Point-slope Form 2.5 Date: 10/03/18 1. Grab your Binder 2. Copy down the Essential Question (EQ). 3. Work on the Warm-up. Essential Question Explain what the point-slope equation is useful and when to use it Warm Up: Write down the purpose This task focus on understanding and using various notations for linear functions. Students can selected the index in two different ways, thus getting two different, but equivalent equations.

Purpose This task focus on understanding and using various notations for linear functions. Students can selected the index in two different ways, thus getting two different, but equivalent equations.

1. What do you think about the strategies that Zac and Sione used 1. What do you think about the strategies that Zac and Sione used? Are either of them correct? Why or why not? Use as many representations as you can to support your answer. Both Zac and Sione are correct. 𝑦=6𝑛+1 and 𝑦=6 𝑛−1 +7 𝑙𝑒𝑡 𝑛=1 𝑦=6(1)+1 𝑦=6 1−1 +7 𝑦=6+1 𝑦=6 0 +7 𝑦=7 𝑦=7

Linear Equations in Two Variables Point-Slope Form Linear Equations in Two Variables

Where m is the slope and b is the intercept What we Know so far…. Slope Intercept Form Where m is the slope and b is the intercept

The New Stuff… Point- Slope Form Point- Slope form is: Where m is the slope and (x1, y1) is the given point

Write the Equation in Point-Slope Form Step 1: Plug it in   Point- Slope Form   Slope – Intercept Form

A linear equation written in the form y – y1 = m(x – x1) is in point-slope form. The graph of this equation is a line with slope m passing through the point (x1, y1). Plot the point, then rise & run from there…. Example: The graph of the equation y – 3 = - (x – 4) is a line of slope m = - passing through the point (4, 3). 1 2 x y 4 8 m = - 1 2 (4, 3)

Using Point-Slope Form We need the slope and a point: 1 2 4 3 5 -1 -2 -3 -4 -5 (2,2) Plug them into the point-slope form Point – Slope Form

Example: Write the point-slope form for the equation of the line through the point (-2, 5) with a slope of 3. Use the point-slope form, y – y1 = m(x – x1), with m = 3 and (x1, y1) = (-2, 5). y – y1 = m(x – x1) y – 5 = 3(x – (-2)) Let m = 3, let (x1, y1) = (-2, 5). y – 5 = 3(x + 2) Simplify into point-slope form

Given two points, write the equation of a line in point-slope form (2,-3) and (4,-2) Steps: 1. Find slope 2. Place a point and the slope in into point slope form 3. Point-Slope form Solve for y & get .. Slope-intercept form

To write point-slope into slope-intercept: Simplify using distributive property, then move the constant to the right side & combine like terms

Example: Write the slope-intercept form for the equation of the line through the points (4, 3) and (-2, 5). 2 1 5 – 3 -2 – 4 = - 6 3 Calculate the slope. m = y – y1 = m(x – x1) Point-slope form Use m = - and the point (4, 3). y – 3 = - (x – 4) 1 3 Solve for y… Slope-intercept form y = - x + 13 3 1

A Challenge Can you write the equation of a line in point-slope form that passes through (-3,6) and (1,-2) 8 2 6 + 2 -3 – 1 = - 4 1 Calculate the slope. m = y – y1 = m(x – x1) Use m = - 2 and the point (1, -2). y + 2 = - 2 (x – 1) Point-Slope Form y = -2(x – 1) - 2 y = -2x + 2 - 2 Slope-Intercept Form y = -2x + 0 y = -2x

Form Follows Function 2.6 Date: 10/04/18 Essential Question Explain how you can use the slope formula to write an equation of a line. Warm Up: Answer the question

Purpose The purpose of the task is to build fluency with the procedural work of linear and exponential functions..

Review of some formula Linear Functions 𝑦=𝒎𝑥+𝑏 Slope Intercept Form 𝒎 is the slope 𝒃 is the y-intercept ( 0, b) 𝑦− 𝑦 1 =𝑚(𝑥− 𝑥 1 ) Point Slope Form (𝑥 1 , 𝑦 1 ) is a point. 𝐴𝑥+𝐵𝑦=𝑐 Standard Form A is an Integer B is an Integer C is an Integer

first point ( 𝑥 1 , 𝑥 2 ) second point ( 𝑥 2 , 𝑦 2 ) Slope formula 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 first point ( 𝑥 1 , 𝑥 2 ) second point ( 𝑥 2 , 𝑦 2 )

Review of some formula Exponential Function Functions 𝑦=𝑓 0 𝑟 𝑥 0-term formula 𝑓 0 = 0 term 𝑟=𝑐𝑜𝑚𝑚𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 𝑦=𝑓 1 𝑟 𝑥−1 1-term formula 𝑓 1 =1 𝑡𝑒𝑟𝑚

How to solve ANY World Problem Step 1: Read the question in full and slow. Step 1.5 ( optional): Draw a Picture. Step 2: Extract all the information and label them. Step 3: Make or find an equation. Step 4: Do some math and solve the problem. Problem with step 4? Option A: You missed some information, go back to Step 1 and Step 3. Option B: You used the wrong equation. Try another equation. Go back to Step 3

1. In his job selling vacuums, Joe makes $500 each month plus $20 for each vacuum he sells. Write the equation that describes Joe’s monthly income 𝐼 as a function of the n, the number of vacuums sold. Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

1. In his job selling vacuums, Joe makes $500 each month plus $20 for each vacuum he sells. Write the equation that describes Joe’s monthly income 𝐼 as a function of the n, the number of vacuums sold. 𝐼 𝑛 =20𝑛+500 Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

Write the equation of the line with a slope of -1 through the point (-2, 5) Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

Write the equation of the line with a slope of -1 through the point (-2, 5) 𝑚=−1 𝑥 1 =−2 𝑦 1 =5 𝑦− 𝑦 1 =𝑚(𝑥− 𝑥 1 ) Point Slope Form 𝑦− 5 =−1(𝑥−(−2)) Plug in 𝑦−5=−1(𝑥+2) Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

Write the equation of the geometric sequence with a constant ratio of 5 and a first term of -3. 𝑟=5 𝑓 1 =−3 𝑦=𝑓 1 𝑟 𝑥−1 1-term formula 𝑦=(−3) 5 𝑥−1 Plug in Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

Write the equation of the function graphed below: Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

Write the equation of the function graphed below: 𝒎= 𝟑 𝟒 𝑏=3 𝑦=𝒎𝑥+𝑏 Slope Intercept Form 𝑦= 𝟑 𝟒 𝑥+3 Plug in Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

4. The population of the resort town of Java Hot Springs in 2003 was estimated to be 35,000 people with an annual rate of increase of about 2.4%. Write the equation that models the number of people in Java Hot Springs, with t = the number of years from 2003? Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

4. The population of the resort town of Java Hot Springs in 2003 was estimated to be 35,000 people with an annual rate of increase of about 2.4%. Write the equation that models the number of people in Java Hot Springs, with t = the number of years from 2003? Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

4. The population of the resort town of Java Hot Springs in 2003 was estimated to be 35,000 people with an annual rate of increase of about 2.4%. Write the equation that models the number of people in Java Hot Springs, with t = the number of years from 2003? 𝑓 0 =35,000 𝑟=1+.024 𝑦=𝑓 0 𝑥 𝑟 0-term formula 𝑦=35,000 𝑥 1.024 Plug in Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither

𝒎=𝟏. 𝟐 𝑥 1 =1 𝑦 1 =1. 7 𝑦− 𝑦 1 =𝑚(𝑥− 𝑥 1 ) Point Slope Form 𝑦−(1. 7)=1 𝒎=𝟏.𝟐 𝑥 1 =1 𝑦 1 =1.7 𝑦− 𝑦 1 =𝑚(𝑥− 𝑥 1 ) Point Slope Form 𝑦−(1.7)=1.2(𝑥−(1)) Plug In 𝑦−1.7=1.2(𝑥−1) Name the form of the equation you wrote and why you chose to use that form. This function is: linear exponential neither (choose one) This function is: continuous discrete neither