1. MDFP Introduction to Mathematics Linear Functions Linear (Straight-Line ) Equations

2. Straight-Line (Linear) Equations Slope-Intercept Form y = mx + c Probably the most useful form of straight-line equations is the slope-intercept form: This is called the slope-intercept form because: m is the slope andslope c gives the y - interceptintercept

4. Let's take a look at the straight line y = x – 4. Its graph looks like this: To find the slope, we need two points from the line. Slope of a Straight Line One of the most important properties of a straight line is its angle from the horizontal. This concept is called "slope“ or gradient.

5. Pick two x values and solve for each corresponding y value. If, say, x = 3, then If, say, x = 9, then So the two points (3, –2) and (9, 2) are on the line y = x – 4

6. To find the slope, you use the following formula: Where and Then If we had taken the coordinates from the points in the opposite order, the result would have been exactly the same value:

7. Let me emphasize: it does not matter which of the two formulas you use or which point you pick to be "first" and which you pick to be "second". The only thing that matters is that you subtract your x-values in the same order as you had subtracted your y-values.

8. NOTE DIRECTION OF POSITIVE AND NEGATIVE GRADIENTS / SLOPE IS: y = mx + c Negative gradient y = mx + c Positive gradient This can help you check your calculations: if you calculate a slope as being negative, but you can see from the graph that the line is increasing (so the slope must be positive), you know you need to re-do your calculations.

9. Increasing lines have positive slopes; decreasing lines have negative slopes. With this in mind, consider the following horizontal line: y = 4 Its graph is shown to the right. Is the horizontal line going up; that is, is it an increasing line? No, so its slope won't be positive. Is the horizontal line going down; that is, is it a decreasing line? No, so its slope won't be negative. What number is neither positive nor negative? Zero! So the slope of this horizontal line is zero.

10. Now consider the vertical line x = -3 Is the vertical line going up on one end? Well, kind of. Is the vertical line going down on the other end? Well, kind of. Is there any number that is both positive and negative? Nope. Verdict: Vertical lines have NO SLOPE. In particular, the concept of slope simply does not work for vertical lines. The slope doesn't exist!

12. Common exercises will give you some pieces of information about a line, and you will have to find the equation of the line. How do you do that? Let us look at some examples. Example 1 – Given the slope and a point on the line Find the equation of the straight line that has slope m = 4 and passes through the point (–1, –6). Okay, they've given us the value of the slope, m = 4, so y = 4x + c We need to find c. We can use the point (-1, -6) to help us do this

13. From the point (-1, -6) we can substitute x = -1 and y = -6 in to the equation, then solve to find c : Then the line equation must be

14. Example 2 – Given two points on a line Find the equation of the line that passes through the points (–2, 4) and (1, 2). If we have two points on a straight line, we can always find the slope; that's what the slope formula is for.slope formula ! We can chose either of the two points (it doesn't matter which one), and use it to solve for c.

15. Using the point (–2, 4), I get: so It doesn't matter which point I choose. Either way, the answer is the same.

16. Complete The Questions 2.1 Linear Functions 1 Any work not completed during class must be completed for homework