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Algebra One Notes for Chapter Four Section 4-‐1 – Graphing Basics Rectangular Coordinates and Lines Label the following 1. quadrant 1 2. quadrant 2 3. quadrant 3 4. quadrant 4 5. origin 6. x-axis 7. y-axis 8. Ordered Pair (x, y) at (2, 1) Graphing Ordered Pairs Start at the origin The first number is the x value – it tells how Left or Right to move Positive moves Right Negative moves Left The second number is the y value – it tells how far Up or Down to move Positive moves Up Negative moves Down Section 4-‐3 – Domain, Range, Function Characteristics Domain and Range Relation: a set of ordered pairs Function: a relation in which each x value maps to exactly 1 y value. Domain: the x values of the relation (without repeat) Range: the y values of the relation (without repeat) Functions Function: a relation in which each x value maps to exactly 1 y value. Vertical Line Test: if a vertical line can hit the graph of a relation more than once it fails the vertical line test and is not a function. Examples Graphically – Function? State Domain and Range. Ordered Pairs, Maps, and Tables Example of a Function: Focus on how no x value does two things. Ordered Pairs Map (-8, 4), (-1, 1), (0, 6), and (5, 4) Table x -8 -1 0 5 Graph y 4 1 6 4 Example of a Non-Function: Focus on how some of the x values do more than one thing. Ordered Pairs (-7, 7), (2, 0), (2, 4), (3, 7), (5, 0), and (5, 8) Map Table x -7 2 2 3 5 5 Graph y 7 0 4 7 0 8 Section 4-‐4 – Domain, Range, Function Characteristics Independent vs. Dependent Variables x is the independent variable y is the dependent variable Example: The equation y = 40x will tell y, the distance travelled, for travelling for x hours and 40 mph. So the distance travelled depends on the time. This shows that y is the dependent variable and x is the independent variable. Solutions to equations of 2 variables. If an ordered pair makes a true statement, then it is part of the solution. To test this – plug in the ordered pair and see if the right is equal to the left. Example: Given the equation y = 4x – 2, are (3, 3) and (0, -2) part of the solution? Plug (3, 3) into y = 4x – 2 3 = 4(3) – 2 3 = 10 NO so (3, 3) is NOT part of the solution plug (0, -2) into y = 4x – 2 -2 = 4(0) – 2 -2 = -2 YES so (0, -2) IS part of the solution Example: create a table to show solutions to the equation y = 2x + 1 Plug -2 in for x Plug 1 in for x Plug 0 in for x Plug 3 in for x y = 2(-2) + 1 and y = -3 y = 2(1) + 1 and y = 3 y = 2(0) + 1 and y = 1 y = 2(3) + 1 and y = 7 for x = -2, 1, 0, 3 x y -2 -3 1 3 0 1 3 7 Section 4-‐5 – Graphing Linear Equations Linear vs. Non-linear • • • • Linear equations can be written as Ax + By = C or y = mx + b The power of x is 1 and the power of y is 1. X and y cannot be multiplied together. X and y not in the bottom of a fraction. There is a consistent change for y as x changes. This is called the slope. The graph of a linear equation is a linear equation. Standard Form: Linear equations can be written as Ax + By = C Y = mx + b form: Linear equations can be written as y = mx + b Examples: Are these linear? yx = 8 y = 4x + 7 y = x2 + 1 3x + 4y = 12 y=4 if yes, write in Standard form: Graphing Constant Functions ( x = • • • • or y = ) When there is only 1 variable, the graph will be vertical or horizontal. X = creates a vertical line (not a function) and y = creates a horizontal line (function). Solve for the variable. The variable is the axis that is crossed and the number is where it crosses. Examples: graph y=4 x = -1 3x + 1 = 7 x=2 If you can make a t-table, you can graph anything! • • • • • Solve for y. Create a table using -2, -1, 0, 1, 2 as the x values (unless x is multiplied by a fraction). If x is multiplied by a fraction, then use the denominator to make the table from the multiples of the denominator. o If y = ½ x table is -4, -2, 0, 2, 4 o If y = 1/3 x table is -6, -3, 0, 3, 6 Find the y value for each x value. Plot the order pairs and connect to make a line. Examples: graph y = 4x + 1 x y -2 -7 -1 -3 0 1 1 5 2 9