Plotting Points

Linear Equations Example 1: Plot the point (5, 1). -8 -6 -4 -2 2 4 6 8 The first number (the “x”) tells you to go left or right. The second number (the “y”) tells you to go up or down.

Linear Equations Example 2: Plot the point (-3, 4). -8 -6 -4 -2 2 4 6 8 Why do I go left instead of right? Why up instead of down?

Linear Equations Example 3: Plot the point (-2, -3). -8 -6 -4 -2 2 4 6

Linear Equations Example 4: Plot the point (0, -4). -8 -6 -4 -2 2 4 6

Linear Equations Example 5: Plot the point (-3½ , 1½). -8 -6 -4 -2 2 4 6 8 For fractions, just approximate somewhere in between.

Slope

Linear Equations Slope Formula.

Linear Equations Find the slope between the given points. 1) (3, 6) and (1, 2) 2) (-3, 4) and (2, -3)

Linear Equations Find the slope between the given points. 3) (-5, -1) and (-1, -3) 4) (4, -1) and (4, -2)

Linear Equations Your turn!!! Find the slope between the given points. (3, 4) and (6, -2) (-3, -2) and (0, -5) (5, 3) and (5, -1) (-2, 4) and (0, 4) (-3, 5) and (6, 4)

Linear Equations Answers: m= -2 m= -1 m= undefined m= 0 m= -1/9

Function Compositions

-3(-2) + 4 = 6 + 4 = 10 (-2, 10) y = Using the function: y = -3x + 4 Evaluate when x = -2: Example 1: y = -3(-2) + 4 = 6 + 4 = 10 (-2, 10)

2(-3) – 5 =-6 – 5 = -11 (-3, -11) g(-3) = Using the function: g(x) = 2x – 5 Evaluate the following: Example 2: g(-3) = 2(-3) – 5 =-6 – 5 = -11 (-3, -11)

2(0) – 5 = 0 – 5 = -5 (0, -5) g(0) = Using the function: g(x) = 2x – 5 Evaluate the following: Example 3: g(0) = 2(0) – 5 = 0 – 5 = -5 (0, -5)

½ (-4) + 5 = -2 + 5 = 3 (-4,3) f(-4) = Using the function: f(x) = ½ x + 5 Evaluate the following: Example 4: f(-4) = ½ (-4) + 5 = -2 + 5 = 3 (-4,3)

(2)2 – 8(2) + 3 =4 – 16 + 3 = -9 (2,-9) f(2) = Using the function: f(x) = x2 – 8x + 3 Evaluate the following: Example 5: f(2) = (2)2 – 8(2) + 3 =4 – 16 + 3 = -9 (2,-9)

Sketching Graphs by Plotting Points

Linear Equations Example 1: Graph each equation. x y -3 (0, -3) 1 -2 -8 -6 -4 -2 2 4 6 8 -3 (0, -3) 1 -2 (1, -2) 2 -1 (2, -1) Pick some x’s and plug them in to see what y goes with them.

Linear Equations Example 2: Graph each equation. x y -5 (0, -5) 1 -3 -8 -6 -4 -2 2 4 6 8 -5 (0, -5) 1 -3 (1, -3) 2 -1 (2, -1) Pick some x’s and plug them in to see what y goes with them.

Linear Equations Example 3: Graph each equation. x y 2 (0, 2) 1 -1 -8 -6 -4 -2 2 4 6 8 2 (0, 2) 1 -1 (1, -1) 2 -4 (2, -4) Pick some x’s and plug them in to see what y goes with them.

Linear Equations Example 4: Graph each equation. x y -1 (0, -1) 3 1 -8 -6 -4 -2 2 4 6 8 -1 (0, -1) 3 1 (3, 1) 6 3 (6, 3) Pick some x’s and plug them in to see what y goes with them.

Linear Equations Example 5: Graph each equation. x y 2 (0, 2) 4 -1 -8 -6 -4 -2 2 4 6 8 2 (0, 2) 4 -1 (4, -1) -4 5 (-4, 5) Pick some x’s and plug them in to see what y goes with them.

Intercepts

Linear Equations Find the intercepts of the given equations. 1) 3x + 2y = 6 y-intercept x-intercept

Linear Equations Find the intercepts of the given equations. 2) 3x – y = -6 y-intercept x-intercept

Linear Equations Find the intercepts of the given equations. 3) y = 2x + 5 y-intercept x-intercept

Linear Equations Find the x-intercept and y-intercept. 1) 3x + 4y =12 4) y = 3x + 6 5) x = 2

Linear Equations Answers: 1) x-int= 4, y-int= 3 2) x-int= 5, y-int= -2 5) x-int= 2, y-int= does not exist

Graphing using the Intercepts

Linear Equations Example 1: Graph each equation using the x- and y-intercepts. -8 -6 -4 -2 2 4 6 8 y-intercept: 2(0) + y = 4 0 + y = 4 y = 4 x-intercept: 2x + 0 = 4 2x = 4 x = 2

Linear Equations Example 2: Graph each equation using the x- and y-intercepts. -8 -6 -4 -2 2 4 6 8 y-intercept: 3(0) – 4y = 12 -4y = 12 y = -3 x-intercept: 3x – 4(0) = 12 3x = 12 x = 4

Linear Equations Example 3: Graph each equation using the x- and y-intercepts. -8 -6 -4 -2 2 4 6 8 y-intercept: 0 - 3y = 6 -3y = 6 y = -2 x-intercept: x – 3(0) = 6 x - 0 = 6 x = 6

Linear Equations Example 4: Graph each equation using the x- and y-intercepts. -8 -6 -4 -2 2 4 6 8 y-intercept: 3(0) – 2y = 4 -2y = 4 y = -2 x-intercept: 3x – 2(0) = 4 3x = 4 x = 4/3

Linear Equations Example 5: Graph each equation. -8 -6 -4 -2 2 4 6 8 For special cases, remember “HOY VUX”. HOY= “H”orizontal line “0” = slope “Y” = # is the equation VUX= “V”ertical line “U”ndefined slope “X” = # is the equation

Linear Equations Example 6: Graph each equation using the x- and y-intercepts. -8 -6 -4 -2 2 4 6 8 y-intercept: y = -3/4(0) – 2 y = 0 – 2 Y = -2 x-intercept: 0 = -3/4x – 2 2 = -3/4x -8/3 = x

Linear Equations Now, you try it! Sketch the graph of each of the following.

Graphing using slopes and y-intercepts

What is the purpose of graphing equations? Linear Equations What is the purpose of graphing equations? It gives a visual representation of the equation, which may give us information about the equation that we need.

Linear Equations If this were the “graph” of someone’s relationship, how would you describe it?

Linear Equations If this were the “graph” of someone’s relationship, how would you describe it?

Linear Equations If this were the “graph” of someone’s relationship, how would you describe it?

Slope is defined to be “rise over run”. Linear Equations Slope is defined to be “rise over run”. This means that the top number in the slope fraction tells you how far “up or down” the line travels vs. how far “left or right” it travels.

Linear Equations Example 1: Graph each equation. slope y-intercept -8 -6 -4 -2 2 4 6 8 slope y-intercept When you get the y by itself, this is called slope-intercept form because you can see the slope and y-intercept.

Linear Equations Example 2: Graph each equation. slope y-intercept -8 -6 -4 -2 2 4 6 8 slope y-intercept Get the y by itself if it isn’t already.

Linear Equations Example 3: Graph each equation. slope y-intercept? -8 -6 -4 -2 2 4 6 8 slope y-intercept? Get the y by itself if it isn’t already.

Linear Equations Example 4: Graph each equation. 1 __ 1 slope? -8 -6 -4 -2 2 4 6 8 1 __ 1 slope? y-intercept Get the y by itself if it isn’t already.

Linear Equations Example 5: Graph each equation. __ 1 slope -8 -6 -4 -2 2 4 6 8 __ 1 slope y-intercept Get the y by itself if it isn’t already.

Linear Equations Example 6: Graph each equation. slope y-intercept -8 -6 -4 -2 2 4 6 8 slope y-intercept Get the y by itself if it isn’t already.

Linear Equations Example 7: Graph each equation. slope y-intercept -8 -6 -4 -2 2 4 6 8 slope y-intercept Get the y by itself if it isn’t already.

Linear Equations Example 8: Graph each equation. -8 -6 -4 -2 2 4 6 8 This is a special case because one of the variables is missing. For special cases, remember “HOY VUX”. HOY= “H”orizontal line “0” = slope “Y” = # is the equation VUX= “V”ertical line “U”ndefined slope “X” = # is the equation

Linear Equations Example 9: Graph each equation. -8 -6 -4 -2 2 4 6 8 For special cases, remember “HOY VUX”. HOY= “H”orizontal line “0” = slope “Y” = # is the equation VUX= “V”ertical line “U”ndefined slope “X” = # is the equation

Linear Equations Example 10: Graph each equation. slope y-intercept? -8 -6 -4 -2 2 4 6 8 slope y-intercept? Is this vertical or horizontal? Neither. It’s only horizontal or vertical if one of the variables is missing.

Linear Equations Now, you try it! Sketch the graph of each of the following.

Relations and Functions

Bellwork:

Linear Equations Answers: #2 x y -2 (0, -2) 5 (5, 0) -5 -4 (-5, -4) -8 -6 -4 -2 2 4 6 8 -2 (0, -2) 5 (5, 0) -5 -4 (-5, -4) Get y by itself first.

Linear Equations Answers: #3 y-intercept: 3(0) – 4y = -4 -4y = -4 -8 -6 -4 -2 2 4 6 8 y-intercept: 3(0) – 4y = -4 -4y = -4 y = 1 x-intercept: 3x – 4(0) = -4 3x = -4 x = -4/3

Linear Equations Answers: #4 slope y-intercept -8 -6 -4 -2 2 4 6 8 slope y-intercept Get the y by itself if it isn’t already.

Linear Equations A relation is a set of ordered pairs, because it tells how “x” and “y” relate to each other. There are three ways that a relation can be represented: a table, a graph, or a mapping. Let’s look at each. Graph:

Linear Equations If I were to give you these ordered pairs, could you write down the three types of representations: table, graph, and mapping? Graph:

Linear Equations The set of first numbers (x’s) of the ordered pairs in a relation is the domain. The set of second numbers (y’s) is the range. In the previous example, the domain of this set would be {0, 5, -3}. The range of this set would be {-2, 0, 4}.

Linear Equations The inverse of any relation is obtained by switching the coordinates in each ordered pair. In other words, the x’s become y’s, and the y’s become x’s. Graph:

Linear Equations A function is a relation in which each element of the domain is paired with exactly one element of the range. This is a function, because the elements of the domain (x’s) have only one element in the range (y’s) that they go to.

Linear Equations Is the following set of ordered pairs a function? This is NOT a function, because one of the elements from the domain (-2) is mapped to two different elements from the range (0 and 3).

Linear Equations How can we tell if the following is a function? This relation is a function, because of the “vertical line test”, which means if there is no place where I can draw a vertical line and hit more than one point, it is a function.

Linear Equations Is the following a function? This is not a function, because you can draw a vertical line and hit two points at x = 2.

Linear Equations How can we tell if the following is a function? This is a function, because I cannot find a place where I can draw a vertical line and cross the graph in more than one place.

Linear Equations Is the following a function? Yes, there is still no place where I can draw a vertical line and hit the function in more than one spot.

Linear Equations Is the following a function? This is not a function, because you can draw a vertical line somewhere and hit two points.

Linear Equations Homework: Page 146/ 9-13 odd, 14-21, 23-29 odd, 30-33

As You Enter Pick up a graphing calculator

Bellwork Solve the following equations/inequalities using your calculator… 2(x+3)-x+7=4(x+2)-2(x-4) 2x-3>4(x-2)+4 3x-5<4 and 2(x-1)>-6

Graphing and The Graphing Calculator

Notes: All graphing menus are found immediately below the screen Y= is where you put in functions to be graphed Window is where you adjust the size of the screen that is being displayed Zoom is where you can quickly change the size of the screen being displayed. Zoom 6 is the standard window Trace allows you to move along the functions by using the left and right arrows. Up/Down arrows will move the cursor from one function to another

Notes: Graph will display the graph of functions stored in Y= screen Stat Plot is where you can graph scatter plots TBL SET is where you setup your table Format is where you can make big changes to the coordinate plane. We will rarely use these features CALC is where we can find specific values of x and y like maximums/minimums, x-intercepts, etc.. TABLE is where the calculator will display a table of x and y values that the function(s) are generating. We can change TBL SET to ask and make our own table.