Section 1.4 Linear Functions and Slope The Slope of a Line Find the slope of the line that passes through (-2,5) and (3,-1) change in y 5 1 6 6 m or change in x 2 3 5 5 Example Find the slope of the line passing through the pair of points. (5,-2) and (-1,7) The Point-Slope Form of the Equation of a Line Write the point-slope form of the equation of the line with slope of 3 that passes through (-1,2). Substitute into the point-slope form; y-y1 m( x x1 ) y 2 3( x 1) y 2 3( x 1) Solving in both forms • A.Write the equation in point slope form of the line with slope 4 that passes through the point (4,-3). B.Then solve the equation for y (slope intercept form) = m(x-x1) y-(-3) = 4(x-4) x1 y1 • y-y1 y+3 = 4(x-4) -3 -3 y= 4(x-4)-3 y= 4x-16-3 Y=4x-19 Substituting the values into the euation 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.) Example Write the point slope form of the equation of the line with slope of -4 that passes through (2,5). Then solve for y. Write the point-slope form of the equation of the line that passes through (-1,2) and (-4,5). Then solve for y. y2 y1 52 3 First: Find the slope. 1 x2 x1 4 1 3 Second: Substitue into the point-slope form. y-y1 m( x x1 ) y 5 1( x 4) Third: Solve for y. y 5 1( x 4) y 5 1x 4 y=-1x+1 Example Write the point slope form of the equation of the line that passes through (2,5) and (-1,0). Then solve for y. The Slope-Intercept Form of the Equation of a Line 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 yintercept 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 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. y (3,6) (0,4) x Example Graph the linear equation y= -3x+5 y x Example 1 Graph the linear equation y= x 3 2 y x Equations of Horizontal and Vertical Lines Example Graph x=4. Graph y=-2 y x The General Form of the Equation of a Line Find the slope and the y intercept of the line. 4 x 5 y 20 0 4 x 5 y 20 -5y=-4x-20 -5y 4 x 20 -5 5 5 4 y= x 4 5 slope Y intercept Example Find the slope and the y intercept of the line whose equation is 2x+5y-10=0. Using Intercepts to Graph Ax + By + C = 0 Find x and y intercepts to graph a line 6x-2y=12 X intercept so let y=0 Y intercept so let x=0 0 0 6x-2(0)=12 6x=12 X=2 (2,0) 6(0)-2y=12 -2y=12 Y=-6 (0,-6) y x X intercept - Let y=0 4x-3 0 6 0 4x-6=0 4x=6 6 3 x= 4 2 3 ,0 2 Y-intercept - Let x=0 4 0-3y-6=0 -3y-6=0 -3y=6 6 y= 2 -3 0, 2 Example Find the x and y intercepts then graph using those points. y X-4y-8=0 x Summary Applications The graph gives the median age of the US population in the indicated year. The data is displayed as a scatterplot 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 the US population in 2010. That means we will substitute in 40 for the x. The reason we use 40 is the initial date was 1970. If we add 40 to 1970 we will get 2010. y=0.265x+27.35 y=0.265(40)+27.35 y=37.95 This means that the median age of the US population will be 37.95 in 2010. 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 this company decides to make an 18 inch pizza, how much should they charge? Diameter 8 10 Price 8.00 9.60 12.80 6.40 12 16 Graphing Calculator-Linear Regression Take the data from the previous pizza problem. Put the data into List1 & List2 in the graphing calculator. To do that Press STAT, then 1 for Edit. Type in the numbers. D 8 10 12 16 $ 6.40 8.00 9.60 12.8 Press STAT, move the cursor to the right to CALC, then press 4 for LinReg. The next screen gives you the values of a and b for the equation. More on the next slide. The equation is y=.8x. Graphing Calculator-Linear Regression continued To see the scatterplot of the data, we need to change the Window. Press WINDOW, and type in what you see at left. Press 2nd Y= to get STAT PLOT. First make certain that all plots are off by pressing 4. Then return to STAT PLOT and press Plot1. On the word ON press ENTER. Cursor down and press the appropriate keys so you get what you see in the picture at left. L1 is obtained by pressing 2nd then 1. L 2 - 2nd then 2. Press the GRAPH key. You will see the scatterplot at left. The equation y=.8x will go through these points. Press Y= and type in the equation. Press GRAPH to see the line and scatterplot. Find the equation of the line in slopeintercept form for a line that passes through (0,-4) and has a slope of -2. y 2 x 4 (b) y 4 2 x (c) y 2 x 4 (d) y 2 4 x (a) Find the equation of the line in slopeintercept form of the line that passes through (-3,-2) and (0,-2). (a) x 3 (b) y 2 (c) y 0 (d) y 2 x 3 What is the slope of the line 3x - 7y – 4 = 0. (a) 7 3 7 4 4 (c) 7 3 (d) 7 (b)