1. Slope Problems © 2002 by Shawna Haider

2. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is given by

3. Slope Problem Examples Determine a value for x such that the line through the points has the given slope. Let's use the slope formula and plug in what we know. (x1,y1)(x1,y1)(x2,y2)(x2,y2) You can cross-multiply to find a fraction-free equation for x to solve.

4. Example when you have a point and the slope A point on a line and the slope of the line are given. Find two additional points on the line. 0 Remember that slope is the change in y over the change in x. The slope is 2 which can be made into the fraction (0,-3) So this point is on the line also. You can see that this point is changing (adding) 2 to the y value of the given point and changing (adding) 1 to the x value. +2+1 To find another point on the line, repeat this process with your new point (0,-3) +1+2 (1,-1) (-1,5)

5. Determine the slope and the y-intercept of the line given by Slope-intercept form slope y intercept m = -1/3 SLOPE b = 2 y - intercept

6. y intercept slope Example of given an equation, find the slope and y intercept Find the slope and y intercept of the given equation and graph it. Now plot the y intercept From the y intercept, count the slope Change in y Change in x Now that you have 2 points you can draw the line

7. y intercept slope Example of given an equation, find the slope and y intercept Find the slope and y intercept of the given equation and graph it. First let's get this in slope- intercept form by solving for y. -3x +4 -4 Now plot the y intercept From the y intercept, count the slope Change in y Change in x Now that you have 2 points you can draw the line

8. These will be linear models. When you read the problem look for two different variables that can be paired together in ordered pairs. Example: The total sales for a new sportswear store were $150,000 for the third year and $250,000 for the fifth year. Find a linear model to represent the data. Estimate the total sales for the sixth year. Total sales depend on which year so let’s make ordered pairs (x, y) with x being the year and y being the total sales for that year. (3, 150,000) (5, 250,000) We now have two points and can determine a line that contains these the points (on the next screen)

9. We’ll want to use the point-slope equation (3, 150,000)(5, 250,000) First we need the slope Now we have all the pieces we need so we can plug them in We now have an equation to estimate sales in any given year. To estimate sales in the sixth year plug a 6 in for x.

10. Slope-Intercept Form Useful for graphing since m is the slope and b is the y-intercept Point-Slope Form Use this form when you know a point on the line and the slope Also can use this version if you have two points on the line because you can first find the slope using the slope formula and then use one of the points and the slope in this equation. General Form Commonly used to write linear equation problems or express answers

11. Graphing Horizontal Lines

12. Horizontal Lines The slope of a horizontal line 0 The graph of any function of the form f(x) = b or y = b is a horizontal line that crosses the y-axis at (0, b).

13. Vertical Lines

14. The slope of a vertical line is undefined The graph of any function of the form x =a is a vertical line that crosses the x-axis at (b, 0).

15. Example of how to find x and y intercepts to graph a line The x-intercept is where a line crosses the x axis (6,0) (-1,0) (2,0) What is the common thing you notice about the x-intercepts of these lines? The y value of the point where they cross the axis will always be 0 To find the x-intercept when we have an equation then, we will want the y value to be zero.

16. Now let's see how to find the y-intercept The y-intercept is where a line crosses the y axis (0,4) (0,1) (0,5) What is the common thing you notice about the y-intercepts of these lines? The x value of the point where they cross the axis will always be 0 To find the y-intercept when we have an equation then, we will want the x value to be zero.

17. Let's look at the equation 2x – 3y = 12 Find the x-intercept. We'll do this by plugging 0 in for y 2x – 3(0) = 12 Now solve for x. 2x = 12 2 2 x = 6 So the place where this line crosses the x axis is (6, 0)

18. 2x – 3y = 12 Find the y-intercept. We'll do this by plugging 0 in for x 2(0) – 3y = 12 Now solve for y. -3y = 12 -3 -3 y = - 4 So the place where this line crosses the y axis is (0, -4) We now have enough information to graph the line by joining up these points (6,0) (0,- 4)

19. Recognizing Linear Equations In every equation of the form Ax + By = C linear? To find out, suppose that A and B are nonzero and solve for y: A, B, and C are constants Adding –Ax to both sides Dividing both sides by B

20. Since the last equation is a slope-intercept equation, we see that Ax +By = C is a linear equation when A≠0 and B≠0. But what if A or B (but not both) is 0, then By = C and y = C/B. If B is 0, then Ax = C and x = C/A. In the first case, the graph is a horizontal line; in the second case, the line is vertical. In either case, Ax + By = C is a linear equation when A or B (but not both) is 0. Standard Form of a Linear Equation Any equation of the form Ax + By = C, where A, B, and C are real numbers and A and B are not both 0, is a linear equation in standard form and has a graph that is a straight line.

21. Point-Slope Form Multiply both sides by Point-Slope Form Any equation of the form is said to be written in point-slope form and has a graph that is a straight line. The slope of the line is m The line passes through.

22. Remember parallel lines have the same slopes so if you need the slope of a line parallel to a given line, simply find the slope of the given line and the slope you want for a parallel line will be the same. Perpendicular lines have negative reciprocal slopes so if you need the slope of a line perpendicular to a given line, simply find the slope of the given line, take its reciprocal (flip it over) and make it negative.