Bell Ringer Write each phrase as a mathematical expression. Thinking with Mathematical Models
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1 Bell Ringer Write each phrase as a mathematical expression. 1. the sum of nine and eight 2. the sum of nine and a number 3. nine increased by a number x 4. fourteen decreased by a number p 5. the product of 9 and a number 6. five more than twice a number 7. three times a number decreased by 11 COPY EACH and SOLVE Thinking with Mathematical Models Investigation 2.2: Exploring Slope Focus: How do you write an equation for a linear function if you are given a graph, a table, or two points?
2 Linear functions are often used as models for patterns we see in data plots! Any linear function can be expressed by an equation in the form: y = mx +b. Now, let's talk about what each of those letters mean! Ponder will you... What do you know about the table and graph of the function with the equation y = 3x + 2? In particular, what do the numbers 3 and 2 tell you about the table and graph values? Today we will write equations using y = mx + b.
3 y = 3x + 2 slope The formula for any linear function is y = mx + b. The coefficient m tells: 1. the rate at which the values of y increase or decrease (the rate of change); 2. the steepness and direction of the graph; y = 3x + 2 y = -3x + 2
4 y = 3x + 2 y-intercept The coefficient b tells: 1. the point at which the graph of the function crosses the y-axis. y = 3x + 2 y = 3x - 2
5 y = mx +b The steepness of a staircase is commonly measured by comparing two numbers, the rise and the run. The rise is the vertical change from one step to the next, and the run is the horizontal change from one step to the next. The steepness of the line is the ratio of rise to run. This ratio is the slope of the line: Vertical Change Horizontal Change = rise run y = mx +b RISE RUN To find the slope, use or y 2 - y 1 x 2 - x 1 y intercept = Equation:
6 Steps to Write Equation 1. Find the slope using 2. Find the y-intercept RISE RUN 3. Write equation in form of y = mx + b. 4. Use a table to check your work. EXAMPLE Graph the line that passes through the points SLOPE - (-1, 1) and (1, 5) Then, use the graph to find the slope. Write the equation and check. INTERCEPT - EQUATION - y intercept = Equation:
7 PRACTICE TOGETHER Graph the line that passes through the points SLOPE - (-2, -2) and (2, 0) Then, use the graph to find the slope. Write the equation and check. INTERCEPT - EQUATION - y intercept = Equation: YOUR TURN Graph the line that passes through the points SLOPE - (2, 2) and (-2, -4) Then, use the graph to find the slope. Write the equation and check. INTERCEPT - EQUATION - y intercept = Equation:
8 Slope can be... Positive Negative Zero Undefined Given any two points on a line (x1, y1) and (x2, y2) slope = y 2 - y 1 x 2 - x 1 Part A pg. 36 For the functions with the graphs below, find the slope and y - intercept. Then write the equations for the lines in the form y = mx + b.
9 Part A pg. 36 Find the slope and y - intercept. Then write the equations for the lines in the form y = mx + b. Vertical Change Horizontal Change = rise run SLOPE - INTERCEPT - EQUATION - Part A pg. 36 Find the slope and y - intercept. Then write the equations for the lines in the form y = mx + b. Vertical Change Horizontal Change = rise run SLOPE - 1/2 = 0.5 INTERCEPT - -1 EQUATION - y = 0.5x - 1
10 Part A pg. 36 Find the slope and y - intercept. Then write the equations for the lines in the form y = mx + b. Vertical Change Horizontal Change = rise run SLOPE - INTERCEPT - EQUATION - Part A pg. 36 Find the slope and y - intercept. Then write the equations for the lines in the form y = mx + b. Vertical Change Horizontal Change = rise run SLOPE - -6/4 = -1.5 INTERCEPT - -1 EQUATION - y = -1.5x - 1
11 Part B 1. Find the equations for the linear functions that give these tables. Write them in the form y = mx +b. 2. For each table, find the unit rate of change of y compared to x. Part B Cont. 3. Does the line represented by this table have a slope that is greater than or less than the equations you found in part 1(a) and part 1(b)?
12 Part C The points (4, 2) and (-1, 7) lie on a line. 1. What is the slope of a line? 2. Find two or more points that lie on this line. Describe your method. 3. Yvonne and Jackie observed that any two points on a line can be used to find the slope. Are they correct? Explain. Part D Kevin said that the line with the equation y = 2x passes through the points (0,0) and (1,2). He also said the line with equation y = -3x passes through the points (0,0) and (1,-3). In general, lines with equations of the form y = mx always passes through the points (0,0) and (1, m). Is he correct? Explain.
13 Part E What is the slope of the horizontal line? What about a vertical line?
slope rise run Definition of Slope
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