Unit 1. Warm-Up – X.X Vocabulary – X.X Holder Holder 2 Holder 3 Holder 4
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Unit 1Section 4.1Section 4.2Section 4.3Section 4.4Section 4.5Section 4.6-4.7Holder
9. 6x + 4y = 16Prerequisite SkillsSKILLS CHECKWrite the equation so that y is a function of x.10. x + 2y = 511. 12x + 6y = 12
3. y = x + 6; when x = 0, 2, 4, 6, and 8Prerequisite SkillsSKILLS CHECKGraph the function on a coordinate plane and give the Input/ Output table.
Vocabulary 4.1DomainSet of INPUTS to a functionSometimes these are considered the X variablesAKA the independent variableRangeSet of OUTPUTS ofa functionSometimes these are considered the YvariablesAKA the dependent variableQuadrants4 regions of a coordinate planeFunctionA numerical relationship where ONE input has EXACTLY ONE output
Notes 4.1 Plot Pts. and Graphs To plot points, move along the X axis first, and then the Y axis You have to run before you jump (or drop!). Domain = Inputs Range = OutputsUsually in TableformatQuadrants labeledWith Roman Numerals
GUIDED PRACTICEfor Example 1 1. Use the coordinate plane in Example 1 to give the coordinates of points C, D, and E.SOLUTION
GUIDED PRACTICEfor Example 1 y-coordinate of any point on the x-axis is 02. What is the y-coordinate of any point on the x-axis?SOLUTION
Make a table by substituting the domain values into the function. STEP 1SOLUTIONGraph the function y = 2x 1 with domain 2, 1, 0, 1, and 2. Then identify the range of the function.EXAMPLE 3Graph a function
STEP 2List the ordered pairs: ( 2, 5),( 1, 3), (0, 1), (1, 1), (2, 3).Then graph the function.EXAMPLE 3Graph a functionIdentify the range. The range consists of the y-values from the table: 5, 3, 1, 1, and 3.
GUIDED PRACTICEfor Examples 2 and 3STEP 1Make a table by substituting the domain values into the function.
GUIDED PRACTICEfor Examples 2 and 3
GUIDED PRACTICEfor Examples 2 and 3STEP 2List the ordered pairs: ( 6, 4),( 3, 3), (0, 2), (3, 1), (6,0). Then graph the function.STEP 3Identify the range. The range consists of the y-values from the table: 0, 1, 2, 3 and 4.
Graph a function represented by a table EXAMPLE 4VOTING
Graph a function represented by a tableEXAMPLE 4
Years before orsince 1920 12 8 404812Votes (millions)15151927293740
Graph a function represented by a table EXAMPLE 4SOLUTION
Graph a function represented by a table EXAMPLE 4SOLUTION
Warm-Up 4.2You may use a calculator on every assignment from this point forward unless otherwise told not to!
Lesson 4.2, For use with pages 215-2221.Graph y = x 2 with domain 2, 1, 0, 1, and 2.ANSWER
Lesson 4.2, For use with pages 215-2222. 3x + 4y = 16 Rewrite the equation so y is a function of x.ANSWER3. 6x 2y = 12 ANSWERy = 3x + 6
Vocabulary 4.2Linear EquationThe graph of the solutions to the function form a STRAIGHT LINE!
Notes 4.2 Graph Linear Equations Standard Form of a Linear Equation looks like this:Ax + By = C, where A, B, and C are real numbers and A and B are not both = 0 There are several ways to sketch graphs of linear equations, but the most common is THISGET Y BY ITSELF!!!Build a table with at least 3 valuesSketch the graphThe graphs of y = constant and x = constant are special cases of linear equations. Well check those out in a minute!
Substitute 3 for x and 4 for y.Simplify.Write original equation.Check whether each ordered pair is a solution of the equation.SOLUTIONWhich ordered pair is a solution of 3x y = 7?EXAMPLE 1Standardized Test PracticeTest (3, 4):
Simplify.Write original equation.Standardized Test PracticeEXAMPLE 1Test (1, 4):Substitute 1 for x and 4 for y.So, (3, 4) is not a solution, but (1, 4) is a solution of 3x y = 7.
Solve the equation for y.SOLUTIONEXAMPLE 2Graph an equationGraph the equation 2x + y = 3. 2x + y = 3y = 2x 3STEP 1Make a table by choosing a few values for x and finding the values of y.STEP 2
x 2 1012y 7 5 3 11
Graph an equation EXAMPLE 2Plot the points. Notice that the points appear to lie on a line.STEP 3
Graph (a) y = 2 and (b) x = 1.Graph y = b and x = a EXAMPLE 3y = 2x = -1
GUIDED PRACTICEfor Examples 2 and 3Graph the equation2. y + 3x = 2Solve the equation for y.SOLUTIONy + 3x = 2y = 3x 2STEP 1
GUIDED PRACTICEfor Examples 2 and 3Make a table by choosing a few values for x and finding the values of y.Plot the points. Notice that the points appear to lie on a line.Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end.STEP 3STEP 4STEP 2
x 2 1012y41 2 5 8
GUIDED PRACTICEfor Examples 2 and 33. y = 2.5SOLUTIONFor every value of x, the value of y is 2.5. The graph of the equation y = 2.5 is a horizontal line 2.5 units above the x-axis.4. x = 4SOLUTIONFor every value of y, the value of x is 4. The graph of the equation x = 4 is a vertical line 4 units to the left of the y-axis.
SOLUTIONSTEP 1Make a table.EXAMPLE 4Graph a linear function
STEP 2STEP 3Connect the points with a ray because the domain is restricted.STEP 4Identify the range. From the graph, you can see that all points have a y-coordinate of 4 or less, so the range of the function is y 4.EXAMPLE 4Graph a linear functionPlot the points.
GUIDED PRACTICEfor Example 4SOLUTIONSTEP 1Make a table.
x0 1 2 3 4y1471013
GUIDED PRACTICEfor Example 4STEP 2STEP 3Connect the points with a ray because the domain is restricted.STEP 4
Daily Homework Quiz For use after Lesson 4.21. Graph y + 2x = 4
Daily Homework Quiz For use after Lesson 4.22. The distance in miles an elephant walks in t hours is given by d = 5t. The elephant walks for 2.5 hours. Graph the function and identify its domain and range.
Vocabulary 4.3X-interceptWhere a graph crosses the X axisY-interceptWhere a graph crosses the Y axis
Notes 4.3 Graph using Intercepts.The primary reason to use the Standard Form of a linear equation is b/c it does make finding the x and y intercepts VERY easy! To find the X-intercept of a function Set Y=0 and solve for X To find the Y-intercept of a function Set X=0 and solve for YSince you only need two points to make a lineGraph the X and Y intercepts and connect them!
Substitute 0 for y.Write original equation.To find the x-intercept, substitute 0 for y and solve for x.SOLUTIONFind the x-intercept and the y-intercept of the graph of 2x + 7y = 28.Find the intercepts of the graph of an equation
EXAMPLE 1Solve for x.2x + 7(0) = 28
2(0) + 7y = 28Find the intercepts of the graph of an equation EXAMPLE 1To find the y-intercept, substitute 0 for x and solve for y.Write original equation.Substitute 0 for x.Solve for y.
Substitute 0 for y.Write original equation.To find the x-intercept, substitute 0 for y and solve for x.SOLUTIONFind the x-intercept and the y-intercept of the graph of the equation.Solve for x.3x + 2(0) = 6x = 23x + 2y = 6GUIDED PRACTICEfor Example 11. 3x + 2y = 6
3(0) + 2y = 6Find the intercepts of the graph of an equation EXAMPLE 1To find the y-intercept, substitute 0 for x and solve for y.Write original equation.Substitute 0 for x.Solve for y.3x +2y = 6y =3 GUIDED PRACTICEfor Example 1
SOLUTIONSTEP 1Use intercepts to graph an equation
EXAMPLE 2Graph the equation x + 2y = 4.x + 2y = 4Find the intercepts.0 + 2y = 4X intercept = (4,0)Y intercept = (0,2)
Use intercepts to graph an equation
EXAMPLE 2STEP 2Plot points. The x-intercept is 4, so plot the point (4, 0). The y- intercept is 2, so plot the point (0, 2). Draw a line through the points.
EVENT PLANNINGSolve a multi-step problemEXAMPLE 4You are helping to plan an awards banquet for your school, and you need to rent tables to seat 180 people. Tables come in two sizes. Small tables seat 4 people, and large tables seat 6 people. This situation can be modeled by the equation.4x + 6y = 180where x is the number of small tables and y is the number of large tables. Find the intercepts of the graph of the equation.
SOLUTIONSTEP 1Solve a multi-step problemEXAMPLE 4 Give four possibilities for the number of each size table you could rent.
Graph the equation.Find the intercepts.
Solve a multi-step problemEXAMPLE 4Since x and y both represent numbers of tables, neither x nor y can be negative. So, instead of drawing a line, draw the part of the line that is in Quadrant I.STEP 2Graph the equation.The x-intercept is 45, so plot the point (45, 0).The y-intercept is 30, so plot the point (0, 30).
Solve a multi-step problemEXAMPLE 4STEP 3Find the number of tables. For this problem, only whole-number values of x and y make sense. You can see that the line passes through the points (0, 30),(15,20),(30, 10), and (45, 0).
Solve a multi-step problemEXAMPLE 4So, four possible combinations of tables that will seat 180 people are: 0 small and 30 large, 15 small and 20 large, 30 small and 10 large,and 45 small and 0 large.
Warm-Up 4.41) Do pages 10-13 from the Classified ads packet as a group.2) You have ~15 minutes to work on this.