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1. LINES

2. The slope of a line is a number that tells us how “steep” the line is and which direction it goes. This one has the greatest slope If you move along the line from left to right and are climbing, it is a positive slope. This one has the smallest slope Slope These are all positive slopes. The “steeper” the line, the larger the slope value.

3. We compute the slope by taking the ratio of how much the line rises (goes up) and how much the line runs (goes over) We could compute the run by looking at the difference between the x values. run (x2, y2) y2 - y1 If we took two points on the line, we could compute the rise by looking at the difference between the y values. rise (x1, y1) x2 - x1 So the slope is the rise over the run This is the slope formula Slope is designated with an m

4. This one has the greatest absolute value slope If you move along the line from left to right and are descending, it is a negative slope. This one has the smallest absolute value slope These are all negative slopes. The “steeper” the line, the larger the absolute value of the slope. (basically this means if you ignore the negative, the larger the number, the steeper the line---but in the negative slope direction). Slope

5. Let’s figure out the slope of this line. We know it should be a positive number. (2, 4) Choose two points on the line. 1 (1, 2) The rise over the run is 2 over 1 which is 2. Let’s compute it with the slope formula. 2 (0, 0) (-2, -4) What if we'd chosen two different points on the line? It doesn't matter which two points we pick, we'll always get a constant ratio of 2 for this line.

6. If we look at any points on this line we see that they all have a y coordinate of 3 and the x coordinate varies. Let's choose the points (-4, 3) and (2, 3) and compute the slope. (-4, 3) (2, 3) (-1, 3) This makes sense because as you go from left to right on the line, you are not rising or falling (so zero slope). The equation of this line is y = 3 since y is 3 everywhere along the line. In general, the equation of a horizontal line is y = b, where b is the y coordinate of any point on the line. In general, the equation of a horizontal line is y = b, where b is the y coordinate of any point on the line.

7. If we look at any points on this line we see that they all have a x coordinate of - 2 and the y coordinate varies. Let's choose the points (-2, 3) and (-2, - 2) and compute the slope. (-2, 3) (-2, 0) (-2, -2) Dividing by 0 is undefined so we say the slope is undefined. You can't go from left to right on the line since there isn't a left and right. The equation of this line is x = - 2 since x is - 2 everywhere along the line. In general, the equation of a vertical line is x = a, where a is the x coordinate of any point on the line. In general, the equation of a vertical line is x = a, where a is the x coordinate of any point on the line.

8. positive slope It is easy to remember undefined slope because you can’t move along from left to right (it is vertical) negative slope undefined slope zero slope (or no slope) It is easy to remember 0 slope because the line does not slope at all (it is horizontal)

9. We often have points on a line but want to find an equation of the line. We'll see how to do this by looking at an example. Find the equation of a line the contains the points (- 2, 4) and (2, - 2). First let's plot the points and graph the line. Now let's compute the slope---we know it will be negative by looking at the line. (x, y) Pick a general point on the line, (x, y). (2, -2) Use the point (2, -2) and this general point in the slope formula subbing in the slope we found. This is an equation for the line. Let's get it in a neater form. If we get rid of parenthesis and fractions and get the x and y on one side (with positive x term) and constants on the other side we'll have standard form.

10. If we get rid of parenthesis and fractions and get the x and y on one side (with positive x term) and constants on the other side we'll have standard form. get rid of parenthesis get rid of fractions by multiplying by - 2 get the x and y terms on one side (with positive x term) general or standard form constants on the other side Choose any x and sub it in this equation and solve for y and you will get a point (x, y) that lies on the line. x = 0 (0, 1) is on the line

11. Let's generalize what we did to get a formula for finding the equation of a line. Let's call the specific point we know on the line (x1, y2). (x - x1) Multiply both sides by x - x1 (x - x1) rearranging a bit we have: This is called the point-slope formula because it will find the equation of a line when you have a specific point (x1, y2)on the line and the slope. We can also use it when we know two points on the line because we could find the slope first and then use one of the points for the specific point.

12. Remember that slope is the change in y over the change in x. The slope is 2 which can be made into the fraction -1 -1 0 0 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. +1 +2 To find another point on the line, repeat this process with your new point (0,-3) (0,-3) +1 +2 So this point is on the line also. You can see that this point is found by changing (adding) 2 to the y value of the given point and changing (adding) 1 to the x value. (1,-1) (-1,5)

13. -1 -1 0 0 A way to do the last problem using the equation of the line A point on a line and the slope of the line are given. Find two additional points on the line. x = 0 If we find the equation of the line using the point-slope formula, we can easily find additional points on the line by subbing in various x values and finding the y values. (-1,5)

14. -1 -1 0 0 Let's take this equation and solve for y. This form of the equation is called slope-intercept form because it contains the slope and the y intercept of the line. slope-intercept form

15. y intercept slope Example: 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 -3x +4 Now plot the y intercept -4 -4 Change in y Change in x From the y intercept, count the slope Now that you have 2 points you can draw the line