Bell Ringer Get out your notebook and prepare to take notes on Chapter 8 From Chapter 1, how do we plot points on a graph? i.e. −2, 4

Linear Functions and Graphing Chapter 8 Linear Functions and Graphing

8.1 – Relations and Functions (Page 404) Essential Question: What is the difference between a function and a relation?

8.1 cont. Relation: A set of ordered pairs Note: { } are the symbol for "set" Examples: { (0,1) , (55,22), (3,-50) } { (0, 1) , (5, 2), (-3, 9) } { (-1,7) , (1, 7), (33, 7), (32, 7) } Any group of numbers is a relation as long as the numbers come in pairs

8.1 cont. DOMAIN: 𝟐, 𝟔, 𝟗, 𝟐 RANGE: 𝟓, 𝟖, 𝟓, 𝟏𝟏 Example 1: What is the domain and range of the following relation: 2,5 , 6,8 , 9,5 , 2,11 ? DOMAIN: 𝟐, 𝟔, 𝟗, 𝟐 RANGE: 𝟓, 𝟖, 𝟓, 𝟏𝟏

8.1 cont. Domain First coordinates of the relation Range Second coordinates of the relation Tip: Alphabetically x comes before y, and “domain” comes before “range” DOMAIN 𝒙,𝒚 RANGE

8.1 cont. Function: Special type of relation Set of ordered pairs containing a domain and range (like a relation) Each member of the domain 𝑥 is paired with EXACTLY one member of the range 𝑦 SOME relations are functions Tip: Think of a function as a number generator

8.1 cont. Mapping diagrams: Shows whether a relation is a function Steps: List domain values and range values in order Draw arrows from domain values to corresponding range values 2 range values for domain value 1 NOT a function 1 range value for each domain value IS a function 1 range value for each domain value IS a function

No, there are two range values for the domain value 2 8.1 cont. Example 2: Is the following relation a function: −2,3 , 2,2 , 2,−2 ? Explain. 3 −2 2 2 −2 No, there are two range values for the domain value 2

This relation is a function! 8.1 cont. Functions can model everyday situations when one quantity depends on another One quantity is a function of the other Example 3: Is the time needed to cook a turkey a function of the weight of the turkey? Explain. The time the turkey cooks (range value) is determined by the weight of the turkey (domain value). This relation is a function!

8.1 cont. Vertical-Line Test Visual way of telling whether a relation is a function If you can find a vertical line that passes through two points on the graph, then the relation is NOT a function

Therefore, the relation is NOT a function. 8.1 cont. Example 4: Graph the relation shown in the following table: Using the vertical line test, you would pass through both 𝟐,𝟎 and 𝟐,𝟑 . Therefore, the relation is NOT a function.

8.1 - Closure What is the difference between a function and a relation? Any set of ordered pairs is a relation A function is a relation with the restriction that no two of its ordered pairs have the same first coordinate

8.1 - Homework Page 407-408, 2-28 even

No, there are two range values for the domain value −𝟐. Bell Ringer Get out your notebook and prepare to take notes on Section 8.2 Draw a mapping diagram for the following relation: −1, 1 , −2, 1 , −2, 2 , 0, 2 Is the relation a function? Explain. −𝟏 −𝟐 𝟎 𝟏 𝟐 No, there are two range values for the domain value −𝟐.

8.2 – Equations With Two Variables (Page 409) Essential Questions: What is the solution of an equation with two variables? How can you graph an equation that has two variables?

8.2 cont. 𝒚=𝟑𝒙+𝟒 Equations With Two Variables: Solution: ordered pair that makes the equation true Can have multiple ordered pair solutions [i.e. 𝑥, 𝑦 = ?, ? ] 𝒚=𝟑𝒙+𝟒

8.2 cont. SOLUTION = −𝟏, 𝟏 𝒚=𝟑𝒙+𝟒 𝒚=𝟑 −𝟏 +𝟒 𝒚=−𝟑+𝟒 𝒚=𝟏 Example 1: Find the solution of 𝑦=3𝑥+4 for 𝑥=−1. 𝒚=𝟑𝒙+𝟒 𝒚=𝟑 −𝟏 +𝟒 𝒚=−𝟑+𝟒 𝒚=𝟏 SOLUTION = −𝟏, 𝟏

8.2 cont. Linear Equations: Any equation whose graph is a line ALL equations in this lesson are linear equations Solutions can be shown in a table or graph

8.2 cont. Example 2: For the following equation, make a table of values to show solutions. Then, graph your results. 𝒚=𝟐𝒙+𝟏 𝒙 𝟐𝒙+𝟏 𝒚 2 0 +1=0+1=1 1 2 1 +1=2+1=3 3 2 2 2 +1=4+1=5 5 2 3 +1=6+1=7 7

8.2 cont. 𝒚=𝟐𝒙+𝟏 Vertical Line Test NOT A FUNCTION!! Every x-value has exactly 1 y-value Therefore, this relation IS a function Linear equations are functions unless its graph is a vertical line NOT A FUNCTION!! 𝒚=𝟐𝒙+𝟏

8.2 cont. 𝒚=−𝟑: YES 𝒙=𝟒: NO Example 3: Graph the following equations: 𝑦=−3 and 𝑥=4. Is each equation a function? 𝒙 𝒚 −3 1 2 3 𝒙 𝒚 4 1 2 3 𝒚=−𝟑: YES 𝒙=𝟒 𝒚=−𝟑 𝒙=𝟒: NO

8.2 cont. 𝟐𝒙+𝒚=𝟑 −𝟐𝒙 −𝟐𝒙 𝒚=𝟑−𝟐𝒙 Example 4: Solve 2𝑥+𝑦=3 for 𝑦. Then, graph the equation. 𝟐𝒙+𝒚=𝟑 −𝟐𝒙 −𝟐𝒙 𝒚=𝟑−𝟐𝒙 𝒙 𝟑−𝟐𝒙 𝒚 3−2 0 =3−0=3 3 1 3−2 1 =3−2=1 2 3−2 2 =3−4=−1 −1 3−2 3 =3−6=−3 −3

8.2 - Closure What is the solution of an equation with two variables? Any ordered pair that makes the equation a true statement How can you graph an equation that has two variables? Make a table of values to show ordered-pair solutions of the equation Graph the ordered pairs, then draw a line through the points

8.2 - Homework Page 412, 2-36 even

Bell Ringer Get out your 8.2 homework assignment Get out your notebook and prepare to take notes on Section 8.3 Is the ordered pair −3, −2 a solution of 4𝑥−3𝑦=6? Why or why not? 𝟒 −𝟑 −𝟑 −𝟐 =𝟔 −𝟏𝟐+𝟔=𝟔 −𝟔≠𝟔

8.3 – Slope and y-intercept (Page 415) Essential Question: What is an easier way to graph linear equations?

8.3 cont. Slope: Ratio that describes the tilt of a line To calculate slope, use the following ratio:

8.3 cont. NEGATIVE SLOPE POSITIVE SLOPE UNDEFINED SLOPE ZERO SLOPE

8.3 cont. Example 1: Find the slope of a line that includes the points −6, 2 and −2, 4 . 𝟒 −𝟐, 𝟒 𝟐 −𝟔, 𝟐 Slope = 𝟐 𝟒 = 𝟏 𝟐

8.3 cont. Horizontal and Vertical Lines:

8.3 cont. = 𝟏−𝟔 𝟖−𝟐 =− 𝟓 𝟔 𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 Example 2: Find the slope of the line through the following pair of points: 2, 6 and 8, 1 How can we find the slope without graphing?? 𝒙 𝟏 , 𝒚 𝟏 𝒙 𝟐 , 𝒚 𝟐 = 𝟏−𝟔 𝟖−𝟐 =− 𝟓 𝟔 𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏

8.3 cont. y-intercept: Slope-intercept form: Point where the line crosses the y-axis y-intercept is the “Constant” in the equation Slope-intercept form: 𝑦=𝑚𝑥+𝑏 Where m is the slope of the line, and b is the y-intercept Slope-intercept form is helpful in graphing equations

8.3 cont. Example 3: Find the slope and y-intercept of 𝑦=− 1 3 𝑥+2; then, graph the equation.

8.3 - Closure USE SLOPE-INTERCEPT FORM!! What is an easier way to graph linear equations? USE SLOPE-INTERCEPT FORM!!

Page 418-419, 2-18 even, 24-38 even 8.3 - Homework

Bell Ringer 𝒚=𝟓+ 𝟐 𝟑 𝒙 Get out your 8.3 homework assignment Get out your notebook and prepare to take notes on Section 8.4 Solve 3𝑦−2𝑥=15 for y. Then identify the slope and the y-intercept of the equation. y-intercept 𝒚=𝟓+ 𝟐 𝟑 𝒙 slope

8.4 – Writing Rules for Linear Functions (Page 422) Essential Question: How can we use tables and graph to write a function rule?

8.4 cont. Function Notation: Function Rule: Use 𝑓 𝑥 instead of 𝑦 𝑓 𝑥 is read “f of x” Function Rule: Equation that describes a function

8.4 cont. Example 1: A long-distance phone company charges its customers a monthly fee of $4.95 plus 9 cents for each minute of a long-distance call. Write a function rule that relates the total monthly bill to the number of minutes a customer spent on long-distance calls. Find the total monthly bill if the customer made 90 minutes of long-distance calls. Let 𝒎= minutes spent on long distance calls Let 𝒕 𝒎 = total monthly bill 𝒕 𝒎 =𝟒.𝟗𝟓+.𝟎𝟗𝒎 Evaluate the function for 𝒎=𝟗𝟎 𝒕 𝟗𝟎 =𝟒.𝟗𝟓+.𝟎𝟗 𝟗𝟎 𝒕 𝟗𝟎 =𝟏𝟑.𝟎𝟓 $𝟏𝟑.𝟎𝟓

8.4 cont. Writing Function Rules From Tables or Graphs Look for a pattern! May need to add, subtract, multiply, divide, or use a power OR a combination of these operations

8.4 cont. Example 2: 𝒇 𝒙 =𝟐𝒙 𝒇 𝒙 =−𝟐𝒙 𝒚=𝟐𝒙+𝟏 Write a rule for each of the following linear function tables: 𝒇 𝒙 =𝟐𝒙 𝒇 𝒙 =−𝟐𝒙 𝒚=𝟐𝒙+𝟏

8.4 cont. Example 3: Write a rule for the linear function graphed below:

8.4 - Closure How can we use tables and graphs to write a function rule? Look for a pattern using a combination of addition, subtraction, multiplication, division, and powers Use slope and y-intercept to write a linear function

8.4 - Homework Page 424-425, 2-22 even

Bell Ringer Plot your given point on the coordinate plane:

8.5 – Scatter Plots (Page 427) Essential Question: How can we make scatter plots and use them to find a trend?

8.5 cont. Scatter Plots: Shows a relationship between two sets of data

8.5 cont. Example 1: Make a scatter plot for the data in the table below: Value (in thousands) Age (in years)

8.5 cont. Example 2: Make a scatter plot for the data below:

8.5 cont. Trends:

8.5 cont. Trend Line: Shows relationship between data sets Allows us to make predictions about data values Possible to have no trend line

Example 3: Use the following scatter plot to predict the height of a tree that has a circumference of 175 in: 88 ft

8.5 - Closure How can we make scatter plots and use them to find a trend? Plot ordered pairs Draw a trend line (positive, negative, or no trend) Predict values

8.5 - Homework P 430-432; 2-30 even