Algebra 1 Ch 4.4 - Slope

Objective Students will find the slope of a line using 2 points.

Before we begin… Continuing our in-depth look at linear equations, in this lesson we will look at the slope of a line… In a future lesson we will plot a linear equation using only the slope and the y-intercept… Before we can do that…we have to understand what slope is…

Vocabulary Rise Slope = Run Slope describes the steepness of a line. Slope is the ratio of the rise (or vertical change) to the run (or the horizontal change) Slope can be expressed verbally as: Vertical Change Rise Slope = Run Horizontal Change

Finding Slope with a Formula You can also find the slope of a line by using any two points on the line and substituting those points into the slope formula In the slope formula m is the symbol for slope. (you will need to know this later when we talk about the slope-intercept form of an equation) The formula for slope is: y2 – y1 x2 – x1 m =

Before using the Formula Of course, since this is a formula – you will use the formula method Write the formula Substitute Simplify When working with the formula’s be careful…you must be proficient with the rules of integers…that is you must know how to work with negative and positive numbers Also you will need to know where you get the x and y values from Let’s look at that…

Example You get the x and y-values from any two points on the line. In this instance the ordered pairs are listed x y (-1, 3) (3,-2) As you know each ordered pair is in a specific format with the x-value listed first and then the y-value (x1,y1) When working with the slope formula, I find it easier to set up the ordered pairs in a table like this: (x2,y2) x y 1 -1 3 2 -2 Once you have set up the ordered pairs…just substitute them into the slope formula to calculate the slope of the line…

Example #1 x y 1 -1 3 2 -2 y2 – y1 x2 – x1 m = 1. Write the formula (-2) – 3 3 – (-1) m = 2. Substitute -5 2 m = 3. Simplify

Slope – What does it mean? One of the characteristics of a line is that the slope is constant. Since the slope is constant, we can plot the next point on the line just by knowing the slope and a point.

Plotting using Slope To plot a point using slope you go up or down depending if the number is positive (up) or negative (down) and always to the right In the previous example the slope was -5 over 2. To plot the next point on the line, using slope, go down 5 and to the right 2.

8/4 can be simplified to 2. Which written in fraction form is 2/1 Example #2 Find the slope of a line passing through the points (-3, -2) and (1, 6) At this point you should be able to identify the x-values as -3 and 1 and the y-values as -2 and 6 y2 – y1 x2 – x1 m = 1. Write the formula 6 – (- 2) 1 – (- 3) m = x y 1 -3 -2 2 6 2. Substitute 8 4 m = 3. Simplify 8/4 can be simplified to 2. Which written in fraction form is 2/1

Example #2 (continued) Now that you know the slope of the line you can plot the next point on the line. In this example the slope was 2/1. x y Before you can use slope you need a starting point. In our example one of the points was (-3, -2) Let’s start there… To plot the next point on the line, since slope was positive go up 2 spaces (-2, 0) Then go over 1 space to the right. That is the next point on the line (-3, -2) Draw the point and connect the dots and you will have the graph of the linear equation. In this case the next point on the line is (-2, 0)

Visual Representation of Slope You can tell the slope of a line just by looking at it… The slope of a line can be either positive, negative, zero or undefined… Let’s see what that looks like…

Positive Slope x y

Negative Slope x y

Zero Slope x y

Undefined Slope x y

Caution… These algebraic concepts that we are talking about are all multi-step processes and require higher level thinking skills To be successful…you must be organized and have a base understanding of integers and fractions… For example, when working with the slope formula you better know that minus a negative means to add. Also, if you have a slope of a whole number like 2 you better know that the fraction form of 2 is 2/1 and that you would go up 2 and to the right 1 to find the next point on the line.

Comments On the next couple of slides are some practice problems…The answers are on the last slide… Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… If you cannot find the error bring your work to me and I will help…

Your Turn Use the slope formula to find the slope of the line. Then choose 1 ordered pair and plot the next point on the line. (4, 5), (2,3) (1, 5), (5, 2) (0, -6), (8, 0) (4, 1), (6, 1) (-6, -1), (-6, 4)

Your Turn Solutions m =1 or 1/1 The line should go upward m = - ¾ The line should go downward m = ¾ m = 0 The line should be horizontal m = Undefined The line should be vertical

Summary A key tool in making learning effective is being able to summarize what you learned in a lesson in your own words… In this lesson we talked about the slope of a line. Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… I will give you credit for doing this lesson…please see the next slide…

Credit I will add 25 points as an assignment grade for you working on this lesson… To receive the full 25 points you must do the following: Have your name, date and period as well a lesson number as a heading. Do each of the your turn problems showing all work Have a 1 paragraph summary of the lesson in your own words Please be advised – I will not give any credit for work submitted: Without a complete heading Without showing work for the your turn problems Without a summary in your own words…