pizza should cost. Tada!!!!

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

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

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