Graphs, Linear Equations, and Functions

1. 3-1 The Rectangular Coordinate System 3-2 Graphs of Equations 3-3 Lines 3-5 Introduction to Functions 3.6 Quadratic Functions 3.7 Operations on Functions 3.8 Inverse Functions 3.9 Variation Graphs, Linear Equations, and Functions

2. The Rectangular Coordinate SystemSection 3.1

3. Identify parts of the rectangular coordinate system. Graph linear equations using an x-y chart and using x and y intercepts. Graph horizontal and vertical lines. Use a graphing calculator to analyze data involving linear equations. Apply the distance and midpoint formulas. Objectives:

4. Plotting ordered pairs. An “ordered pair” of numbers is a pair of numbers written within parenthesis in which the order of the numbers is important. Example 1: (3,1), (-5,6), (0,0) are ordered pairs. Note: The parenthesis used to represent an ordered pair are also used to represent an open interval. The context of the problem tells whether the symbols are ordered pairs or an open interval. Graphing an ordered pair requires the use of graph paper and the use of two perpendicular number lines that intersect at their 0 points. The common 0 point is called the “origin”. The horizontal number line is referred to as the “x-axis” or “abscissa” and the vertical line is referred to as the “y-axis” or “ordinate”. In an ordered pair, the first number refers to the position of the point on the x-axis, and the second number refers to the position of the point on the y-axis. 3-1 The Rectangular Coordinate System

5. Plotting ordered pairs. The x-axis and the y-axis make up a “rectangular or Cartesian” coordinate system. Points are graphed by moving the appropriate number of units in the x direction, than moving the appropriate number of units in the y direction. (point A has coordinates (3,1), the point was found by moving 3 units in the positive x direction, then 1 in the positive y direction) The four regions of the graph are called quadrants. A point on the x-axis or y-axis does not belong to any quadrant (point E). The quadrants are numbered. 3-1 The Rectangular Coordinate System

6. Finding ordered pairs that satisfy a given equation. To find ordered pairs that satisfy an equation, select any number for one of the variables, substitute into the equation that value, and solve for the other variable. Example 2: For 3x – 4y = 12, complete the table shown: Solution 3-1 The Rectangular Coordinate System

7. The point where the line intersects the x-axis is (x, 0) and is called the x-intercept. To find the value of x, substitute 0 in for y and solve for x. The point where the line intersects the y-axis is (0, y) and is called the y-intercept. To find the value of y, substitute 0 in for x and solve for y. Intercepts of a Line

8. Graph the linear equation using x and y intercepts • Graph -2x + 4y = 8

9. Graph using either method • 2(x -1) = 6 - 8y

10. If a and b are real numbers then: The graph of x = a is a vertical line with x-intercept (a, 0). x = 3, x = -2, x = 7/9, x = -7.5 The graph of y = b is a horizontal line with y-intercept (0, b). y = 5, y = -1, y = 2/3, y = -3.4 Equations of Horizontal and Vertical lines

11. Graph x = 2 Graph y = -3

12. Used to find the distance between any two points in a rectangular coordinate system. The distance formula can be derived by plotting two points (x1, y1) and (x2, y2), then form a right triangle, and apply the Pythagorean theorem. The Distance Formula

13. Find the distance between P(-2, -5) and Q(3, 7)

14. The Midpoint Formula • The point M that is half way or midway between points P(x1, x2) and Q(y1,, y2) is called the midpoint. • The midpoint is average of the x-coordinates and the average of the y-coordinates

15. Find the midpoint of the segment joining the points P(-7, -8) and Q(1, -4)

16. If the midpoint of the segment is M(2, -5) and one endpoint is P (6, 9), find the coordinates of the other endpoint Q.

17. The graph of a linear equation is a straight line. Know how to graph a linear equation using an x-y chart and using x and y intercepts. Know that x = constant is a vertical line. Know that y = constant is a horizontal line. Know the distance formula and the midpoint formula. Important Information

18. Graphing lines: Example 3: Draw the graph of 2x + 3y = 6 Step 1: Find a table of ordered pairs that satisfy the equation. Step 2: Plot the points on a rectangular coordinate system. Step 3: Draw the straight line that would pass through the points. Step 1 Step 2 Step 3 3-1 The Rectangular Coordinate System

19. Finding Intercepts: In the equation of a line, let y = 0 to find the “x-intercept” and let x = 0 to find the “y-intercept”. Note: A linear equation with both x and y variables will have both x- and y-intercepts. Example 4: Find the intercepts and draw the graph of 2x –y = 4 x-intercept: Let y = 0 : 2x –0 = 4 2x = 4 x = 2 y-intercept: Let x = 0 : 2(0) – y = 4 -y = 4 y = -4 x-intercept is (2,0) y-intercept is (0,-4) 3-1 The Rectangular Coordinate System

20. Recognizing equations of vertical and horizontal lines: An equation with only the variable x will always intersect the x-axis and thus will be vertical. An equation with only the variable y will always intersect the y-axis and thus will be horizontal. Example 6: A) Draw the graph of y = 3 B) Draw the graph of x + 2 = 0 x = -2 A) B) 3-1 The Rectangular Coordinate System

21. Graphing a line that passes through the origin: Some lines have both the x- and y-intercepts at the origin. Note: An equation of the form Ax + By = 0 will always pass through the origin. Find a multiple of the coefficients of x and y and use that value to find a second ordered pair that satisfies the equation. Example 7: A) Graph x + 2y = 0 3-1 The Rectangular Coordinate System

22. 3.1 Homework Answers 15. 17. 19. 21. 23. 25. 27. 29. 38. Radius = 5, Center = (0,0) Pick a pt. Q(0,y) on y-axis Since y-coord. is – in the third quad. a = -1. (-2,-1)

23. 3.2 Solving Polynomial Equations by Graphing

24. Finding the slope of a line given two points on the line: The slope of the line through two distinct points (x1, y1) and (x2, y2) is: Note: Be careful to subtract the y-values and the x-values in the same order. Correct Incorrect 3-2 The Slope of a Line

25. Quadratic - has the form ax2 + bx + c = 0 Highest exponent is two (this is the degree) The most real solutions it has is two. Types of Equations

26. Cubic - has the form ax3 + bx2 + cx + d = 0 Highest exponent is three (this is the degree) The most real solutions it has is three. Types of Equations

27. Quartic - has the form ax4 + bx3 + cx2 + dx + e = 0 Highest exponent is four (this is the degree) The most real solutions it has is four. Types of Equations

28. These keep on going up as the highest exponent increases. You don’t need to know the names above quartic, but you do need to be able to give the degree. Types of Equations

29. CIRCLES

30. The standard form of the equation of a circle with its center at the origin is r is the radius of the circle so if we take the square root of the right hand side, we'll know how big the radius is. Notice that both the x and y terms are squared. Linear equations don’t have either the x or y terms squared. Parabolas have only the x term was squared (or only the y term, but NOT both).

31. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Let's look at the equation This is r2 so r = 3 The center of the circle is at the origin and the radius is 3. Let's graph this circle. Count out 3 in all directions since that is the radius Center at (0, 0)

32. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 If the center of the circle is NOT at the origin then the equation for the standard form of a circle looks like this: The center of the circle is at (h, k). This is r2 so r = 4 Find the center and radius and graph this circle. The center of the circle is at (h, k) which is (3,1). The radius is 4

33. If you take the equation of a circle in standard form for example: This is r2 so r = 2 (x - (-2)) Remember center is at (h, k) with (x - h) and (y - k) since the x is plus something and not minus, (x + 2) can be written as (x - (-2)) You can find the center and radius easily. The center is at (-2, 4) and the radius is 2. But what if it was not in standard form but multiplied out (FOILED) Moving everything to one side in descending order and combining like terms we'd have:

34. If we'd have started with it like this, we'd have to complete the square on both the x's and y's to get in standard form. Move constant to the other side Group x terms and a place to complete the square Group y terms and a place to complete the square 4 16 4 16 Complete the square Write factored and wahlah! back in standard form.

35. Now let's work some examples: Find an equation of the circle with center at (0, 0) and radius 7. Let's sub in center and radius values in the standard form 0 0 7

36. Find an equation of the circle with center at (0, 0) that passes through the point (-1, -4). Since the center is at (0, 0) we'll have The point (-1, -4) is on the circle so should work when we plug it in the equation: Subbing this in for r2 we have:

37. Find an equation of the circle with center at (-2, 5) and radius 6 Subbing in the values in standard form we have: -2 5 6

38. Find an equation of the circle with center at (8, 2) and passes through the point (8, 0). Subbing in the center values in standard form we have: 8 2 Since it passes through the point (8, 0) we can plug this point in for x and y to find r2.

39. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Identify the center and radius and sketch the graph: 9 9 9 To get in standard form we don't want coefficients on the squared terms so let's divide everything by 9. Remember to square root this to get the radius. So the center is at (0, 0) and the radius is 8/3.

40. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Identify the center and radius and sketch the graph: Remember the center values end up being the opposite sign of what is with the x and y and the right hand side is the radius squared. So the center is at (-4, 3) and the radius is 5.

41. Find the center and radius of the circle: We have to complete the square on both the x's and y's to get in standard form. Move constant to the other side Group x terms and a place to complete the square Group y terms and a place to complete the square 9 4 9 4 Write factored for standard form. So the center is at (-3, 2) and the radius is 4.

42. 3.2 day 2 Homework Answers 33. 41. r = 3 43. (-4,4) 45. Midpt = (1,2) = center 61. If distance from P to C is less than r, greater than r, or = to r 63. x – int. (0 for y) y – int. (0 for x) 65.

43. Finding the slope of a line given two points on the line: Example 1) Find the slope of the line through the points (2,-1) and (-5,3) 3-3 LinesThe slope of a line

44. Finding the slope of a line given an equation of the line: The slope can be found by solving the equation such that y is solved for on the left side of the equal sign. This is called the slope-intercept form of a line. The slope is the coefficient of x and the other term is the y-intercept. The slope-intercept form is y = mx + b Example 2) Find the slope of the line given 3x – 4y = 12 3-3 The Slope of a Line

45. Finding the slope of a line given an equation of the line: Example 3) Find the slope of the line given y + 3 = 0 y = 0x - 3 The slope is 0 Example 4) Find the slope of the line given x + 6 = 0 Since it is not possible to solve for y, the slope is “Undefined” Note: Being undefined should not be described as “no slope” Example 5) Find the slope of the line given 3x + 4y = 9 3-3 The Slope of a Line

46. Graph a line given its slope and a point on the line: Locate the first point, then use the slope to find a second point. Note: Graphing a line requires a minimum of two points. From the first point, move a positive or negative change in y as indicated by the value of the slope, then move a positive value of x. Example 6) Graph the line given slope = passing through (-1,4) Note: change in y is +2 3-3 The Slope of a Line

47. Graph a line given its slope and a point on the line:Locate the first point, then use the slope to find a second point. Example 7) Graph the line given slope = -4 passing through (3,1) Note: A positive slope indicates the line moves up from L to R A negative slope indicates the line moves down from L to R 3-3 The Slope of a Line

48. Using slope to determine whether two lines are parallel, perpendicular, or neither: Two non-vertical lines having the same slope are parallel. Two non-vertical lines whose slopes are negative reciprocals are perpendicular. Example 8) Is the line through (-1,2) and (3,5) parallel to the line through (4,7) and (8,10)? 3-3 The Slope of a Line

49. Using slope to determine whether two lines are parallel, perpendicular, or neither: Two non-vertical lines having the same slope are parallel. Two non-vertical lines whose slopes are negative reciprocals are perpendicular. Example 9) Are the lines 3x + 5y = 6 and 5x - 3y = 2 parallel, perpendicular, or neither? 3-3 The Slope of a Line

50. Solving Problems involving average rate of change:The slope gives the average rate of change in y per unit change in x, where the value of y depends on x. Example 10) The graph shown approximates the percent of US households owing multiple pc’s in the years 1997-2001. Find the average rate of change between years 2000 and 1997. 3-3 The Slope of a Line