1 Review Linear relationships

2 Let’s investigate the relationship between x and y denoted by y = -2x – 2. We’ll complete the table and graph it. xy = -2x – In words, x was multiplied by -2, then 2 was subtracted to get y. The equation shows the relationship between the x and y values of these points.

3 xy = -2x – Notice for every increase of 1 in x, y goes down by 2. You can see it in the graph too. From one point to the next, you go down 2 and over to the right 1.

4 To find slope, you can think of it as going down 2 and over to the right 1, or going up 2 and over to the left 1. When you go left or down, count it as negative. When you go right or up, count it as positive. For example our slope is because we went down 2 and right 1. The relationship y = -2x – 2 graphs as a straight line so it is called a linear relationship. Two things define a linear relationship, the slope and the y intercept. We know the slope of our line, let’s find the y intercept.

5 The slope tells you how slanted the line is. The slope of our line is -2. The y-intercept is where the graph crosses the y-axis. From the graph, we see that is at -2. Our slope is negative so the line slopes downward left to right. Slope/intercept form y = mx + b m = slope b = y-intercept (x, y) = general point on line Remember the equation y = -2x – 2 y = -2x + -2 m = -2 b = -2

6 undefine d slope Horizontal and vertical lines All lines can be written in the form y = mx + b except horizontal and vertical lines. Every point on a vertical line has the same x value. (x = -1, x = 2, etc.) zero slope Every point on a horizontal line has the same y value. (y = -2, y = 3, etc.)

7 Return to slope expl: Find the slope of the line between (4, -3) and (6, 2). Going from one point to the next, rise is how much you go up or down and run is how much you go left or right. First, plot the points! Notice the slope will be positive because the line would slope upward.

8 Start at (4, -3). You go up 5 units and over to the right 2 units. Alternatively, start at (6, 2). You go down 5 and over to the left 2.

9 Slope formula You could subtract 2 – -3 and 6 – 4 to get the rise and run that compose slope… …Or you could subtract -3 – 2 and 4 – 6 to get the alternative rise and run. first point (x 1, y 1 ) and second point (x 2, y 2 )

10 expl: Find equation of line Now we’ll find the equation of the line that connects the points (4, -3) and (6, 2). As we’ve said before, lines are defined by their slope and their y-intercept. We know the slope between the two points is 2.5. Now let’s find the y- intercept and the equation of this line. We already know its slope. All we need to know is its y-intercept.

11 All lines must be of the form y = mx + b where m is the slope and b is the y-intercept. Put our known slope in. Put one of the given points in for x and y. Now we’ve got an equation we can solve for b. So b is -13. Put the line in slope-intercept form.

12 expl: Point-slope form of line We could have found the equation using a slightly different formula. We will derive it. Start with slope formula. Multiply both sides by denominator. Exchange the point (x, y) for (x 2, y 2 ). We end up with the Point-slope form of a line.

13 We will find the equation of the line between (4, -3) and (6, 2) using this new formula. Start with general formula. Put in values for slope and one point. Simplify the left and solve for y.

14 Graphing lines Method 1: Solve for y to get the equation in slope-intercept form. Plot its y-intercept. Then use the slope to go up or down and left or right to get to the next point. Continue using the slope to plot another point. Then connect them using a straight edge. Method 2: Find the x and y- intercepts. Plot and connect them with a straight edge.

15 x and y intercepts x and y intercepts occur on the axes. The one thing you know about any point on an axis, is that the other coordinate is zero. We’ll use that with the equation to find the x and y intercepts.

16 expl: Find intercepts of 3y – 4x = 24. y intercept: Set x to 0, then solve for y. x intercept: Set y to 0, then solve for x.

17 Perpendicular lines The slope of the red line is - 5/2. The slope of the blue line is 2/5. The product of the slopes is -1. The slopes of perpendicular lines (lines at right angles) are negative reciprocals.

18 Parallel lines are slanted at the same angle. Since slope tells you how slanted a line is, parallel lines will have equal slopes. Parallel lines The slope of the blue line is 1/4. The slope of the red line is 1/4.

19 Worksheets “Linear function applications” provides practice in graphing and finding equations of lines. The first problem gives you slope and y-intercept information that you put into the slope-intercept form to find the lines’ equations. It then discusses the break-even point. The second problem gives you two points from which you can find the line’s equation. “Understanding slope” works through several examples that help us understand the pattern of points that create a straight line and what slope tells us about the points.