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Chapter 1

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  1. Chapter 1 • Functions & Graphs

  2. 1.1 • Graphs & Graphing Utilities

  3. Objectives • Plot points in the rectangular coordinate system • Graph equations in the rectangular coordinate system • Interpret information about a graphing utility’s viewing rectangle or table • Use a graph to determine intercepts • Interpret information from graphs

  4. Rectangular coordinate system • X-axis: horizontal (right pos., left neg.) • Y-axis: vertical (up pos., down neg.) • Ordered pairs: (x,y) • Graph of an equation: infinitely many ordered pairs that make a true statement • X-intercept: point where y=0, (x,0) • Y-intercept: point where x=0, (0,y)

  5. Graphing an Equation • List ordered pairs that make your equation true (plug values in for x and find the resulting y’s). Include the x-intercept & y-intercept among your points. • Plot several points and look for a trend. • Use the graphing utility on your calculator and compare graphs. Do they always match? Why or why not?

  6. Intercepts • Intercepts are key points to plot when graphing an equation. Remember, an intercepts is a POINT (x,y), and not just a number! • Look at the x-axis: What is true of EVERY point on the axis? (y-value is always 0) • Look at the y-axis: What is true of EVERY point on the axis? (x-value is always 0)

  7. Horizontal and Vertical Lines • What if the y-value is ALWAYS the same, regardless of the x-value (it could be anything!). (i.e. (4,2), (5,2), (-14,2), (17,2)) It’s a horizontal line! The x isn’t part of the equation, because its value is irrelevant: y = k (k is a constant) • What if the x-value is ALWAYS the same, regardless of the y-value (it could be anything!). (i.e. (2,4), (2,5), (2,-14), (2,17)) It’s a vertical line! The y isn’t part of the equation, because its value is irrelevant: x = c (c is a constant)

  8. 1.2 • Basics of Functions & Their Graphs

  9. Objectives • Find the domain & range of a relation. • Determine whether a relations is a function. • Determine is an equation represents a function. • Evaluate a function. • Graph functions by plotting points. • Use the vertical line test to identify functions. • Obtain information from a graph. • Identify the domain & range from a graph. • Identify intercepts from a graph.

  10. Domain & Range • A relation is a set of ordered pairs. • Domain: first components in the relation (independent) • Range: second components in the relation (dependent, the value depends on what the domain value is) • Functions are SPECIAL relations: A domain element corresponds to exactly ONE range element.

  11. EXAMPLE • Consider the function: eye color • (assume all people have only one color, and it is not changeable) • It IS a function because when asked the eye color of each person, there is only one answer. • i.e. {(Joe, brown), (Mo, blue), (Mary, green), (Ava, brown), (Natalie, blue)} • NOTE: the range values are not necessarily unique.

  12. Evaluating a function • Common notation: f(x) = function • Evaluate the function at various values of x, represented as: f(a), f(b), etc. • Example: f(x) = 3x – 7 f(2) = 3(2) – 7 = 6 – 7 = -1 f(3 – x) = 3(3 – x) – 7 = 9 – 3x – 7 = 2 – 3x

  13. Graphing a functions • Horizontal axis: x values • Vertical axis: y values • Plot points individually or use a graphing utility (calculator or computer algebra system) • Example:

  14. Table of function values

  15. Graphs of functions

  16. Can you identify domain & range from the graph? • Look horizontally. What all x-values are contained in the graph? That’s your domain! • Look vertically. What all y-values are contained in the graph? That’s your range!

  17. What is the domain & range of the function with this graph?

  18. Can you determine if a relation is a function or NOT from the graph? • Recall what it means to be a function: an x-value is assigned ONLY one y-value (it need not be unique). • So, on the graph, if the x value is paired with MORE than one y value there would be two points directly on a vertical line. • THUS, the vertical line test! If 2 or more points fall on a vertical line that would cross any portion of your graph, it is NOT the graph of a function.

  19. Finding intercepts: • X-intercept: where the function crosses the x-axis. What is true of every point on the x-axis? The y-value is ALWAYS zero. • Y-intercept: where the function crosses the y-axis. What is true of every point on the y-axis? The x-value is ALWAYS zero. • Can the x-intercept and the y-intercept ever be the same point? YES, if the function crosses through the origin!

  20. 1.3 • More of Functions and Their Graphs

  21. Objectives • Find & simplify a function’s difference quotient. • Understand & use piecewise functions. • Identify intervals on which a function increases, decreases, or is constant. • Use graphs to locate relative maxima or minima. • Identify even or odd functions & recognize the symmetries. • Graph step functions.

  22. Difference Quotient • Useful in discussing the rate of change of function over a period of time • EXTREMELY important in calculus, (h represents the difference in two x values)

  23. Find the difference quotient

  24. What is a piecewise function? • A function that is defined differently for different parts of the domain. • Examples: You are paid $10/hr for work up to 40 hrs/wk and then time and a half for overtime.

  25. Increasing and Decreasing Functions • Increasing: Graph goes “up” as you move from left to right. • Decreasing: Graph goes “down” as you move from left to right. • Constant: Graph remains horizontal as you move from left to right.

  26. Even & Odd Functions • Even functions are those that are mirrored through the y-axis. (if –x replaces x, the y value remains the same) (i.e. 1st quadrant reflects into the 2nd quadrant) • Odd functions are those that are mirrored through the origin. (if –x replaces x, the y value becomes –y) (i.e. 1st quadrant reflects into the 3rd quadrant)

  27. Determine if the function is even, odd, or neither. • Even • Odd • Neither

  28. 1.4 • Linear Functions & Slope

  29. Objectives • Calculate a line’s slope. • Write point-slope form of a line’s equation. • Write & graph slope-intercept from of a line’s equation. • Graph horizontal or vertical lines. • Recognize & use the general form of a line’s equation. • Use intercepts to graph. • Model data with linear functions and predict.

  30. What is slope? The steepness of the graph, the rate at which the y values are changing in relation to the changes in x.How do we calculate it?

  31. A line has one slope • Between any 2 pts. on the line, the slope MUST be the same. • Use this to develop the point-slope form of the equation of the line. • Now, you can develop the equation of any line if you know either a) 2 points on the line or b) one point and the slope.

  32. Find the equation of the line that goes through (2,5) and (-3,4) • 1st: Find slope of the line m= 2nd: Use either point to find the equation of the line & solve for y.

  33. Slope-Intercept Form of the Equation of the Line • It is often useful to express the line in slope-intercept form, meaning that the equation quickly reveals the slope of the line and where it intercepts the y-axis. • It is REALLY use of the point-slope form, except that the point is the intercept, (0,b). • y - b=m(x - 0) becomes y = mx + b • This creates a quick equation to graph.

  34. Horizontal & Vertical lines • What is the slope of a vertical line? • It is INFINITELY steep, it only rises. It is SO steep, we can’t define it, therefore undefined slope. • Look at points on a line, i.e. (-4,8),(-4,9),(-4,13),(-4,0). We don’t care what value y has, all that matters is that x = - 4. Therefore, that is the equation of the line! • What is the slope of a horizontal line? • There is no rise, it only runs, the change in y is zero, so the slope = 0. • Look at points on the line, i.e., (2,5), (-2,5), (17,5), etc. We don’t care what value x has, all that matters is that y = 5. Therefore, that is the equation of the line!

  35. 1.5 More on SLOPE • Objectives • Find slopes & equations of parallel & perpendicular lines • Interpret slope as a rate of change • Find a function’s average rate of change

  36. Parallel lines • Slopes are equal. • If you are told a line is parallel to a given line, you automatically know the slope of your new line (same as the given!). • Find the equation of the line parallel to y=2x-7 passing through the point (3,-5). • slope = 2, passes through (3,-5) y - (-5) = 2(x – 3) y + 5 = 2x – 6 y = 2x - 11

  37. Perpendicular lines • Lines meet to form a right angle. • If one line has a very steep negative slope, in order to form a right angle, it must intersect another line with a gradual positive slope. • The 2 lines graphed here illustrate that relationship.

  38. What about the intersection of a horizontal line and a vertical line? They ALWAYS intersect at a right angle. Since horizontal & vertical lines are neither positive or negative, we simply state that they are, indeed, ALWAYS perpendicular. • What about all other lines? In order to be perpendicular, their slopes must be the negative reciprocal of each other. (HINT: think about the very steep negative-sloped line perpendicular to the gradual positive-sloped line)

  39. Find the equation of the line perpendicular to y = 2x – 7 through the point (2,7) . • y = ½ x + 7 • y = - 2 x + 1/7 • y = - ½ x + 4/3 • y = - ½ x + 8

  40. Average Rate of Change • Slope thus far has referred to the change of y as related to the change in x for a LINE. • Can we have slope of a nonlinear function? • We CAN talk about the slope between any 2 points on the curve – this is the average rate of change between those 2 points!

  41. 1.6 Transformation of Functions • Recognize graphs of common functions • Use vertical shifts to graph functions • Use horizontal shifts to graph functions • Use reflections to graph functions • Use vertical stretching & shrinking to graph functions • Use horizontal stretching & shrinking to graph functions • Graph functions w/ sequence of transformations

  42. Vertical shifts • Moves the graph up or down • Impacts only the “y” values of the function • No changes are made to the “x” values • Horizontal shifts • Moves the graph left or right • Impacts only the “x” values of the function • No changes are made to the “y” values

  43. Recognizing the shift from the equation, examples of shifting the function f(x) = • Vertical shift of 3 units up • Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

  44. Combining a vertical & horizontal shift • Example of function that is shifted down 4 units and right 6 units from the original function.

  45. Reflecting • Across x-axis (y becomes negative, -f(x)) • Across y-axis (x becomes negative, f(-x))

  46. Horizontal stretch & shrink • We’re MULTIPLYING by an integer (not 1 or 0). • x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x values by 3, thus it’s compressed horizontally.

  47. VERTICAL STRETCH (SHRINK) • y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

  48. Sequence of transformations • Follow order of operations. • Select two points (or more) from the original function and move that point one step at a time. f(x) contains (-1,-1), (0,0), (1,1) 1st transformation would be (x+2), which moves the function left 2 units (subtract 2 from each x), pts. are now (-3,-1), (-2,0), (-1,1) 2nd transformation would be 3 times all the y’s, pts. are now (-3,-1), (-2,0), (-1,3) 3rd transformation would be subtract 1 from all y’s, pts. are now (-3,-2), (-2,-1), (-1,2)

  49. Graph of Example

  50. 1.7 Combinations of Functions;Composite Functions • Objectives • Find the domain of a function • Combine functions using algebra. • Form composite functions. • Determine domains for composite functions. • Write functions as compositions.