1. Tues., Sept. 15th Chapter 2.1 Functions Target: Students will analyze relations and functions Agenda: ◦ 2.1 Function Introduction ◦ 2.1 Homework

2. Relations and Functions! relation A relation is a set of ordered pairs ◦ A mapping or pairing of input values with output values function A function is a special type of relation in which each element of the domain is paired with exactly one element of the range Domain The Domain of a relation is the set of all first coordinates (x-coordinates or independent variable) Range The Range of a relation is the set of all second coordinates (y-coordinates or dependent variable)

3. Domain & Range Domain – all the X values Range – all the Y values Function is if there are NO duplicate X values x y 0 2 1 4 2 8

4. Functions Is every function a relation? ◦ Yes Is every relation a function? ◦ No A function is called one to one if every element of the range is also paired to exactly one element of the range

5. Vertical Line Test You can use the vertical line test to determine whether a relation is a function If no vertical line intersects a graph in more than one point, the graph represents a function Function Not

6. Solving Functions Notation Notation: for equations, in function notations – it can be written f(x) Yf(x) Y = x + 3 is written as f(x) = x + 3 f(x) is read as ‘f of x’ f(x) is another way to write y What is the value of y when x=3 is also called f(3) independentinside The independent variable is inside the ( ) dependentoutside The dependent variable is outside f(3) is written as ‘f of 3’ It means evaluate the function when x = 3 (plug 3 into the equation for x) Any letters can be used c(x), f(x), a(x)… where ‘x’ is the independent variable

7. Solving Functions Find f(-3) (to solve – replace the f function with the value of -3 f(-3) = -3 + 3 f(-3) = 0 X coordinate value = -3 Y coordinate value = 0 (-3,0)

8. Solving Functions g(x) = 4x 2 – 2x If g(x) = 4x 2 – 2x Find g(2) (to solve – replace the g function with the value of 2 g(2) = 4(2) 2 – 2(2) g(2) = 16-4 g(2) = 12 X coordinate value = 2 Y coordinate value = 12 (2,12)

9. Discrete & Continuous Functions Discrete functions are functions with points that are not connected Continuous Functions are functions with an infinite number of points

10. Homework Page 60-62 ◦ #17 – 33 eoo, 35-37, 47-53 odd, 54, 55, 66-69 all

11. Thurs., Sept 17 th Chapter 2.1 Functions Target: Students will identify and write linear equations in standard form and graph them. Students will find and use the slope of a line and graph lines using slope and ordered pairs Agenda: ◦ 2.1 Homework ◦ 2.2 Linear Equations ◦ 2.3 Slope ◦ Homework assignment Quiz – Tues., Sept. 22 nd 2.1 Functions Quiz – Tues., Sept. 22 nd 2.1 Functions

12. Linear Equations Linear equation: ◦ has no operations other than addition, subtraction, and multiplication of a variable by a constant – such as: 5x- 3y = y may not ◦ variables may not be multiplied together or appear in a denominator  xy = 5 or 1/x=5 are not linear ◦ does not contain variables with exponents other than 1 … x 2 = 16 is not linear

13. Linear Function Is a function whose ordered pairs satisfy a linear equation f(x) = mx + b Any linear function can be written in the form of f(x) = mx + b

14. Standard Form Any linear equation can be written in standard form  Ax + By = C where A, B, and C are integers whose greatest common factor is 1 Ex: y= -2x + 3 2x + y = 3 (add 2x to each side) A = 2, B = 1, C = 3

15. Intercepts Y-intercept ◦ The y-coordinate of the point at which a graph crosses the y-axis ◦ Input x=0 ◦ Input x=0 to find y-intercept X-intercept ◦ The x-coordinate of the point at which a graph crosses the x-axis ◦ Input y=0 ◦ Input y=0 to find the x-intercept

16. Using Intercepts to graph a line Find the x-intercept (input y=0 into equation, solve for what x equals)  (X,0) Find the y-intercept (input x=0 into equation, solve for what y equals)  (0,Y) Now you have 2 points – you can plot and graph 3x – 4y + 12 = 0 X-intercept (find X when y=0)Y-intercept (find Y when x=0) 3x – 4(0) + 12 = 0 3x + 12 = 0 3x = -12 X = -4 (-4, 0) 3(0) – 4y + 12 = 0 -4y + 12 = 0 -4y = -12 y = 3 (0, 3)

17. 2.2 Worksheet Practice Walk through together 2.2 worksheet for practice

18. 2.3 Slope The slope of a line is the ration of the change in y-coordinates to the corresponding change in the x- coordinates Rise / Run ∆Y/∆X (y 1 – y 2 ) (change in y-coordinates) M= -------------------------------- (x 1 – x 2 ) (change in x-coordinates)

19. Slope-intercept form of a line Y = mx + b ◦ Where: m=slope, and b = y intercept (0,b) Switching from Standard form to Slope intercept form ◦ Take equation X + Y = 8 ◦ Solve for Y  Y = -X + 8 (m = -1, b = 8)

20. Using Slope to Graph a Line If (-4, -3) and slope of 2/3 1. Plot point on graph 2. Use slope (Rise/Run) to move to next spot ◦ Up 2 on the Y axis, Over right 3 on the X axis Connect two points with line

21. 2.3 Practice Practice on the 2.3 worksheet together

22. Homework Pg. 66 #15 – 25 (Odd), 39-47 (Odd) Pg. 73 #15, 19, 23, 27, 37, 43, 47, 52

23. Bell Ringer Problem… Can you figure this out:

24. Tues., Sept. 22 nd Chapter 2.4 Functions Target: Students will use the Slope Intercept form and the Point-Slope form to create linear equations Agenda: ◦ Quiz – 2.1Functions ◦ Homework, Bathroom & Seating Chart Discussion ◦ 2.2 & 3 Homework Check ◦ 2.4 Writing Equations of Lines ◦ 2.4 Homework Test – Chapter 2 Wed., Sept. 30th Test – Chapter 2 Wed., Sept. 30th

25. Seating Chart Starting next class… you can sit anywhere If there is a sub – sit in the current seating chart

26. Bathroom No leaving room when instructing unless absolute emergency, and then please ask During work time – please sign out on board – but also please let me know you are leaving

27. Homework Take out when you arrive in class Homework will get stamped for completion each day BEFORE we do homework check Mark up/grade homework in COLORED pen – (pencils are away) Put grade on top of page and turn in We will move to Homework Quizzes as our next step

28. Writing Equations of Lines Using Slope-Intercept Form 1. Use Slope-intercept form equation: a)Y=mx+b b)Find y-intercept (b) by… 1.Choice a point (x,y) 2.Use the slope, x-, and y-values to substitute into y=mx+b 3.Solve for b c)Rewrite y=mx+b with the slope and y- intercept (b) that you found

29. Writing equations of lines Using Point-Slope form 1. Use point-slope form equation (y - y 1 ) = m(x - x 1 ) a)This formula is derived from the slope formula b)Find slope c)Use a point (x 1, y 1 ) and slope to substitute into y - y 1 =m(x - x 1 ) d)Solve for y. State your equation in y=mx+b

30. Homework Pg. 78 #14 – 44 even