STEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is

Size: px

Start display at page:

Download "STEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is"

Alexander Baker
5 years ago
Views:

1 EQUATIONS OF LINES 1. Writing Equations of Lines There are many ways to define a line, but for today, let s think of a LINE as a collection of points such that the slope between any two of those points is the same. You can write the equation of a line using either of the following two formulas: SLOPE/INTERCEPT y = mx + b POINT/SLOPE y y 1 = m(x x 1 ) Note: It doesn t matter which formula you use, but it s generally easier to use the formula named after the information you know about your line: If you know the slope and a point, then use POINT/SLOPE If you know the slope and the y-intercept, then use SLOPE/INTERCEPT When the question says, Write the equation of the line... STEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is points on the line. parallel/perpendicular to another given line. Good! Write Not bad! Calculate That s OK! Find the down m = the slope using: slope of THAT line m = y 2 y 1 by rearranging into x 2 x 1 y = mx + b so you can then write down m = see its slope STEP 2: Ask Do I know the Y-INTERCEPT of the line? PERPENDICULAR PARALLEL Use the negative Use the exact reciprocal slope same slope Then write down m = YES! NO! But I know a different point instead. Good, write b = OK, write x 1 = and y 1 = USE y = mx + b y y 1 = m(x x 1 ) Fill in m, and b, and you re done! Fill in m, x 1 and y 1 then rearrange into y = mx+b

2 Let s try some examples... Write the equation of the line that (a) passes through the point P(2, 3) and has a slope of 4 (b) has a slope of -3 and a y-intercept of 12 (c) passes through (1, 4) and (2, 8) (d) passes through (-2, -5) and is parallel to 4x 3y = 8 (e) passes through (1, 7) and is perpendicular to x 2y 3 = 0

3 Let s try some examples... Write the equation of the line that (a) passes through the point P(2, 3) and has a slope of 4 (b) has a slope of -3 and a y-intercept of 12 (c) passes through (1, 4) and (2, 8) (d) passes through (-2, -5) and is parallel to 4x 3y = 8 (e) passes through (1, 7) and is perpendicular to x 2y 3 = 0

4 Note to teacher/parent: Students often have difficulty with the notion of lines written in standard form because, unlike the other formula they know (POINT/SLOPE and SLOPE/INTERCEPT) they cannot begin a question using this formula. Yet, many textbook questions are simply phrased, Write the equations of the following lines in standard form, allowing students to forget that they must use a different form to get the question started! It must be emphasized that one arrives at standard form through a process of steps and that standard form is really just a conventional notation for their final answer. What about STANDARD FORM? What s up with that formula? When do I use it? STANDARD FORM is simply a standard way of expressing your final answer. You do not begin a question by using standard form! You get there by first using one of the earlier two formulas and then rearranging! It looks like this: Ax + By + C = 0 or like this: 0 = Ax + By + C But, unlike the other two formulas we use, the A, B, and C themselves don t represent anything on the graph. (You ll notice, over time, that combinations of A, B, and C have meaning, but based on what we know now, A, B and C don t mean very much.) Remember, m represents the slope of the line, b represents the y-intercept of the line while x 1 and y 1 represent the x and y values of a specific point on your line. Some combination of these properties of the graph will be given to you in the question, making it very easy to plug into either SLOPE/INTERCEPT or POINT/SLOPE to get an equation. So, you should begin your question with one of these two formula in mind. You see, you don t get values for A, B and C from your question you simply rearrange one of the other two formulas until it looks like Ax + By + C = 0. This formula is really just meant to be a guide showing you what your final answer should look like. Specifically: You ll know your equation is in standard form when it satisfies the following criteria 1. All terms must be moved to one side so that one side is left empty. ( = 0 ) 2. The terms must be written in alphabetical order according to the variables. (the x term comes first, then the y term, then the constant) 3. There can be no fractions or decimals. (multiply the entire equation or use common denominators to clear fractions) 4. The x term must be positive. (If necessary, multiply/divide by -1 throughout OR move all terms to the other side of the equation so that x is positive) Let s convert our answers from the previous examples into standard form: y = 4x 5 Why is this not standard form? needs to = 0 0 = 4x 5 y Why is this not standard form? needs terms in proper order 0 = 4x y 5 y = -3x +12 Why is this not standard form? needs to = 0 0 = -3x +12 y Why is this not standard form? needs terms in proper order 0 = -3x y +12 Why is this not standard form? needs positive coefficient of x 0 = 3x + y 12

5 How about some more? y = ½x ¾ Why is this not standard form? needs to = 0 0 = ½x ¾ - y Why is this not standard form? needs terms in proper order 0 = ½x y ¾ Why is this not standard form? no fractions allowed 0 = ½x y ¾ *use your preferred method for clearing fractions usually one of: Multiply each term by the common Rewrite every term with a common denominator (on both sides, including OR denominator (on both sides, including the 0) the 0). Then, reduce/cancel all and then simply cancel out the denominators individual fractions separately 0 = ½x y ¾ 0 = ½x y ¾ 0 (4) = ½x(4) y(4) ¾(4) 0/4 = 2/4 x 4/4 y -3/4 0 = 2x 4y 3 0 = 2x 4y 3 y = -3/4x + 2/3 Why is this not standard form? needs to = 0 0 = -3/4x + 2/3 y Why is this not standard form? needs terms in proper order 0 = -3/4x y + 2/3 Why is this not standard form? no fractions allowed 0/12 = -9/12 x 12/12 y + 8/12 OR 0(12) = -3/4x(12) y(12) + 2/3(12) 0 = -9x 12y + 8 Why is this not standard form? needs positive coefficient of x Move terms to other side OR Multiply or Divide every term by -1 9x + 12y 8 = 0 OR 0(-1) = -9x(-1) 12y(-1) + 8(-1) 9x + 12y 8 = 0 OR 0 = 9x + 12y 8 y = 9 3x Why is this not standard form? needs to = 0 0 = 9 3x y Why is this not standard form? needs terms in proper order 0 = -3x y + 9 Why is this not standard form? needs positive coefficient of x 0 = 3x + y 9 2x 4y = 7 Why is this not standard form? needs to = 0 2x 4y 7 = 0 (Notice, it doesn t matter which side = 0) In Conclusion: Standard form is a way of expressing your final answer You must first have an equation of a line before you can put it in standard form You ll know your answer is in standard form when it satisfies the checklist: o One side equals zero o Terms are in proper order (x, then y, then constant) o No fractions or decimals o Starts with a positive coefficient of x Some equations will take more work than others to put in standard form When you begin to notice patterns, then you can take shortcuts and combine steps

6 SUMMARY Now you should be able to write the equation of a line when given o A point and the slope (You have x 1 and y 1 and m, so use y y 1 = m(x x 1 ) o The slope and the y-intercept (You have m and b so use y = mx + b o Two points (You have x 1, y 1, x 2 and y 2 so use m = y 2 y 1 / (x 2 x 1 ) and then y = mx + b o A point and told you are parallel to another given line (You have x 1 and y 1 and you will have m when you rearrange the given line into y = mx + b and use the same slope as that line so use y y 1 = m(x x 1 ) o A point and told you are perpendicular to another given line (You have x 1 and y 1 and you will have m when you rearrange the given line into y = mx + b and use the negative reciprocal slope so use y y 1 = m(x x 1 ) And, you can then rearrange any of these equations into standard form Ax + By + C = 0 which requires that... o One side equals zero o Terms are in proper order (x, then y, then constant) o No fractions or decimals o Starts with a positive coefficient of x

7 2. Finding x- and y-intercepts Remember: an intercept is the value where the line crosses through the axis. The x-intercept is the value of x where the line crosses the x-axis. The y-intercept is the value of y where the line crosses the y-axis. Note: The intercepts occur where the other variable equals zero So, to find intercepts, we take turns setting each variable equal to zero. 4x + 3y = 12 For the x-intercept, let y = 0 For the y-intercept, let x = 0 TRICK! Cover up the entire TRICK! Cover up the entire y term with your finger because x term with your finger, because it s zero it s gone! It s zero it s gone! 4x = 12 3y = 12 x = 3 y = 4 Therefore, the x-intercept is 3 and the y-intercept is 4. This means that the line crosses the x-axis at 3 and the y-axis at 4. 2x 5y = 20-6x + 5y = 90 For the x-intercept For the y-intercept For the x-intercept For the y-intercept let y = 0 let x = 0 let y = 0 let x = 0 (cover up y!) (cover up x!) (cover up y!) (cover up x!) 2x = 20-5y = 20-6x = 90 5y = 90 x = 10 y = -4 x = -15 y = 18 ½ x 3y = 12 4x 5y 20= 0 For the x-intercept For the y-intercept For the x-intercept For the y-intercept let y = 0 let x = 0 let y = 0 let x = 0 (cover up y!) (cover up x!) (cover up y!) (cover up x!) ½ x = 12-3y = 12 4x 20 = 0 5y 20= 0 x = 24 y = -4 4x = 20 5y = 20 x = 5 y = -4

8 3. Graphing Lines This brings us to graphing lines when given their equations. There are 3 common ways to graph a line (You can usually choose the method you prefer for each individual question) 1. Intercept Graphing a. Find both intercepts b. Plot both intercepts c. Connect the two points with a straight line Let s use intercept graphing for the equations on the previous page: 2x 5y = 20-6x + 5y = 90 For the x-intercept For the y-intercept For the x-intercept For the y-intercept let y = 0 let x = 0 let y = 0 let x = 0 (cover up y!) (cover up x!) (cover up y!) (cover up x!) 2x = 20-5y = 20-6x = 90 5y = 90 x = 10 y = -4 x = -15 y = 18 ½ x 3y = 12 4x 5y 20= 0 For the x-intercept For the y-intercept For the x-intercept For the y-intercept let y = 0 let x = 0 let y = 0 let x = 0 (cover up y!) (cover up x!) (cover up y!) (cover up x!) ½ x = 12-3y = 12 4x 20 = 0 5y 20= 0 x = 24 y = -4 4x = 20 5y = 20 x = 5 y = -4

9 2. SLOPE/INTERCEPT Graphing using y = mx + b a. Make sure your equation is written in y = mx + b form b. Plot the b value (remember, this is just the y-intercept) c. Write your slope as a fraction so you can clearly see the rise and run Examples: y = 2x 4 y = 0.5x 7 y = -x/5 + 8 d. From the y-intercept you plotted, move according to your slope (do the rising and the running ) and then plot a second point where you end up e. Connect the two points with a straight line Let s plot the previous examples using y = mx + b 2x 5y = 20-6x + 5y = 90 ½ x 3y = 12 4x 5y 20= 0

10 3. Graphing using a Table of Values a. Make sure your equation is written in y = mx + b form b. Draw a Table of Values as shown below c. Choose values for x (you can really choose any you want, but if you re not sure, choose -3, -2, -1, 0, 1, 2, 3 until you are comfortable picking values) d. Sub each of the the x-values into the equation to get their corresponding y-values. Note, each pair of x and y values is really a point on your graph. e. Plot the points and connect them to make a straight line. Let s plot the previous examples using tables of values 2x 5y = 20-6x + 5y = 90 ½ x 3y = 12 4x 5y 20= 0

11 Graph the following using a method of your choice. (a) y = 3/2x 4 (b) x + 2y = 6 (c) 5x 2y = 10

12 Graph the following using a method of your choice. (a) y = 3/2x 4 (b) x + 2y = 6 (c) 5x 2y = 10

13 m = 0 m = undefined Don t forget about HORIZONTAL and VERTICAL lines! (We don t even need a formula because they re so easy!) You ll recognize them because they only contain one variable, for example: y = 2 x = 5 When you see these equations, remind yourself that they must be pretty simple, because there s only one variable. Instead of freaking out because they look different, just take a moment to remember that we have a different strategy for equations that are so easy. These equations are simply telling you what will always be true for any point on the line. STRATEGY: Just find a few points that satisfy the condition, then plot them! y = 2 x = 5 List some points where y = 2 List some points where x = 5 (0, 2) (3, 2) (-4, 2) (6, 2) (5, 0) (5, 3) (5, -2) (5, 1) Can you do it backwards? Can you write the equations of the lines graphed below simply by checking to see what condition is always true?

Pre-calculus is the stepping stone for Calculus. It s the final hurdle after all those years of

Pre-calculus is the stepping stone for Calculus. It s the final hurdle after all those years of Chapter 1 Beginning at the Very Beginning: Pre-Pre-Calculus In This Chapter Brushing up on order of operations Solving equalities Graphing equalities and inequalities Finding distance, midpoint, and slope