4.1 Coordinates Objective: To plot points and name points in the coordinate plane. A coordinate plane is formed by two real number lines that intersect at the origin. (x-axis and y-axis) is a point in the coordinate plane represented by real numbers. The x-coordinate is the first number. The y-coordinate is the second number. Ex. (3,6) An ordered pair (x, y)  (right or left, up or down) How do you tell?

Coordinate plane (x, y) y-axis Quadrant II (-, +) Quadrant I (+, +) Origin (0,0) x- axis Quadrant III (-, -) Quadrant IV (+, -)

Plotting points To plot a point: (3,4) Start at (0,0) Move 3 to the right (positive) Then 4 up (positive) make a point Plot these points: (-2, -4) (0, 3) (-1,0) (6,-2) (-4, 5)

Practice Name the following points and give the quadrant or axis where they lie. A: B: C: D: A D C B Plot the following points and label each!! E (-3, 4) F(-5, 0) G(-3, -1) H(0, 0)

The Coordinate Plane Steps to Make a Scatter Plot: Determine what will be x and y. x – is in charge, it changes automatically y – depends on x, is not automatic Determine units of each axis and label. Find range of variable Divide range by number of squares Always round up to “nice” unit Plot points.

Example Make a Scatter Plot The age (in years) of seven used cars and the price (in thousands of dollars) paid for the cars are recorded in the table. Make a scatter plot and explain what it indicates. Age 4 5 3 6 7 Price 6.9 6.1 7.5 5.2 4.2 7.1 3.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 price in $1,000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 age of car How much would a 2-year old car cost?

Example Make a Scatter Plot The amount (in millions of dollars) spent in the United States on snowmobiles is shown in the table. Make a scatter plot and explain what it indicates. Year 90 91 92 93 94 95 96 Spent 322 362 391 515 715 924 970

Reminders Math Lab Tomorrow - Stocks 4.1-4.3 Quiz on Monday, Oct. 7th Homework: P. 206-207 #’s 10-26 EVEN, 35-37 EXTRA CREDIT – Halloween “Goblin’ Goblin” Grid

What is represented by this BrainBat? TOOL O O O O LOOT

4.2 Graphing Linear Equations Objective: To graph linear equations using a table of values. Note (1) All the Eqs. in Chap 4 refer 2 variable linear Eqs. (2) The graph of each linear eq. is a LINE The solution to a linear equation -- is an ordered pair (x, y). There are many solutions to a linear equation and all of the solutions together form a straight line, 

Find out if the ordered pairs are solutions. HOW? A) -5x – 8y = 15 (-3, 0) B) -2x – 9y = 7 (-1, -1)

Graph a line -1 5 3 1 1 x 2x + y = 3 y (x, y) Steps to graph a line 5 Steps to graph a line 3 1. Pick three values for x 1 Plug in values for x then solve for y Solve for y, then evaluate y for all input x 3. Graph the ordered pairs 4. Connect these order pairs. This should form a straight line! y = -2x + 3

Function form  When an equation is solved for y = -1 1 A) 3x – y = 2 What are the advantages of putting the equation into function form? Graph the given linear equations. A) 3x – y = 2 Solve for y: x y (x, y) -1 1 We select three x values, and evaluate the corresponding y values.

Graph the given linear equations. B) 2x – 2y = 10 Solve for y, We select three x values, and evaluate the corresponding y values. What are the “good” x values we should select? The x values should make “nice” y values. (or, no fraction values for y)

Make a table of values and graph the following line: 6x – 3y = 12 2y = 4x + 1

MEMORIZE THESE!!!! It’s easy! Special Linear Equations x = #  always a vertical line y = #  always a horizontal line MEMORIZE THESE!!!! It’s easy! x = # , label the # on x-axis, then “cut” there y = # , label the # on y-axis, then “cut” there Ex 4) y = -3 Ex 5) x = 4 Ex 7) y = 0 Ex 6) x = -2

Make a table of values and graph the following line: y = -3 x = 4

Summary A two variables linear equation represents a line in x-y coordinate plan. An ordered pair is a solution to a two variable linear equation, then the point represented by the ordered pair is on the line represented by the linear equation, and vice versa. Remember the two types of special line by an easy way: x = #  no y  cut x-axis at that #  parallel to y-axis y = #  no x  cut y-axis at that #  parallel to x-axis

Summary When graphing a linear equation, remember the 4 steps: Pick a few x values Solve for y, then evaluate y for all input x Graph ordered pairs Connect ordered pairs with a line

Lesson 4.2 DHQ Decide whether the given ordered pair is a solution of 2x – 3y = 8. (-2, -4) b. (7, -2) Rewrite 4x – 2y = 18 in function form.

Reminders 4.1-4.3 Quiz on Monday, Oct. 7th Homework: P. 214-215 #’s 15-20, 30-32, 36-37, 60

What is represented by this BrainBat? G G E G E G E G G

4.3 Quick Graphs Using Intercepts Objective: To graph lines using x and y-intercepts. What is an x-intercept? What is a y-intercept? Where the line or curve crosses the x-axis. This should be written as the point (x, 0) . (WHY?) Where the line or curve crosses the y-axis. This should be written as the point (0, y) . (WHY?) y-intercept? ( , ) 0 3 x-intercept? ( , ) 2 0 Note x or y intercept is a point!!!

Remember – TWO POINTS CAN MAKE A LINE! Ex 1) Find the x and y-intercepts To find the y-intercept; let x = 0 and solve for y. To find the x-intercept; let y = 0 and solve for x. 6x + 3y = 12 6( 0 ) +3y = 12 6x + 3( 0 ) = 12 3y = 12 6x = 12 y = 4 x = 2 The y intercept is (0, 4) The x intercept is (2, 0)

1. 2x – y = 4 2. 3y – 2x = -6 You try these: Calculate the x and y-intercept. Then graph each line. 1. 2x – y = 4 2. 3y – 2x = -6

What happens with horizontal and vertical lines? Find the x and y-intercepts (if possible). Graph each line. Ex 2) y = 4 Ex 3) x = -1 You can set x = 0 but end with “No solution”  can not find x, or no x-intercept. You can set y = 0 but end with “No solution”  can not find y, or no y-intercept. y-int. (0, 4) x-int. (-1, 0) Variable x does not show up  no x-intercept  line is parallel to x-axis Variable y does not show up  no y-intercept  line is parallel to y-axis

You try these: Calculate the x and/or y-intercept. Then graph each line. 4. y = 2x + 4 5. y = -3

6. Horizontal line passing (-3, 4) and (4, 4). Graph and Write the equation of the special line 6. Horizontal line passing (-3, 4) and (4, 4). 7. Vertical line passing through (-2, 3)

Summary x-intercept (y-intercept) is a point where the line or curve crosses the x(y)-axis. To find x-intercept (y-intercept), just setting y = 0 (x = 0) in an equation. 3. When graph a line, just find x and/or y-intercept and then connect two intercepts with a line.

Graphing with Intercepts The point where line(curve) crosses the x-axis. What is an x-intercept? How do you find an x-intercept? (Why does this work?) How should you write the x-intercept? set y = 0 and then solve for x. In an order pair (x, y). The point where line(curve) crosses the y-axis. What is a y-intercept? How do you find a y-intercept? (Why does this work?) How should you write the y-intercept? set x = 0 and then solve for y. In an order pair (x, y). Find the intercepts and graph. 3) 2x – 4y = –8 4) x – 4y = 2 – 4y = –8 – 4y = 2 y = 2 y = – 1/2 2x = –8 x = 2 x = –4 So – If I gave you a quiz over this material, how would you do?

Lesson 4.3 DHQ Give the x- and y- intercepts of the graph of 2x – y = -4. 2. Graph 2x – 3y = 6 using x and y intercepts.

Reminders 4.1-4.3 Quiz on Monday, Oct. 7th Homework: P. 221-222 #’s 35-37, 44-49, 56-57

What is represented by this BrainBat? C H I M A D E N A

4.4 The Slope of a Line Slope is:    Objective: To calculate the slope of a line using 2 points, to read slope from a given line and to understand some applications of slope. Slope is: 1) The measurement of the steepness and direction of a line (m) (ORDER MATTERS!)  To read the slope from a graph, choose 2 lattice points and write a ratio of the vertical to horizontal change. Explain. Slope: m = m = m =  

The slope formula – finding slopes from ordered pairs If you are given 2 points  you have 2 x-values and 2 y-values. You need memorize it!!! 1) (2, 4) and (3, -2) 2) (0, -5) and (1, 3) 3) (4, 5) and (4, -3) *** 4) (-2, -5) and (2, -5)

6) Describe how to move a slope of…….  a) -2  b) 5a) Sketch a positive slope. c) Sketch a 0 slope. d) Sketch an undefined slope. b) Sketch a negative slope. 6) Describe how to move a slope of…….  a) -2  b) c) d)

Ex 4) The store plane descends 100 feet for every 2000 feet it travels Ex 4) The store plane descends 100 feet for every 2000 feet it travels. Find the slope of decent. Ex 5) The road rises 2 feet for every 50 yards. Find the slope of the road. * REMEMBER TO CONVERT YDs to FT

There are 3 different formulas to calculate the slope: Summary There are 3 different formulas to calculate the slope: (order matters!!)

Lesson 4.4 DHQ Find the slope of each line. Use formula to find the slope. (-2, 2), (0, 4) (1, 1), (4, 2) 2. Find the slope of the line.

Reminders Homework: 4.4+4.6 Quiz on Wednesday, Oct. 16th P. 230-231 #’s 23-28, 35-36 ***38

What is represented by this BrainBat? sdraw

4.6 Graph Using Slope-Intercept Form Objective: 1. To use slope intercept form to graph a line. 2. Tell the slope of two parallel lines. Slope-Intercept Form is : Memorize this form!! y = mx + b (m = slope & b = y-intercept.) The equation must be y = to use this short-cut m is always the coefficient of y-intercept (b) is always x Constant – (no variable)

Solve the following equations for y =  then pick out the m and b for each. 2x + y = -5 m= b = 2) 3y – 6 = 2x 3) 2x – 5y + 10 = 0 4) 2x + 3y = -9 -2x -2x y = –2x - 5 -2 -5

Use the slope (m) and y-intercept (b) to graph each line. 2x + y = 5 You need to graph a) the y-intercept first and then b) use the slope to get the second (third) point (move the slope) c) connect the two (three) points by a line.

One more for you to try.  3x – 2y = –6 b = Can you solve for y? Can you locate the m and b? Can you put this on the graph paper? 3x – 2y = –6 -3x -3x

Summary The slope-intercept form is the form in which y is solved. In the slope-intercept, the number in front of x is the slope m and the number right after x is the y-intercept. When you graph a line of slope-intercept form, you have to graph a) the y-intercept first and then b) use the slope to get the second (third) point c) connect the two (three) points by a line. 4. Parallel lines have equal slope.

Lesson 4.6 DHQ Write x + y + 3 = 0 in slope-intercept form. Then graph the equation.

Reminders Homework: 4.4+4.6 Quiz on Wednesday, Oct. 16th P. 244 #’s 13-18, 29-31

Slope-Intercept Form Organization Check:   Organization Check: We can now graph an equation in three different ways: Using an x, y table Using x-intercepts, y-intercepts Slope-intercept form   Where a line crosses the y-axis.

Write the equation in slope-intercept form, then graph.   m = b =

Write the equation in slope-intercept form, then graph.   m = b =

Write the equation in slope-intercept form, then graph.   m = b =

Write the equation in slope-intercept form, then graph.   m = b =

Write the equation in slope-intercept form, then graph.   m = b =

Write the equation in slope-intercept form, then graph.   m = b =

Write the equation in slope-intercept form, then graph.   m = b =

Parallel Lines Two different lines in the same plane are _______ if they do not intersect. parallel

Parallel Lines Lines are parallel if they have the same ______. slope

Are the lines parallel? y = 2x + 3 y = 3 + 2x

Are the lines parallel? y = -2x + 3 y = 3 + 2x

Are the lines parallel? 4x + 2y = 8 y = -2x + 2

Reminders Homework: 4.4+4.6 Quiz on Wednesday, Oct. 16th P. 247 QUIZ 2 #’s 1-5, 13-14

If it takes six men one hour to dig six holes, how long does it take one man to dig half a hole?

Identify when a relation is a function. Daily Objective: Identify when a relation is a function.

4.8 Functions Recall: Vocabulary Relation: Domain: Objective: To evaluate functions at given values Recall: Vocabulary Relation: Domain: Range: Function:____________________________________ Any set of points, equation or graph The inputs ; the x-values from the points or graph The outputs ; the y-values from the points or graph A relation where all the inputs are different and any input has one and only one output  Ex. { (-3, 3), (1, 1), (3, 1), (4, -2) } Is this a function? Why? Domain: Range: {-3, 1, 3, 4} Yes – all x-values are different {3, 1, -2}

Remember? one For every input, there is exactly _____ output!

Is it a function? Input 1 3 Output 2 Domain ? Range ?

Is it a function? Input 1 3 Output 2 Domain ? Range ?

Is it a function? Input 1 2 3 Output 4 5 6 Domain ? Range ?

However… There are rules that associate more than one output for each input. But, we can’t call it a function anymore. We call it a…

Relation

Any set of ordered pairs! What is a relation? Any set of ordered pairs! It doesn’t matter how many different outputs an input has!

Decide if a relation (set of ordered points) is a function. What we need to do is… Decide if a relation (set of ordered points) is a function.

Is the relation a function?

Besides looking at a table and deciding, we are going to look at graphs as well! Vertical Line Test – A relation is a function, if NO vertical line passes through two or more points!

Is the relation a function? Relation – Not a function

Is the relation a function? Relation – Not a function

Function Notation f(x) f(x) means the value of f at x. You can replace it with y. Does not mean f times x. It means f at x.

Evaluate the function for the given variable. f(x) = 2x – 3 when x = -2, 0, and 3 Steps: Write the original function. Substitute -2 for x. Simplify.

Evaluate the function for the given variable. g(x) = -5x + 3 when x = -3, 0, and 1

Graph f(x) = -2x + 3 Graph the Function! Steps for using slope-intercept form: Rewrite the function as “y = …” Find y-intercept and slope. Graph and connect.

Graph f(x) = x + 4 Graph the Function! Steps for using intercepts: Find the x-intercept by setting y = 0. Find the y-intercept by setting x = 0. Plot the intercepts on the coordinate plane and graph the line.

Reminders Ch. 4 Test on Tuesday, Oct. 22nd Homework: P. 259-260 #’s 16-17, 20-21, 37-40