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DQ1 and DQ 2 for MAT/116
DQ1 and DQ 2 for MAT/116
Please be patient… this is a challenging question (maybe the most challenging of the term), there’s a lot of material and examples to follow, but you’ll get through it. Just take your time, and ask me any questions you have. Graphing is the most important tool you’ll learn all term, and you’ll continue with this in MAT/117. First, please read the Week 4 Preview and “Everything You Need to Know About Graphing.” Please read them slowly and don’t pass out, okay? Then read this discussion and study the example (and the graph attached that supports the example) before answering this question (please don’t hyperventilate; take your time).
There are six parts to this DQ, and you need to answer all parts to earn the points. Take it one step at a time; don’t rush it! There’s a lot of material to cover this week, and this is going to be a challenge, but I know all of you can do it; I’ve seen each of you do great work so far!
Think of this question as a worksheet that has fill-in-the-blank type of responses. Each step builds to the next one, so don’t skip one part to get to another; by the time you hit part 5, you’ll have the result and be ready to use your critical-thinking skills to respond to part 6. In other words, you cannot answer part 2 until you’ve answered part 1; you cannot answer part 3 until you’ve answered part 2, and so on.
This is the method to find the equation of any line just by looking at a graph; and, given an equation, you can graph any line. From either pages 199-200, problems 19-30, or on page 202, problems 87-88, choose one problem and construct the slope-intercept equation of the line following these steps, in order. Don’t follow the instructions given in the text; follow the instructions below. I attached the problem I would like answered, please see attachment.
a. Find the slope. Be sure to show your work as to how you found the slope of the equation of the line. Once you find the slope, answer the appropriate question, as follows: i. If the slope is positive, how is the line moving? ii. If the slope is negative, how is the line moving? iii. If the slope is 0, what does that tell you about the line? iv. If the slope is undefined, what does that tell you about the line?
b. Find the y-intercept (written as an ordered pair) by reading it from the graph. c. Write the equation in slope-intercept form. All you need is the slope and y-intercept to write the slope-intercept equation of the line. The format is y = mx + b; remember the sign goes with the number for both m and b. d. Find the x-intercept (written as an ordered pair). Remember that intercepts are always written as ordered pairs. e. Find another point on the line using the slope (“Linda’s Shortcut”); see p. 10 of the “Everything” handout for the description of “Linda’s Shortcut.” f. What is the slope-intercept equation of any vertical line? What is the slope-intercept equation of any horizontal line? (See pp. 14-15 of the “Everything” handout to help you answer this question.)
Attached is an example (“Graph for Week 4 DQ1 example”) so you see what is expected in parts 1-5. Take it slowly, one step at a time, and work it out on paper first. When you type your response, be sure to proofread it and ensure that it’s in y = mx + b form (slope-intercept). For part 6, refer to the handouts provided in the recommended tasks area. _________________________________________
EXAMPLE: a. Find the slope: Slope = m = (y2 – y1)/(x2 – x1), using points (0, -2) and (2, -1). The subscripts for x and y refer to the value from the ordered pairs. In other words, (x1, y1) is the first ordered pair and (x2, y2) is the second ordered pair. m = [-1 – (-2)]/(2 – 0) Notice my use of grouping symbols to make it clear what the numerator and denominator are. m = (-1 + 2)/2 m = 1/2 The slope is positive, so the line rises from left to right. b. Find the y-intercept: The y-intercept is (0, -2), which was given as one of the points. c. Write the equation in slope-intercept form: y = mx + b I calculated the slope as 1/2, and the y-intercept is (0, -2), so the slope-intercept equation is: y = (1/2)x – 2 Notice the sign goes with the number in the y-intercept and slope. d. Calculate the x intercept: To find the x-intercept, let y = 0 in the slope-intercept equation you wrote, as y is always 0 when the line crosses the x-axis. y = (1/2)x – 2 0 = (1/2)x – 2 2 = (1/2)x – 2 + 2 2 = (1/2)x 2(2/1) = (2/1)(1/2)x 4 = x The x-intercept is (4, 0) e. Find another point on the line using the slope (“Linda’s shortcut”): Slope = 1/2; y-intercept is (0, -2). Then, using “Linda’s shortcut”): (0 + 2, -2 + 1) = (2, -1) is another point on the line.
Note: I used the fraction of the slope, 1/2, because the numerator is the rise (add to the y-value) and the denominator is the run (add to the x-value); I find that to be a lot easier than having to do more calculations to find another point. In other words, I avoid the Table of Values at all costs; it was my biggest struggle when learning how to graph way back in my day.
PART TWO Discussion Question 2
This question is best answered after responding to Question 1 and reading the materials provided. If a line has no y-intercept, what can you say about the line? What if a line has no x-intercept? Think of a real-life situation where a graph would have no x- or y-intercept. Will what you say about the line always be true in that situation?
1 Everything You Need to Know About Graphing Linda J. Ferlaak Certified Advanced Faculty CALCULATING ORDERED PAIRS We’ve calculated solutions when solving a one-variable equation. Now, we’re going to be working in two variables. We had practice in solving for variables last week. This week, we’ll be using what we learned in weeks 2 and 3 to set up equations to graph straight lines (linear equations). In the table below, we are given the x- or y-value of an ordered pair (first column) and need to calculate the missing value (second column) to complete the ordered pair (third column). By substituting the value given in the equation, we can calculate the missing value. The result (solution) is the ordered pair of one point on the line. An ordered pair is always written as (x, y), where x is the x-value (horizontal axis) of that point and y is the y-value (vertical axis) of that point. NOTE: The variables in an ordered pair are always alphabetical: (a, b), (c, g), (t, v), (x, y), and so on. The first value (a, c, t, or x) is the value on the horizontal axis; the second value (b, g, v, or y) is the value on the vertical axis. Example 1: Fill in the missing values to complete the ordered pair. Incomplete ordered pair Our work to find the missing value Completed ordered pair (0, ) y 3 4 12 3(0) 4 12 4 12 4 12 4 4 3 x y y y y y (0, 3) 3 , 4 x 3 4 12 3 3 4 12 4 3 3 12 3 3 3 12 3 3 9 3 9 3 3 3 x y x x x x x x 3 3, 4 2 Incomplete ordered pair Our work to find the missing value Completed ordered pair ( ,0) x 3 4 12 3 4(0) 12 3 12 3 12 3 3 4 x y x x x x (4, 0) 8 , 3 y 3 4 12 8 3 4 12 3 8 4 12 8 8 4 12 8 4 4 4 4 4 4 1 x y y y y y y y 8 ,1 3 Here are the calculated points on a graph (x is the horizontal axis and y is the vertical axis; always read the x-values first when plotting ordered pairs): 3 We also use the same method to calculate conversions and other data of comparison. Example 2 (from week 3): Convert the Celsius temperature to Fahrenheit using the equation 9 32 5 F C . Celsius Our work to convert the values Fahrenheit -10o 9 ( 10) 32 5 9 5 F F 1 ( 10 2 ) 32 18 32 14 F F 14o 0 0 9 (0) 32 5 0 32 32 F F F 32o 100o 9 (100) 32 5 9 5 F F 1 (10020) 32 180 32 212 F F 212o On a conversion graph, the ordered pairs would be (-10, 14), (0, 32), and (100, 212): 4 Example 3: Other common applications of graphing are for analyzing, forecasting, budgeting, and recognizing trends, such as in finance, marketing, crime data, and research data in health care and management. Let’s say we are looking at the break-even price to manufacture a certain number of thingamajigs (where costs = revenues). A simplified equation is: 1 75 2 p n where p = price, n = number of thingamajigs sold, and the base cost to manufacture one thingamajig is $75. Thingamajigs Sold Our work to find the price Price 2 1 (2) 75 2 1 75 74 p p p $74.00 7 1 (7) 75 2 7 75 2 3.50 75 71.50 p p p p $71.50 9 1 (9) 75 2 9 75 2 4.50 75 70.50 p p p p $70.50 11 1 (11) 75 2 11 75 2 5.50 75 69.50 p p p p $69.50 In this example, the ordered pairs are then (2, $74), (7, $71.50), (9, $70.50), and (11, $69.50). This tells us that the thingamajigs sold are on the x-axis and the price for that quantity is on the y-axis. Notice the importance of using the proper units. 5 PLOTTING POINTS We need only two points to draw a straight line. You’ll be given the points or you will need to work with an equation to find the points to plot so you can draw a line. The x-value always comes first; the y-value comes second (always). The x-axis is the horizontal line; the y-axis is the vertical line. The point where the x- and y-axes intersect is called the “origin,” and the ordered pair at that point is (0, 0). Example 4: Plot the point (2, 3). Count two points to the right (the x-value) and then count three points up (the y-value). 6 Example 5: Plot the point (-5, -2). The negative x-numbers are to the left of the origin; negative y-numbers are below the origin. DRAWING A GRAPH Now that we know how to calculate an ordered pair given one of the values and how to plot points, let’s learn how to draw a graph (such as examples 1-3 at the beginning of this handout). As you know, a straight line contains an infinite number of points. But, to draw that straight line, we need only two points. Example 6: Given 5 2 10 x y , solve for y to determine solutions. We may as well draw the graph, too. This is the first part of your introduction to the most powerful graphing tools you’ll learn in this course. We start with the slope-intercept form of the equation of a line, y mx b , where m is the slope and b is the y-value of the y-intercept. This simple equation will tell you everything you need to know about a graph and how to analyze it. You’ll memorize this simply because of its simplicity and power (and practice). Yep, I’m excited about this; it’s the only thing I understood before I took algebra the second time! By the third time, I had Algebrish! But I’m getting ahead of myself. Let’s start with the equation above (5x + 2y = 10) and solve for y to determine ordered pairs to plot and then draw the line. (At this point, you may want to review the Week 4 Preview handout.) 7 We solve for y to put any given equation into the proper format. Remember, we learned how to solve for variables in week 3. To put an equation into slope-intercept form, always solve for y: Subtract 5 from both sides to isolate . Divide both sides, all terms, by 2 to get by itself. We’ve solved for y, so we st 5 2 10 5 5 2 5 op her 10 2 5 10 2 5 10 2 2 2 5 5 2 e for now. x y x x y x y x y x y x y y x We’ve now solved for y, but we still need to determine solutions. This means that we need to find values for y based on different values of x. There are two ways to do this. We can construct a “table of values” (lots of calculations!) or use the slope-intercept equation we already have. Let’s start with the table of values so you can see how much easier the slope-intercept equation is. The table of values is actually a table of ordered pairs, or coordinates, that determine the line of a graph. We need only two points to draw a straight line, but I’ll do 3 just for fun. I always choose x = 0 for one of my values; x = -1 for another; and then an easy number for the third, such as 2 for x (it gets rid of the fraction in this example). My table then looks like this: x My work, using 5 5 2 y x Ordered pairs 0 5 (0) 5 2 0 5 5 y y y (0, 5) -1 5 ( 1) 5 2 5 10 2 2 15 2 y y y 15 1, 2 2 5 (2) 5 2 5 5 0 y y y (2, 0) 8 Remember: The x-coordinate always comes first in an ordered pair, and ordered pairs are always written in parentheses: (x, y). Now we just need to plot our points and draw our line. I used Graph here, but you can also use Excel. See the GRAPH thread in the CMF for information
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