Finding Equations of Lines How to find the equation of a line from a graph. Method 1: 1) Look at the graph. Locate 2 coordinates on the line and find the slope by using the y y1 rise ratio of or the slope formula of: m= 2 . x2 x1 run 2) Find the coordinate where the line crosses the y-axis: (0,b) 3) Substitute your slope “m” and y-intercept “b” into the slope-intercept formula: y=mx+b. Example: 2 coordinates: (0,0) and (3,4). Slope= 4 3 y-intercept=(0,0) 4 4 The equation of the line is: y= x+0 or y= x 3 3 Method 2: 1) Look at the graph. Locate 2 coordinates on the line and find the slope by using the y y1 rise ratio of or the slope formula of: m= 2 . x2 x1 run 2) Use the point slope formula y-y1=m(x-x1) Substitute slope value for M and pick either point to get the values for y1 and x1. Solve the equation for y. Example: using the graph above 4 4 The slope is and use the coordinate (0,0): y-0= (x-0) 3 3 4 4 The equation of the line is y= x+0 or y= x. 3 3 How to find the equation of a line from a table. 1) Look at the table. Locate 2 coordinates in the table and find the slope by using the y y1 slope formula of: m= 2 . x2 x1 2) Use the point slope formula y-y1=m(x-x1) Substitute slope value for M and pick either point to get the values for y1 and x1. 3) Solve the equation for y. Example: 86 1) 2 coordinates: (3,6) and (4,8): m= =2, m=2 x y 43 2) Substitute (3,6) and m=2 into y-y1=m(x-x1): 3 6 y-6=2(x-3) 4 8 y-6=2x-6 5 10 +6 +6 3) The equation of the line is y=2x+0 or y=2x How to find the equation of a line (closed form) from arithmetic sequences 1) Take the arithmetic sequence and place into a t-chart. The x values are the term positions (1, 2, 3, etc…), the y values are the sequence numbers in that term position. 2) Look at the table. Locate 2 coordinates in the table and find the slope by using the y y1 slope formula of: m= 2 . x2 x1 3) Use the point slope formula y-y1=m(x-x1) Substitute slope value for M and pick either point to get the values for y1 and x1. 4) Solve the equation for y. Example: Sequence: -4, 1, 6, 11, 16, … x y 1 -4 2 1 3 6 4 11 5 16 6 1 =5 32 2) Substitute (3,6) and m=5 into y-y1=m(x-x1): 1) (2,1) and (3,6) : m= 3) Solve for y: y-6=5(x-3) y-6=5x-15 y=5x-9 How to find the equation of a line in context Step 1: Identify the rate of change (slope or m) and the starting value (y-intercept or b). Step 2: Write a verbal model. Then write an equation. Step 3: Calculate the rate of change, also known as slope, also known as m Step 4: Evaluate using the point-slope formula. Example 1: Your gym membership costs $35 per month after an initial membership fee. A member has paid a total of $250 after 6 months. Write an equation that gives the total cost of a gym membership as a function of the length of membership (in months). Find the total cost of membership after 10 months. 1) Rate of change: monthly cost= $35 per month Starting value: initial membership fee … the point is (0,b) 2) Total Cost (C)=Monthly cost ($35) X Number of months (t)+Membership Fee (b) C=35t+b 3) Substitute: 250=35(6)+b; b=40 4) Substitute to evaluate: C=35(t)+40;when t=10, C=390 Example 2: The initial fee to have a website set up using a server is $48. It costs $44 per month to maintain the website. 1) Rate of change: $44 Starting value: set up=$48… so our point is (0,48) 2) Total Cost (C)=Monthly cost ($44) x Number of months (t)+Set up fee($48) C=44t+48 How to find the best fit line (equation of a line) from a scatterplot Step1: Make a scatter plot of data or use an existing one. Step 2: Decide whether the data can be modeled by a line (has a positive or negative correlation) Step 3: Draw a line that appears to fit the data closely. There should be approximately as many points above the line as below it. Step 4: Write an equation by choosing 2 coordinates on the line. Find the slope and the y-intercept: A. Look at the graph. Locate 2 coordinates on the line and find the slope by using the ratio of the slope formula of: m= y2 y1 . x2 x1 B. Substitute 1 coordinate and the slope into the point-slope form: rise or run y-y1=m(x-x1):, then solve for y. Example: A. 2 coordinates: (1, 1.5) and (2.5, 2.5)= 2.5 1.5 1 10 2 = or 2.5 1 1.5 15 3 2 B. y-y1=m(x-x1):; y-1.5= (x-1) 3 2 5 C. y= x+ 3 6 Homework: Create one of each. Directions: Create one of each type of method for finding an equation of a line. Show an example, the method for finding the equation, and give the equation. You will need graph paper, knowledge of graphs, scatterplot, sequences, and tables, and a lot of creativity for this assignment.