Section 3.2A – Slope of a Line

Section 3.2A – Slope of a Line Chapter 3: Lines Slope of a Line 𝒓𝒊𝒔𝒆 𝒓𝒖𝒏 ▪ The slope of a line is 𝒎 = ▪ ▪ ▪ The rise measures the vertical change. The run measures the horizontal change. Slopes can be: A positive number A negative number Zero Undefined For Example: Find the slope of the line pictured below. • • • The rise between the marked points is positive 2 as the graph gets 2 units higher as you move up to get from left marked point to the right marked. The run is 4. The slope of this line is positive because the rise is positive. 𝑠𝑙𝑜𝑝𝑒 = 𝑟𝑖𝑠𝑒 2 1 = = 𝑟𝑢𝑛 4 2 Answer: Slope of the line is 1 2 For Example: Find the slope of the line pictured below. • • • The rise will be negative −3 because the line gets 3 units lower as you move down from left marked point to the right marked point. The run is 5. The slope of this line is negative because the rise is negative. 𝑟𝑖𝑠𝑒 −3 = 𝑟𝑢𝑛 5 3 Answer: Slope of line is − 𝑠𝑙𝑜𝑝𝑒 = 5 Section 3.2A – Slope of a Line For Example: Find the slope of the line pictured below. • • • The rise between the marked points is 0 as the graph does not get higher nor lower as you move from left marked point to the right marked point. The rise is 0. The run between the marked points is 4. The slope of this line is zero because the rise is zero. 𝑠𝑙𝑜𝑝𝑒 = 𝑟𝑖𝑠𝑒 0 = =0 𝑟𝑢𝑛 4 Chapter 3: Lines For Example: Find the slope of the line pictured below. • The run between the marked points is 0 as the graph does not get further left nor further as you move bottom marked point to the top marked point. The run is 0. • The rise of this line is 3. As the bottom and the top marked points are 3 units apart. • The slope of this line is undefined because the run is zero. Answer: Slope of the line is 0 𝑠𝑙𝑜𝑝𝑒 = 𝑟𝑖𝑠𝑒 3 = = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑟𝑢𝑛 0 (Fractions with 0 in the denominator are undefined.) Answer: Slope of the line is undefined Section 3.2A – Slope of a Line Horizontal line properties: 𝑦 = 𝑏 • • All horizontal lines have an equation in the form 𝑦 = 𝑏 (We learned this in the previous section) All horizontal lines have a slope of 0. Because the rise of all horizontal lines is 0. Find the equation of the HORIZONTAL line below, and then find its slope. Horizontal lines have equations of the form 𝑦 = 𝑏 Chapter 3: Lines Vertical line properties: 𝑥 = 𝑎 • • All vertical lines have an equation in the form 𝑥 = 𝑎 (We learned this in the previous section) All vertical l lines have a slope of undefined. Because the run of all vertical lines is 0. Find the equation of the VERTICAL line below, and then find its slope. Vertical lines have equations of the form 𝑥 = 𝑎 Each point on the graph has a 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of Each point on the graph has a 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of 3. 4. Slopes of all horizontal lines are 0. Slopes of all vertical lines are undefined. Answer: Answer: • Equation of the line 𝑦 = 3 • Equation of the line: 𝑥 = 4 • Slope = 0 (This is the line from the • Slope = undefined (this is a graph from a previous example) previous example) Section 3.2A – Slope of a Line Chapter 3: Lines How to graph a line given a point and the slope. Step 1. Plot the given point. Step 2. Use the slope formula 𝑚 = 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 to identify the rise and the run. Step 3. Starting at the given point, count out the rise and run to mark the second point. Step 4. Connect the points with a line. 3 For Example: Graph a line that has a point (1,-1) and a slope = 4 Step 1. Plot the point (1, -1) Step 2. Rise = 3, Run = 4 Step 3. Start at the point (1,-1) and go up 3 and right 4 and mark a second point. Step 4. Connect the points with a line. Section 3.2A – Slope of a Line Chapter 3: Lines Slope Intercept Form of an Equation of a Line: 𝑦 = 𝑚𝑥 + 𝑏 The slope–intercept form of an equation of a line with 𝑠𝑙𝑜𝑝𝑒 m 𝑎𝑛𝑑 y − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡, (0, 𝑏) is 𝑦 = 𝑚𝑥 + 𝑏 Example: Given the equation of the line: 𝑦 = 3𝑥 + 2 • Identify the slope • Identify the y-intercept • sketch a graph. Solution: The equation is written in slope-intercept form so the number in front of the 𝑥 is the slope and the constant is the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡. Answer: Slope: 𝑚 = 3 y-intercept (0,2) To graph follow these steps: Step 1. Plot the point (0, 2) 𝑟𝑖𝑠𝑒 3 Step 2. 𝑅𝑖𝑠𝑒 = 3, 𝑅𝑢𝑛 = 1 (the slope can be written as 𝑚 = 𝑟𝑢𝑛 = 1) Step 3. Start at the point (0,2) and go up 3 and right 1 and mark a second point. Step 4. Connect the points with a line. Section 3.2A – Slope of a Line Chapter 3: Lines How to find the slope of a line between two points The formula to find the slope of the line between two points (𝑥1 , 𝑦1 )and (𝑥2 , 𝑦2 ) is: 𝑚 = 𝑦2 −𝑦1 𝑥2 −𝑥1 (Note the letter 𝑚 is often used to represent the slope of a line.) 𝑠𝑙𝑜𝑝𝑒 = 𝑦 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 − 𝑦 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 𝑥 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 − 𝑥 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 Steps to find the slope of the line that connects two points. Step 1: Identify 𝑥1 , 𝑦1 (These are the 𝑥 𝑎𝑛𝑑 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 of the first point) Step 2: Identify 𝑥2 , 𝑦2 (These are the 𝑥 𝑎𝑛𝑑 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 of the second point) Step 3: Plug the numbers into the formula and simplify. For Example: Find the slope of the line that passes through the points. (5, −2) and (−6,1) Step 1: 𝑥1 = 5, 𝑦1 = −2 Step 2: 𝑥2 = −6, 𝑦2 = 1 Step 3: 𝑚 = 𝑦2 −𝑦1 𝑥2 −𝑥1 = 1−(−2) −6−5 = 1+2 −11 = 3 −11 𝐼𝑡 𝑤𝑜𝑢𝑙𝑑 𝑏𝑒 𝑜𝑘𝑎𝑦, 𝑏𝑢𝑡 𝑜𝑑𝑑, 𝑡𝑜 𝑙𝑒𝑎𝑣𝑒 𝑡ℎ𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 . 3 Answer: The slope of the line between the points is: 𝑚 = − 11 Section 3.2A – Slope of a Line Chapter 3: Lines There are at least 4 ways to graph a line. We have used each method in the two sections we covered in this chapter. This table gives a quick review of what we have already done. The table below suggests a method that can be used to graph the line next to it. Although there are generally more than one way to graph a line. For Example: Graph 𝑥 = 2 For Example: Graph 𝑦 = −1 The equation only has an 𝑥. I know this graph will be a vertical line and it must pass through 2 on 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠. To graph simply draw a vertical line through 2 on the 𝑥 − 𝑎𝑥𝑖𝑠. The equation only has a 𝑦. I know this graph will be a horizontal line and it must pass through −1 on the 𝑦 − 𝑎𝑥𝑖𝑠. To graph simply draw a vertical line through −1 on the 𝑦 − 𝑎𝑥𝑖𝑠. Section 3.2A – Slope of a Line For Example: Graph −𝑥 + 2𝑦 = 6 Chapter 3: Lines This equation is written in general form. I usually find x and y-intercepts to graph a line if the equation is written in general form. Step 1: Find 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 (𝑙𝑒𝑡 𝑦 = 0) −𝑥 + 2(0) = 6 −1𝑥 = 6 −1𝑥 6 = −1 −1 𝑥 = −6 (𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 (−6,0)) Step 2: Find 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 (𝑙𝑒𝑡 𝑥 = 0) −0 + 2𝑦 = 6 2𝑦 = 6 𝑦 = 3 (𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 (0,3) Step 3: Find a third point, pick a value for 𝑥 and solve for 𝑦. Let 𝑥 = 2 −2 + 2𝑦 = 6 +2 +2 2𝑦 = 8 𝑦=4 𝑇ℎ𝑖𝑟𝑑 𝑝𝑜𝑖𝑛𝑡 (2, 4) Step 4: Plot the points Step 5: Connect the points with a line Section 3.2A – Slope of a Line Chapter 3: Lines For example: Graph 𝑦 = −𝑥 + 4 This equation is in slope-intercept form. First find the relevant information to be able to create the graph. −1 1 • Identify the slope: 𝑚 = −1 = • Identify the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 (0,4) To graph follow these steps: Step 1. Plot the point (0, 4) Step 2. 𝑅𝑖𝑠𝑒 = −1 𝑅𝑢𝑛 = 1 Step 3. Start at the point (0,4) and go down 1 and right 1 and mark a second point. Step 4. Connect the points with a line. Section 3.2A – Slope of a Line given line. Chapter 3: Lines #1-8: Find the slope of the 1) 2) 3) 4) Section 3.2A – Slope of a Line of the given line. 5) 7) Chapter 3: Lines #1-8 continued: Find the slope 6) 8) #9-16: Sketch the graph of the line. State the value of the slope and of the y-intercept, state if there is no y-intercept. 3 2 −3 𝑥 2 9) 𝑦 = 4 𝑥 − 5 10) 𝑦 = 3 𝑥 − 1 11) 𝑦 = 13) 𝑥 = 3 14) 𝑥 = 5 15) 𝑦 = −2 +2 12) 𝑦 = −1 𝑥 4 +3 16) 𝑦 = −4 #17 – 18: Find the equation of the line. Write your equation in slope-intercept form when possible. 17a) 17b) 17c) 18a) 18b) 18c) Section 3.2A – Slope of a Line Chapter 3: Lines #19-26: Sketch the graph of a line passing through the given point with the indicated slope. 19) 𝑝𝑜𝑖𝑛𝑡 (3, −2) 𝑠𝑙𝑜𝑝𝑒 = 1 2 20) 𝑝𝑜𝑖𝑛𝑡 (−4,1 ) 𝑠𝑙𝑜𝑝𝑒 = 3 2 21) 𝑝𝑜𝑖𝑛𝑡 (−1, 2) 𝑠𝑙𝑜𝑝𝑒 = −3 4 22) 𝑝𝑜𝑖𝑛𝑡 (4, −3 ) 𝑠𝑙𝑜𝑝𝑒 = −2 3 23) 𝑝𝑜𝑖𝑛𝑡 (5,6) 𝑠𝑙𝑜𝑝𝑒 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 24) 𝑝𝑜𝑖𝑛𝑡 (2,7) 𝑠𝑙𝑜𝑝𝑒 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 25) 𝑝𝑜𝑖𝑛𝑡 (0,6) 𝑠𝑙𝑜𝑝𝑒 = 0 26) 𝑝𝑜𝑖𝑛𝑡 (2,4) 𝑠𝑙𝑜𝑝𝑒 = 0 #27 – 34: Find the slope of the line that passes through the two points. 27) 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 (1,5) 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 (3, −6) 28) 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 (1,4) 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 (7, −5) 29) 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 (−2, −3) 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 (4,7) 30) 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 (−3, −4) 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 (2,8) 31) 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 (−2,5) 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 (9,5) 32) 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 (−3,2) 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 (4,2) 33) 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 (−2,0) 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 (−2,9) 34) 𝑓𝑖𝑟𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 (−3,0) 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 (−3,5)