3.5 Graphing Linear Equations in Slope-Intercept Form

Slope Describes the rate of change between any two points on a line. It is the measure of the steepness of the line Slope = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 = 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 Slope = 1/2 Slope = 2 4 2 4

Slope Describes the rate of change between any two points on a line. It is the measure of the steepness of the line Slope = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 = 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 Slope = m = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1

Example 1: Finding the Slope of a Line Describe the slope of each line. Then find the slope using the formula. a.) The slope rises from left to right. So, the slope is positive. m = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 4 Let ( 𝑥 1 , 𝑦 1 )=(2,2) Let ( 𝑥 2 , 𝑦 2 )=(-1,-2) m = −2 −2 −1 −2 3 m = −4 −3 m = 4 3

Example 1: Finding the Slope of a Line Describe the slope of each line. Then find the slope using the formula. b.) The slope falls from left to right. So, the slope is negative. (-3,3) m = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 -4 Let ( 𝑥 1 , 𝑦 1 )=(-3,3) Let ( 𝑥 2 , 𝑦 2 )=(3,-1) 6 (3,-1) m = −1 −3 3 −−3 m = −4 6 m = −2 3

You try! m = 1−3 1−−4 m = 3−−1 3−−3 m = −3−4 2−5 m = −2 5 m = 4 6 Let ( 𝑥 1 , 𝑦 1 )=(-4,3) Let ( 𝑥 2 , 𝑦 2 )=(1,1) Let ( 𝑥 1 , 𝑦 1 )=(-3,-1) Let ( 𝑥 2 , 𝑦 2 )=(3,3) Let ( 𝑥 1 , 𝑦 1 )=(5,4) Let ( 𝑥 2 , 𝑦 2 )=(2,-3) m = 1−3 1−−4 m = 3−−1 3−−3 m = −3−4 2−5 m = −2 5 m = 4 6 m = −7 −3 m = 2 3 m = 7 3

Example 2: Finding the Slope From a Table The points represented in the table lie on a line. How can you find the slope of the line from the table? What is the slope of the line? a.) YOU CAN CHOOSE ANY TWO POINTS FROM THE TABLE AND USE THE SLOPE FORMULA! m = 14−20 7−4 m = −6 3 Let ( 𝑥 1 , 𝑦 1 )=(4,20) Let ( 𝑥 2 , 𝑦 2 )=(7,14) m = -2

Example 2: Finding the Slope From a Table The points represented in the table lie on a line. How can you find the slope of the line from the table? What is the slope of the line? b.) m = 2−2 1−−1 m = 0 2 Let ( 𝑥 1 , 𝑦 1 )=(-1,2) Let ( 𝑥 2 , 𝑦 2 )=(1,2) m = 0

Example 2: Finding the Slope From a Table The points represented in the table lie on a line. How can you find the slope of the line from the table? What is the slope of the line? c.) m = 0−−3 −3−−3 m = 3 0 Let ( 𝑥 1 , 𝑦 1 )=(-3,-3) Let ( 𝑥 2 , 𝑦 2 )=(-3,0) The slope is undefined

You try! 4.) The points represented by the table lie on a line. What is the slope of the line? x 2 4 6 8 y 10 15 20 25 m = 5 2

Using the Slope-Intercept Form of a Linear Equation A linear equation written in the form y = mx + b is in slope-intercept form. The slope of the line is m and the y-intercept of the line is b. y = mx + b Slope y-intercept

Example 3: Identifying Slopes and y-Intercepts Find the slope and y-intercept of the graph of each linear equation. y = 3x – 4 y = 6.5 c) -5x – y = -2 y = 3x + (-4) Slope = 3 y intercept = -4 y = 0x + 6.5 Slope = 0 Y-intercept = 6.5 -5x – y = -2 +5x +5x - y = 5x – 2 -1 -1 y = -5x + 2 Slope = -5 y-intercept = 2

You try! Find the slope and y-intercept of the graph if the linear equations. 5.) y = -6x + 1 6.) y = 8 7.) x + 4y = -10 Slope = -6 y-intercept= 1 y = 0x + 8 Slope = 0 y-intercept = 8 x + 4y = -10 -x -x 4y = -x – 10 4 4 y = −1 4 x – 10 4 y = −1 4 x – 2 1 2 Slope= -1/4 y-intercept= -2 1 4

Example 4: Using Slope-Intercept Form to Graph Graph 2x + y = 2. Identify the x-intercept. Step 1: Rewrite the equation in slope-intercept form. 2x + y = 2 -2x -2x y = -2x + 2 Step 2: Find the slope and y-intercept. m= -2 y-intercept= 2 Step 3: The y-intercept is 2. So, plot (0,2) Step 4: Use the slope to find another point on the line. slope= 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 = −2 1 Plot a point 2 units down and 1 unit to the right from the y- intercept) The line crosses the x-axis at (1,0). So, the x-intercept is 1.

Example 5: Graphing from a Verbal Description a.) A linear function g models a relationship in which the dependent variable increases 3 units for every 1 unit the independent variable increases. Graph g when g(0)=3. Identify the slope, y-intercept and x-intercepts of the graph. Slope: Since the graph is linear it has a constant rate of change. Let x represent the independent variable and y represent the dependent variable. When the dependent variable increases by 3, the change in y is +3. When the independent variable increases by 1, the change in x is +1. So the slope is 3 1 =3.

Example 5: Graphing from a Verbal Description a.) A linear function g models a relationship in which the dependent variable increases 3 units for every 1 unit the independent variable increases. Graph g when g(0)=3. Identify the slope, y-intercept and x- intercepts of the graph. Slope= 3 y-intercept: The statement g(0) = 3 indicates that when x=0, y=3. So, the y-intercept is 3.

Example 5: Graphing from a Verbal Description a.) A linear function g models a relationship in which the dependent variable increases 3 units for every 1 unit the independent variable increases. Graph g when g(0)=3. Identify the slope, y-intercept and x- intercepts of the graph. Slope= 3 y-intercept= 3 Graph: Plot the y-intercept and use the slope to find another point on the line.

Example 5: Graphing from a Verbal Description a.) A linear function g models a relationship in which the dependent variable increases 3 units for every 1 unit the independent variable increases. Graph g when g(0)=3. Identify the slope, y-intercept and x- intercepts of the graph. Slope= 3 y-intercept= 3 x-intercept: The line crosses the x-axis at (-1,0). So, the x-intercept is -1.

You try! Graph the linear equation and identify the x-intercept. 8) y = 4x – 4 9) x + 2y = 6 The line crosses the x-axis at (1,0). So, the x-intercept is 1. Slope= 4 y-intercept= -4 x + 2y = 6 -x -x 2y = -x + 6 2 2 y = - 1 2 x + 3 Slope =- 1 2 The line crosses the x-axis at (6,0). So, the x-intercept is 6. y-intercept= 3

You try! 10) A linear function h models a relationship in which the dependent variable decreases by 2 units for every 5 units the independent variable increases. Graph h when h(0) = 4. Identify the slope, y-intercept and x-intercept of the graph. Slope = −2 5 y-intercept = 4 x-intercept =10

Example 6: Solving Real-Life Problems In most real-life problems, slope is interpreted as rate, such as miles per hour, dollars per hour or people per year.

Example 6: Solving Real-Life Problems A submersible that is exploring the ocean floor begins to ascend to the surface. The elevation h (in feet) of the submersible is modeled by the function h(t) = 650t – 13,000, where t is the time (in minutes) since the submersible began to ascend. a) Graph the function and identify its domain and range. (see board) Domain: 0 < t < 20 Range:-13,000 < h < 0

Example 6: Solving Real-Life Problems A submersible that is exploring the ocean floor begins to ascend to the surface. The elevation h (in feet) of the submersible is modeled by the function h(t) = 650t – 13,000, where t is the time (in minutes) since the submersible began to ascend. b) Interpret the slope and the intercepts of the graph. The slope is 650. So, the submersible ascends 650 feet per minute. The h-intercept is -13,000. So, the elevation of the submersible at 0 minutes is -13,000 feet. The t-intercept is 20. So, the submersible takes 20 minutes to reach an elevation of 0 feet (sea level).

You try! 11) The elevation of the submersible is modeled by h(t) = 500t – 10,000. Graph the function and identify its domain and range. Interpret the slope and intercepts of the graph.