Math 083 – Intermediate Algebra Mr. Bianco Please take out a pencil and paper to take notes. All Cell Phones OFF and AWAY for the entire class. Thank You!

Chapter 3 Graphs and Functions

§ 3.1 Graphing Equations

Vocabulary Ordered pair – a sequence of 2 numbers where the order of the numbers is important Axis – horizontal or vertical number line Origin – point of intersection of two axes Quadrants – regions created by intersection of 2 axes Location of a point residing in the rectangular coordinate system created by a horizontal (x-) axis and vertical (y-) axis can be described by an ordered pair. Each number in the ordered pair is referred to as a coordinate

Graphing and Ordered Pairs y-axis Quadrant II Quadrant I (0, 0) x-axis origin Quadrant III Quadrant IV

Graphing an Ordered Pair y-axis Quadrant II Quadrant I B A C (0, 0) x-axis D origin E Quadrant III Quadrant IV

Graphing an Ordered Pair y-axis Quadrant II Quadrant I B A C 3 units up (0, 0) x-axis D 5 units right origin E Quadrant III Quadrant IV

Graphing an Ordered Pair y-axis (x, y) Quadrant II Quadrant I B A (5, 3) C 3 units up (0, 0) x-axis D 5 units right origin E Quadrant III Quadrant IV

Graphing an Ordered Pair y-axis Quadrant II Quadrant I (0, 5) Note that the order of the coordinates is very important, since (-4, 2) and (2, -4) are located in different positions. (5, 3) (-4, 2) 3 units up (0, 0) x-axis (-6, 0) 5 units right origin (2, -4) Quadrant III Quadrant IV

Graphing an Ordered Pair y-axis Quadrant II Quadrant I (0, 5) Note that the order of the coordinates is very important, since (-4, 2) and (2, -4) are located in different positions. (5, 3) (-4, 2) 3 units up (0, 0) x-axis (-6, 0) 5 units right origin (2, -4) Quadrant III Quadrant IV

Graphing an Ordered Pair y-axis Quadrant II Quadrant I (0, 5) Note that the order of the coordinates is very important, since (-4, 2) and (2, -4) are located in different positions. (5, 3) (-4, 2) 3 units up (0, 0) x-axis (-6, 0) 5 units right origin (2, -4) Quadrant III Quadrant IV

Graphing and Ordered Pairs y-axis Quadrant II Quadrant I (-x,y) (x,y) (0, 0) x-axis origin (-x, -y) (x, -y) Quadrant III Quadrant IV 12

Solutions of an Equation Example: Determine whether (3, – 2) is a solution of 2x + 5y = – 4. Let x = 3 and y = – 2 in the equation.

Solutions of an Equation Example: Determine whether (– 1, 6) is a solution of 3x – y = 5. Let x = – 1 and y = 6 in the equation.

Linear Equations Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form Ax + By = c where A and B are not both 0. This is called standard form.

Solutions of an Equation Example: Determine whether (3, – 2) is a solution of 2x + 5y = – 4. Let x = 3 and y = – 2 in the equation. 2x + 5y = – 4 2(3) + 5(– 2) = – 4 Replace x with 3 and y with –2. 6 + (– 10) = – 4 Compute the products. – 4 = – 4 True So (3, -2) is a solution of 2x + 5y = – 4

Solutions of an Equation Example: Determine whether (– 1, 6) is a solution of 3x – y = 5. Let x = – 1 and y = 6 in the equation. 3x – y = 5 3(– 1) – 6 = 5 Replace x with – 1 and y with 6. – 3 – 6 = 5 Compute the product. – 9 = 5 False So (– 1, 6) is not a solution of 3x – y = 5

Linear Equations Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form Ax + By = C where A and B are not both 0. This is called standard form.

Graphing Linear Equations Example: Graph the linear equation 2x – y = -4. We find two ordered pair solutions (and a third solution as a check on our computations) by choosing a value for one of the variables, x or y, then solving for the other variable. We plot the solution points, then draw the line containing the 3 points. Continued.

Graphing Linear Equations Example continued: Graph the linear equation 2x – y = – 4. Let x = 1. Then 2x – y = – 4 becomes 2(1) – y = – 4 Replace x with 1. 2 – y = – 4 Simplify the left side. – y = – 4 – 2 = – 6 Subtract 2 from both sides. y = 6 Multiply both sides by – 1. So one solution is (1, 6) Continued.

Graphing Linear Equations Example continued: Graph the linear equation 2x – y = – 4. For the second solution, let y = 4. Then 2x – y = – 4 becomes 2x – 4 = – 4 Replace y with 4. 2x = – 4 + 4 Add 4 to both sides. 2x = 0 Simplify the right side. x = 0 Divide both sides by 2. So the second solution is (0, 4) Continued.

Graphing Linear Equations Example continued: Graph the linear equation 2x – y = – 4. For the third solution, let x = – 3. Then 2x – y = – 4 becomes 2(– 3) – y = – 4 Replace x with – 3. – 6 – y = – 4 Simplify the left side. – y = – 4 + 6 = 2 Add 6 to both sides. y = – 2 Multiply both sides by – 1. So the third solution is (– 3, – 2) Continued.

Graphing Linear Equations Example continued: x y (1, 6) (0, 4) (– 3, – 2) Now we plot all three of the solutions (1, 6), (0, 4) and (– 3, – 2). And then we draw the line that contains the three points.

Graphing Linear Equations Example: Graph the linear equation y = x + 3. Since the equation is solved for y, we should choose values for x. To avoid fractions, we should select values of x that are multiples of 4 (the denominator of the fraction). Continued.

Graphing Linear Equations Example continued: Graph the linear equation y = x + 3. Let x = 4. Then y = x + 3 becomes y = (4) + 3 Replace x with 4. y = 3 + 3 = 6 Simplify the right side. So one solution is (4, 6) Continued.

Graphing Linear Equations Example continued: Graph the linear equation y = x + 3. For the second solution, let x = 0. Then y = x + 3 becomes y = (0) + 3 Replace x with 0. y = 0 + 3 = 3 Simplify the right side. So a second solution is (0, 3) Continued.

Graphing Linear Equations Example continued: Graph the linear equation y = x + 3. For the third solution, let x = – 4. Then y = x + 3 becomes y = (– 4) + 3 Replace x with – 4. y = – 3 + 3 = 0 Simplify the right side. So the third solution is (– 4, 0) Continued.

Graphing Linear Equations x y Example continued: (4, 6) (0, 3) (– 4, 0) Now we plot all three of the solutions (4, 6), (0, 3) and (– 4, 0). And then we draw the line that contains the three points.

Intercepts Intercepts of axes (where graph crosses the axes) Since all points on the x-axis have a y-coordinate of 0, to find x-intercept, let y = 0 and solve for x Since all points on the y-axis have an x-coordinate of 0, to find y-intercept, let x = 0 and solve for y

Intercepts Example: Find the y-intercept of 4 = x – 3y Let x = 0. Then 4 = x – 3y becomes 4 = 0 – 3y Replace x with 0. 4 = – 3y Simplify the right side. = y Divide both sides by – 3. So the y-intercept is (0, )

Intercepts Example: Find the x-intercept of 4 = x – 3y Let y = 0. Then 4 = x – 3y becomes 4 = x – 3(0) Replace y with 0. 4 = x Simplify the right side. So the x-intercept is (4,0)

Graph by Plotting Intercepts Example: Graph the linear equation 4 = x – 3y by plotting intercepts. We previously found that the y-intercept is (0, ) and the x-intercept is (4, 0). Plot both of these points and then draw the line through the 2 points. Note: You should still find a 3rd solution to check your computations. Continued.

Graph by Plotting Intercepts Example continued: Graph the linear equation 4 = x – 3y. Along with the intercepts, for the third solution, let y = 1. Then 4 = x – 3y becomes 4 = x – 3(1) Replace y with 1. 4 = x – 3 Simplify the right side. 4 + 3 = x Add 3 to both sides. 7 = x Simplify the left side. So the third solution is (7, 1) Continued.

Graph by Plotting Intercepts x y Example continued: Now we plot the two intercepts (0, ) and (4, 0) along with the third solution (7, 1). (4, 0) (7, 1) (0, ) And then we draw the line that contains the three points.

Introduction to Functions § 3.2 Introduction to Functions

Relations Equations in two variables define relations between the two variables. There are other ways to describe relations between variables. Set to set Ordered pairs A set of ordered pairs is also called a relation.

Domain and Range Recall that a set of ordered pairs is also called a relation. The domain is the set of x-coordinates of the ordered pairs. The range is the set of y-coordinates of the ordered pairs.

Domain and Range Example: Find the domain and range of the relation {(4,9), (–4,9), (2,3), (10, –5)} Domain is the set of all x-values, {4, –4, 2, 10} Range is the set of all y-values, {9, 3, –5}

Some relations are also functions. A function is a set of order pairs that assigns to each x-value exactly one y-value.

Given the relation Functions Example: Given the relation {(4,9), (– 4,9), (2,3), (10, –5)}, is it a function?

Functions DOMAIN IS IN PURPLE IT’S THE X-VALUES Example: Given the relation {(4,9), (–4,9), (2,3), (10, –5)}, is it a function? Since each element of the domain is paired with only one element of the range, it is a function. Note: It’s okay for a y-value to be assigned to more than one x-value, but an x-value cannot be assigned to more than one y-value (has to be assigned to ONLY one y-value).

Vertical Line Test Relations and functions can also be described by graphing their ordered pairs. Graphs can be used to determine if a relation is a function. If an x-coordinate is paired with more than one y-coordinate, a vertical line can be drawn that will intersect the graph at more than one point. If no vertical line can be drawn so that it intersects a graph more than once, the graph is the graph of a function. This is the vertical line test.

Vertical Line Test Example: y Example: Use the vertical line test to determine whether the graph to the right is the graph of a function. Since no vertical line will intersect this graph more than once, it is the graph of a function.

Vertical Line Test Example: y Example: Use the vertical line test to determine whether the graph to the right is the graph of a function. Since no vertical line will intersect this graph more than once, it is the graph of a function.

Vertical Line Test Example: y Example: Use the vertical line test to determine whether the graph to the right is the graph of a function. Since vertical lines can be drawn that intersect the graph in two points, it is NOT the graph of a function.

Vertical Line Test Since the graph of a linear equation is a line, all linear equations are functions, except those whose graph is a vertical line Note: An equation of the form y = c is a horizontal line and IS a function. An equation of the form x = c is a vertical line and IS NOT a function.

Functions Example: Is the relation y = x2 – 2x a function? Since each element of the domain (the x-values) would produce only one element of the range (the y-values), it is a function.

Functions Example: Is the relation x2 – y2 = 9 a function? Since each element of the domain (the x-values) would correspond with 2 different values of the range (both a positive and negative y-value), the relation is NOT a function.

Function Notation Specialized notation is often used when we know a relation is a function and it has been solved for y. For example, the graph of the linear equation y = –3x + 2 passes the vertical line test, so it represents a function. We often use letters such as f, g, and h to name functions. We can use the function notation f(x) (read “f of x”) and write the equation as f(x) = –3x + 2. Note: The symbol f(x) is a specialized notation that does NOT mean f • x (f times x).

Function Notation When we want to evaluate a function at a particular value of x, we substitute the x-value into the notation. For example, f(2) means to evaluate the function f when x = 2. So we replace x with 2 in the equation. For our previous example when f(x) = – 3x + 2, f(2) = – 3(2) + 2 = – 6 + 2 = – 4. When x = 2, then f(x) = – 4, giving us the order pair (2, – 4).

Function Notation Example: Given that g(x) = x2 – 2x, find g(– 3). Then write down the corresponding ordered pair. g(– 3) = (– 3)2 – 2(– 3) = 9 – (– 6) = 15. The ordered pair is (– 3, 15).

3.1 # 1- 25 Odd 3.2 # 1-9 Odd, 13- 17 Odd, 23-39 Odd, 55-79 Odd Assignment #1 3.1 # 1- 25 Odd 3.2 # 1-9 Odd, 13- 17 Odd, 23-39 Odd, 55-79 Odd Quiz on Monday 3.1 & 3.2