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§2.1 Graphing Equations The following is the Rectangular Coordinate System, also called the Cartesian Coordinate System. y II (-,+) I (+,+) Origin x III (-,-) IV (+,-) A coordinate is a number associated with the x or y axis. An ordered pair is a pair of coordinates, an x and a y, read in that order. An ordered pair names a specific point in the system. Each point is unique. An ordered pair is written (x,y) The origin is where both the x and the y axis are zero. The ordered pair that describes the origin is (0,0). The quadrants are the 4 sections of the system labeled counterclockwise from the upper right corner. The quadrants are named I, II, III, IV. These are the Roman numerals for one, two, three, and four. It is not acceptable to say One when referring to quadrant I, etc. Plotting Points When plotting a point on the axis we first locate the x coordinate and then the y coordinate. Once both have been located, we follow them with our finger or our eyes to their intersection as if an imaginary line were being drawn from the coordinates. Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 1 Example: a) (2,-2) Plot the following ordered pairs and note their quadrants b) (5,3) c) (-2,-5) d) (-5,3) e) (0,-4) y f) (1,0) x Also useful in our studies will be the knowledge of how to label points in the system. Labeling Points on a Coordinate System We label the points in the system, by following, with our fingers or eyes, back to the coordinates on the x and y axes. This is doing the reverse of what we just did in plotting points. When labeling a point, we must do so appropriately! To label a point correctly, we must label it with its ordered pair written in the fashion – (x,y). Example: Label the points on the coordinate system below. y x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 2 Linear Equation in Two Variable is an equation in the following form, whose solutions are ordered pairs. A straight line graphically represents a linear equation in two variables. ax + by = c a, b, & c are constants x, y are variables x & y both can not = 0 Since a linear equation’s solution is an ordered pair, we can check to see if an ordered pair is a solution to a linear equation in two variables by substitution. Since the first coordinate of an ordered pair is the x-coordinate and the second the y, we know to substitute the first for x and the second for y. (If you ever come across an equation that is not written in x and y, and want to check if an ordered pair is a solution for the equation, assume that the variables are alphabetical, as in x and y. For instance, 5d + 3b = 10 – b is equivalent to x and d is equivalent to y, unless otherwise specified.) Example: Is (0,5) a solution to: y = 2x 3 ? Notice that we said above that the solutions to a linear equation in two variables! A linear equation in two variables has an infinite number of collinear (solutions on the same line) solutions, since the equation represents a straight line, which stretches to infinity in either direction. An ordered pair can represent each point on a straight line, and there are infinite points on any line, so there are infinite solutions. The trick is that there are only specific ordered pairs that are the solutions! Graphing a Linear Equation Step 1: Choose 3 “easy” numbers for either x or y (easy means that your choice either allows the term to become zero, or a whole number) Step 2: Solve the equation for the other value that you did not choose in step one. You will solve for three values. Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 3 Example: Graph y = 2x y = 2x – 3 & y x In this example, I have chosen to have you graph two equations with the same slope and different y-intercepts, so that I can bring up a concept in graphing called translation. Translation is a way of describing graphs based upon a general form of a family of graphs, being moved about the coordinate system. Once we learn the basic concept it makes graphing all types of equations easier, because we can see their basic characteristic graph being moved by horizontal, vertical, reflection, and stretching translations. In this section we will discuss, three of these translations – horizontal, vertical and reflection. The last example is a vertical translation of the line with slope of 2. Just imagine if the line y = 2x were a piece of wire and we simply picked it up and moved it down the vertical axis three units, but nothing else about the line changes. Not all equations produce a line. Those that don't are called non-linear functions or equations. Martin-Gay/Greene suggest that we learn to graph these non-linear functions just as we have with a linear equation. I am going to show you some tips about 5 special non-linear functions (Martin-Gay/Greene discuss 3 at this time and bring the other 2 up during the next few sections, so I’ll just get them all out of the way now) and how to graph them. We should also realize that choosing random points and solving the equation/function for those point will yield ordered paired solutions, but without the knowledge of what a function’s graph looks like, we may never get a true picture of the function’s graph. Following are a few of the non-linear functions are some of their characteristics. Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 4 Quadratic (2nd Degree) Form a type of graph called a parabola Form of equation we'll be dealing with in are: y = ax2 + d & y = a(x – c)2 + d Sign of a determines opens up or down "+" opens up "" opens down The vertex (where the graph changes direction) is at: (0, d) in y = ax2 + d (c, d) in y = a (x – c)2 + d Symmetric around a vertical line called a line of symmetry Goes through the vertex, and has the equation x = “x-coordinate of vertex” Example: Graph y = x2, y = -x2 and y = -x2 + 2 on this graph, by making a t-table of points and while thinking about the vertex, the line of symmetry and 4 points. Use the same points for each graph. After we will discuss the shape, the translations and how those translations effected the points that we graphed. y x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 5 Absolute Value Functions Form a V shaped graph Form of equation we'll be dealing with are: y = a | x | + d and y = a | x – c | + d Sign of “a” determines opens up or down "+" opens up "" opens down The vertex (where the graph changes direction) is at: (0, d) in y=a|x|+d (c, d) in y=a|x–c|+d Symmetric around a vertical line called a line of symmetry Goes through the vertex, and has the equation x = “x-coordinate of vertex” Example: Graph y = | x | and y = | x – 2 | on this graph, by making a t-table of points and while thinking about the vertex, the line of symmetry and 4 points. Use the same points for each graph. After we will discuss the shape, the translations and how those translations effected the points that we graphed. y x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 6 Cubic Functions Form a lazy "S" shaped graph Type we'll deal with are: y = ax3 + d and y = a(x – c)3 + d Sign of "a" determines curves up and to right or down and to right "+" curves up and to right "" curves down and to right Point of Inflection is where the graph starts to have opposite slope (0, d) in y = ax3 + d (c, d) in y = a(x – c)3 + d Choose opposites on either side of the inflection point to graph Still has symmetry allowing for easy graphing Example: Graph y = x3 & y = x3 1 on this graph, by making a t-table of points and while thinking about the “inflection point”, and 2 positive x’s and their opposites. Use the same points for each graph. After we will discuss the shape, the translations and how those translations effected the points that we graphed. y x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 7 Square Root Functions Looks like half a parabola opening up or down from the x-axis Only half because principle square root is only consideration Type we'll deal with are: y = a x + d and y = a x – c Sign of "a" tells up or down from x-axis "+" is up "" is down Vertex is still (-c,0) similar to a parabola Example: Graph y = √ x , y = x y = -√ x – (-3), x 0 on this graph, by making a t-table of points and while thinking about the vertex, the line of symmetry and 4 points. Use the same points for each graph. After we will discuss the shape, the translations and how those translations effected the points that we graphed. y x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 8 Reciprocal Functions Look like a wide parabola in 1st & 3rd or in 2nd & 4th quadrants Never touch or cross an axis Because x 0, but as it gets close it is very small which makes f(x) get big Like wise as x gets big, f(x) gets small but never gets to zero! Type we'll deal with in this chapter are: y = a/x Sign of "a" tells which quadrants "+" is in 1st & 3rd "" is in 2nd & 4th Trick in graphing is choosing both small & large values of x that are both positive and negative Example: Graph y = 2/x using 8 points. Use 2 small & 2 large positive & negative values. y x x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 9 §2.2 Introduction to Functions A relation is any set of ordered pairs. A function is a relation for which every value of the independent variable (the values that can be inputted) has one and only one value for the dependent variable (the values that are output, dependent upon those input). All the possible values of the independent variable form the domain and the values given by the dependent variable form the range. Think of a function as a machine and once a value is input it becomes something else, thus you can never input the same thing twice and have it come out differently. This does not mean that you can't input different things and have them come out the same, however! Your textbook shows some great pictures of functions on page 97. There are many ways to show a function's domain and range. One way is to draw a picture and show the mapping (each element of the domain maps to only one member of the range in a function) of the domain onto the range. Another way is to list the domain as a set using roster or set builder notation and the range as a set using roster or set builder notation. If the domain and range are both finite, they can be listed together as a set of ordered pairs. We can also graph a function, which shows both the domain and the range. 2 3 5 15 25 5 These are maps. Left is f(n) Rt. Is not an f(n) 2 6 7 Is a Relation a Function? 1) For every value in the domain is there only one value in the range? a) Looking at a map – If any x have lines to more than one y, then not a function b) Looking at ordered pairs – If no x’s repeat then it’s a function c) Looking at a graph – Vertical line test (if any vertical line intersects the graph in more than one place the relation is not a function) d) Think about the domain & range values – If input of any x will give different y’s, then not a function (probably a graph is still best!) Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 10 Example: a) Which of the following are functions? What are the domain & range of each? {(3,5), (2,5), (4,5)} b) y = x, {x| x 0, x} c) 2 2 2 5 7 9 y d) 5 15 -15 x -5 e) Input (Eye Color) Output (people in class) blue green brown Jorge Sally Maria Katerina Fritha Manesh Tung Note: When a set is finite it is easiest to use roster form to list the domain & range, but when the sets are infinite or subsets of the , set builder and interval notation are better for listing the elements. Function notation may have been discussed in algebra, but if it wasn't you didn't miss much. It is just a way of describing the dependent variable as a function of the independent. It is written using any letter, usually f or g and in parentheses the independent variable. This notation replaces the dependent variable. f (x) Read as f of x The notation means evaluate the equation at the value given within the parentheses. It is exactly like saying y=!! Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 11 Example: a) Evaluate f(2) f(x) = 2x + 5 at Example: a) For h(g) = 1/g h(1/2) Example: Let’s look to our book for evaluating a function represented on a graph. p. 135 #82, 86, & 90 Let’s also see if we can write this function’s domain and range. b) f( -1/2) find b) h(0) When using function notation in everyday life the letter that represents the function should relate to the dependent variable's value, just as the independent variable should relate to its value. Example: a) b) Y. Butterworth The perimeter of a rectangle is P = 2l + 2w. If it is known that the length must be 10 feet, then the perimeter is a function of width. Write this function using function notation Find the perimeter given the width is 2 ft. Write this using function notation. Ch. 2 Notes – Martin-Gay/Greene 12 §2.3 Graphing Linear Functions Recall from §2.1: Linear Equation in Two Variable is an equation in the following form, whose solutions are ordered pairs. A straight line can graphically represent a linear equation in two variables. ax + by = c a, b, & c are constants x, y are variables x & y both can’t = 0 Also Recall from Beginning Algebra: Solving an equation for y is called putting it in slope-intercept form. This is a special form, a function, with the following properties. y = mx + b m = slope b = y-intercept The other great thing about this form is that it allows us to use function notation and eliminate the need to write the dependent variable. Hence, y = mx + b becomes f(x) = mx + b since y is a function of x. We reviewed graphing a linear equation in 2 variables in §2.1, based upon 3 randomly chosen points. In this section graphing is discussed in terms of the intercepts. We will still need a 3rd random point, but this allows us to more quickly graph using points. I won’t be spending a lot of time on this as I feel the “plug and chug” method, whether it is with 3 random points or intercepts and a random point is still an ineffective method which wastes our valuable time! However, we do need to know what intercepts are and how they can be used to graph a line. An intercept is where a graph crosses an axis. There are two types of intercepts for any graph, an x-intercept and a y-intercept. An x-intercept is where the graph crosses the xaxis and it has an ordered pair of the form (x, 0). A y-intercept is where the graph crosses the y-axis and it has an ordered pair of the form (0, y) most often written (0, b). Finding the Y-intercept (X-intercept) Step 1: Let x = 0 (for x-intercept let y = 0) Step 2: Solve the equation for y (solve for x to find the x-intercept) Step 3: Form the ordered pair (0,y) where y is the solution from step two. [the ordered pair would be (x, 0)] Note: When finding both intercepts in a single problem, you are expected to clearly mark you work as: The x-intercept and The y-intercept and then show work for each under these headings finishing with your final answer as an ordered pair. Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 13 Example: Find the x & y intercepts and give them as ordered pairs a) x 2y = 9 b) y = 2x Note1: In part a) will the y-intercept be easy to graph? Note2: In part b)notice that the x & y intercepts are the same. This is because this is a line through the origin, so it crosses the x & y axis in the same place. If there is no b in slope-intercept form or c = 0 in std. form then the line goes through the origin. To Graph a Linear Function Step 1: Choose 3 values of x (appropriate values) Now we have 2 special points that are easy to find, the x and y-intercepts & still find the 3rd one Step 2: Substitute and solve for f(x) [e.g. y] 3 times Remember that two points make a line, but 3 gives a check! Also recall that the since the domain and range of a linear function are all real numbers it makes it possible to choose any value of x and know that there will be a corresponding value for f(x). Example: Graph f(x) = 1/3 x + 1 y x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 14 Now, we need to discuss another set of special lines. They don't appear to be linear equations in 2 variables because they are written in 1 variable, but this is because the other variable can be anything. The lines in question are vertical and horizontal lines. Here is a table of facts about these special types of lines that you need to know forward, backward, and upside-down and inside out. Type Horizontal Equation y = # Slope Zero Type of ordered Pairs (#1, y), (#2, y), (#3, y); y agrees with equation & #1, #2 & #3 can be anything Vertical x=# Undefined (x, #1), (x, #2), (x, #3); x agrees with equation & #1, #2,& #3 Intercepts y-intercept: (0,y) no x-intercept No y-intercept x-intercept: (x, 0) Vertical Lines x = a for a A vertical line is not a function. A vertical line has no y-intercept, unless it is the y-axis (x = 0) x-intercept is value of x For whatever y chosen, x will equal “a” (thus all y's are valid) Horizontal Lines y = b or f(x) = b for b A constant function A horizontal line has no x-intercept, unless it is the x-axis (y = 0) y-intercept is the value of y For whatever x chosen, y will equal “b” (thus all x’s are valid) Example: Graph the following two equations on the same coordinate system. y a) x = -1 b) f(x) = 1/3 x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 15 Remember our non-linear functions from the previous sections. In this section we learn that they also have y-intercepts. The y-intercept of the general function, for all of our functions except the reciprocal function, is the origin, just as it is for the general form of a linear equation in 2 variable (that general function is y = ax, btw). When we make a vertical translation (that’s adding in the d, in all the functions above), then the y-intercept is whatever the “d” is. This is just like a straight line where we see that when written in function notation (slope-intercept form) that the constant that is added to the independent variable is the yintercept. The reason that this is the case is because when we let x = 0, this makes the variable term “go away” and we are simply left with the constant, and we know that mathematically, finding the y-intercept is letting x = 0 and solving for y, so all the concepts begin to come together here. Example: Match the following 3 functions with their graphs a) f(x) = -x2 b) f(x) = x2 + 2 c) i) ^ ^ ii) iii) v Y. Butterworth f(x) = -x2 – 3 v Ch. 2 Notes – Martin-Gay/Greene v v 16 §2.4 The Slope of a Line Slope is the ratio of vertical change to horizontal change. m = rise = y2 y1 = y run x2 x1 x Rise is the amount of change on the y-axis and run is the amount of change on the x-axis. A line with positive slope goes up when viewing from left to right and a line with negative slope goes down from left to right. When asked to give the slope of a line, you are being asked for a numeric slope found using the equation from above. The sign of the slope indicates whether the slope is positive or negative, it is not the slope itself! Knowing the direction that a line takes if it has positive or negative slope, gives you a check for your calculations, or for your plotting. 3Ways to Find Slope 1) Formula given above Example: Use the formula to find the slope of the line through (5,2) & (-1,7) 2) Geometrically using m = rise/run Choose points, create rise & run triangle, count & divide Example: #70 p.161 Martin-Gay/Greene 3) From the slope-intercept form of a linear function y = mx + b, where m, the numeric coefficient of x is the slope Solve the equation for y, give numeric coeff. of x as the slope (including the sign) Example: Find the slope of the line 2x 5y = 9 Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 17 Special Lines We have already discussed horizontal and vertical lines, but now we need to discuss them in terms of their slope. Horizontal Lines, recall, are lines that run straight across from left to right. A horizontal line has zero slope. This is so since it has zero vertical change (rise) and zero divided by anything is zero. m = rise = 0 = zero run # Example: Find the slope of the horizontal line through the points (0,5);(10,5) Note: It is always acceptable to use the formula to find the slope when given points, but if the ycoordinates are the same then the line is horizontal and has zero slope and if the x-coordinates are the same the line is vertical (recall my table from the last section) and has undefined slope. Vertical Lines, recall, are lines that run straight up and down. A vertical line has undefined slope. This is so since it has zero horizontal change (run) and anything divided by zero is undefined. m = rise = # = undefined run 0 Note: Some authors will use the very weak, no slope, to mean either undefined or zero slope. I will not accept "no slope" for either answer. The reason for this is that no slope can be translated to mean none or zero and a vertical line does not have zero slope, it has undefined slope, as you will see from the following example! Also, since this confusion exists, I want it eliminated completely!! Example: Find the slope of the line below y x 3 Note: When finding the slope of a horizontal or vertical line that has been graphed, the formula approach can still be used, but it is easier to remember that a vertical line has undefined slope and a horizontal line has zero slope. Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 18 Let's go back to the discussion of graphing a line. There are 3 methods, let's recap them and then focus on the one of interest. Graphing a Line (A linear equation in two variables) Method 1: Choose 3 random x's and solve for y and graph the 3 ordered pairs Method 2: Find x & y intercepts & a third point as in Method 1 and graph Method 3: Graph y-intercept & use geometric approach of slope to get 2nd & 3rd points. Slope & y-intercept come from slope-intercept form of line. Way back in Pre-Algebra you may have learned Method 1 and we reviewed it earlier in this chapter, and then you learned about the 2 special points – the intercepts and how that is just method 1 with a couple of points that are pretty easy to find. Now we will focus our attention on Method 3. Graphing a Line Using Slope-Intercept Form Step 1: Put the equation into slope-intercept form (solve for y) Step 2: Plot the y-intercept point (0, b) Step 3: Count up/down and over left/right from the intercept point as indicated by the slope, to find a second point. Step 4: Repeat Step 3, with a different iteration of the slope (e.g. +/ is same as /+ or +/+ is same as /). If you went up and left then go down and right, and if you went up and right then go down and left. Example: Graph the following using the method just described 2x + 3y = -9 x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 19 Special Pairs (Groups) of Lines There are two types of special pairs. There are perpendicular lines and parallel lines. Perpendicular Lines are lines that meet at right (90) angles. Perpendicular lines have slopes that are negative reciprocals of one another. Example: Give the slope of the line perpendicular to each of the following. a) y = 1 /2 x + 3 b) 2x + 3y = 9 c) Thru the points (0, 3) & (5, 2) Parallel Lines are lines that never cross. At all points, parallel lines are equidistant. Parallel lines have the same slope, but they do not have the same y-intercept. These are lines that are vertical translation of the same line. Example: a) y = c) -2 Give the slope of the line parallel to each of the following: /3x – 3 b) 5x + 4y = 9 Thru the points (5, -1) & (-2, -3) Parallel or Perpendicular? Based upon the slope of the line, we can judge whether or not a pair (group) of lines are parallel or perpendicular. Assessing Whether Perpendicular or Parallel Step 1: Find the slope of the lines, [if equation then put into slope-intercept form (solve for y)] Step 2: a) For parallel lines -- Are the slopes the same? b) For perpendicular lines -- Are the slopes negative reciprocals? Note: The product of negative reciprocals is –1, or take the reciprocal and then the opposite of one slope and see if it is the other slope. Example: Are the lines parallel, perpendicular or neither? Why or why not? a) 2x y = -10 b) x + 4y = 7 c) f(x) = 5x – 6 2x + 4y = 2 2x = -5y + 3 g(x) = 5x + 2 Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 20 Applications of Slope The slope of a line is the same thing as pitch of a roof and the grade of a climb. It is exactly the same calculation for pitch as for slope and in grade it is simply converted to a percentage. Example: From the middle of the ceiling to the point of the roof (apex of the roof) it is 5 feet. From the middle of the ceiling to the outer wall, where the roof connects, is 10 feet. Find the pitch of the roof. Example: The train climbed 2580 vertical meters from the bottom of the hill to the top, but the climb took 6450 horizontal meters. What is the grade of the climb? Example: The following example came from p. 207, Beginning Algebra, 9th Edition, Lial, Hornsby and McGinnis That’s the y-intercept! Slope ! Note: Remember our discussions about linear equations in Ch. 1.7, this is an example of one of those linear equations. The baseline is the y-intercept and the “Amount per” is the slope. This is also known as a rate of change. Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 21 §2.5 Equations of Lines The equation for a line can be written in 3 different forms. First we learned the standard form, then we introduced the slope-intercept form and finally we'll learn the point-slope form. Each way of writing the equation has its drawbacks and its benefits, but the slopeintercept is the most informative and therefore the way that we most typically write the equation for a line. We will start out learning how to write the equation for a line by using this form. If we have any of the following scenarios we can use slope-intercept form. Slope-Intercept Form Scenario 1: We have the slope and the intercept both given (intercept may be given as an ordered pair (0, b)) Scenario 2: We have two points and one is the intercept point (we can calculate the slope from the formula) Scenario 3: We have a visual line and we can determine two integer ordered pairs, one of which is the y-intercept (you can’t guess) Under scenario 1 we have the easiest case. All we have to do is to plug in the slope for “m” and the intercept for “b” (if it is an ordered pair, pick off the y-coordinate to use as b). Recall that the general form of the equation in slope-intercept form is: y = mx + b m = slope b = y-intercept Here are some Scenario 1 examples: Example: a) Use the given information to write an equation for the line described in slope-intercept form. m = 2 and b = 3 b) m = 0 & (0,2/3) c) m = undefined & ( -1/2, 0) Note: Parts b & c are special cases since they are the slopes for horizontal and vertical lines. In the case of a horizontal line (slope is zero) there is no problem in plugging into the slope-intercept form, except that it wastes time. However, you can’t plug into the slope-intercept form for a vertical line since a vertical line is not a function! You really need to study the facts from the table in §2.1 so that you can recognize a vertical or horizontal line and give it’s equation no matter what information that you are given! Under scenario 2 we have a little more work, but it still isn't bad. All we must do is calculate the slope and then plug into the slope-intercept form as described under the first scenario. Example: Find the slope of the following lines described by the points. a) (0, 5) & (-1, 7) b) (2, 4) & (0, 0) Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 22 c) (2, -5) & (2, 0) d) (0, 7) and (5, 7) Note: The last three examples are special cases. B) is a line through the origin, C) is a vertical line and D) is a horizontal line. C) is the only one that doesn’t fit the scenario, but I threw in the xintercept point to throw you off! Scenario 3 is just about the same as scenario 2 except we will find the slope by visual inspection. Example: Give the equation of the line shown below. y x Now, let me give you one of each type to try on your own! Your Turn Example: Find the equation of each of the following lines. a) m = ½ & b = 4 b) m = -5 & thru (0, -1) c) Y. Butterworth Thru (-1, 5) & (0, 4) d) Ch. 2 Notes – Martin-Gay/Greene Thru (2,0) and (2, 9) 23 e) m = 0 & (2, 1) y f) If there is no y-intercept given (or if it is not an integer ordered pair) then we must use the point-slope form. There are also 3 scenarios here. They are as follows: Point-Slope Form Scenario 1: You are given the slope & a point that isn’t the y-intercept – not (0, b) Scenario 2: You are given two points neither of which is the y-intercept, (0, b) Scenario 3: You are given a graph & the y-int. isn’t an integer ordered pair. You need only plug into the point-slope form: y y1 = m(x x1) m = slope (x1, y1) is a point on the line x & y are variables (don’t substitute for those) Under scenario 1 our job is the easiest! Example: Find the equation of the line described by the point and slope given. a) (-2, 5) m = -1 b) (1, -3), m = ½ Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 24 Example: In the following case, why can't point-slope form be used to write the equation of the line? Why doesn't it matter? (0, -5), m = undefined Scenario 2 just increases the number of steps in the process. We must find the slope in addition to plugging into the point-slope form and solving the equation for y. Example: a) Find the equation of each line through the given points. Use the point-slope form. (5, 2) and (2, 5) b) (2, 0) and (8, -2) c) (-3, -1) and (-4, 2) c) (1/2, 5) and (1, 1) Scenario 3 just makes us visually find the slope and then pick a point from the graph so that we can use the point-slope form. Example: Find the slope of the line shown below. y x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 25 Now I’ll give you a chance to try these type of problems too. Your Turn Example: Give the equation of the lines described below using point slope form. a) m= 5 thru (5, 2) b) Thru (2, -5) & (-2, 7) c) y x x Writing An Equation with Special Requirements Finally, we need to discuss how to write the equation of a line given certain requirements. Requirement 1 involves the equation of a line that is perpendicular or parallel and a point that lies on the line for which you are graphing an equation. When you have these requirements, you can easily find the slope and then use the point-slope form to give the equation of your new line. Let’s look at a couple of these now. Example: Find the equation of the line described. a) Parallel to y = 2/3x + 9 through (-9, 7) b) Perpendicular to 2x + y = 3 through (4, -3) Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 26 Requirement 2 involves vertical and horizontal lines. Remember the table that I gave you on page 15? You need to review the equations, slopes and how all ordered pairs look on horizontal and vertical lines for these examples. Example: Find the equation of the line described. a) Parallel to the line y = 7 through the point (2, -1/4) Note: That is parallel to a horizontal line so it is also horizontal and therefore the only thing I’m interested in is the y-coordinate of my point since the equation looks like y = y-coordinate of the ordered pair. b) Perpendicular to the line y = -2/5 through the point (7, -3) Note: That is perpendicular to a horizontal line so it is vertical and therefore the only thing I’m interested in is the x-coordinate of my point since the equation looks like x = x-coordinate of the ordered pair. c) With m = 0 through the point (2, -178) Note: We’ve already seen this type, but just to reiterate – the slope is zero so you know that it is a horizontal line, and therefore its equation must look like y= y-coordinate of the point. d) Through the point (1 5/8, 0) with undefined slope. Note: We’ve already seen this type, but just to reiterate – the slope is undefined so you know that it is a vertical line, and therefore its equation must look like x= x-coordinate of the point. Now, I’ll give you some to try on your own. Most of these should take a few seconds, because they deal with the special cases. There are only 2 that should require a little longer to figure out. We may or may not have class time to do these and they may be assigned as homework. Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 27 Your Turn Example: Find the slope of the following lines. a) Parallel to the line y = 4 and passing through (2,-2) b) Perpendicular to x = 1 and passing through (8,111) c) Perpendicular to 3x + 6y = 10 through (2,-3) d) Vertical through (-1000, 2) e) Horizontal through (1239,1/4) f) With slope, -4; y-intercept, -2 g) With undefined slope through (-3, 1) h) With zero slope through (1/3,7.8) i) Through (5,9) parallel to the x-axis j) Through (4.1,-92) perpendicular to the x-axis Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 28 Applications of Slopes and Finding Equations Remember our application problems where we were given the baseline and the rate of change? Well, those times are gone in this section! Instead, you’ll be given information for 2 independent values and their corresponding dependent values. From this information you will have two ordered pairs, and you will be able to build the linear equation based on the slope (rate of change) and y-intercept (baseline), using methods that we have just developed. Example: Let’s look at problem #78 on p. 172 of Martin-Gay/Greene’s text The value of a building bought in 1990 appreciate, or increases as time passes. Seven years after the building was bought, it was worth $165,000; 12 years after it was bought, it was worth $180,000. a) If this relationship between years past 1990 and value continues in a linear fashion, write an equation describing it. b) Use this equation to estimate the value of the building in the year 2010. Note: The baseline is the value in the year 1990. The rate of change is the increase per year over the original price in 1990. Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 29 §3.6 Graphing Linear Inequalities in Two Variables I always cover this section with graphing, because it is simply an application of graphing. A linear inequality in two variables is the same as a linear equation in two variables, but instead of an equal sign there is an inequality symbol (, , , or ). Ax + By C A, B & C are constants A & B not both zero x & y are variables Determining if an ordered pair is the solution set to a linear inequality is just like determining if it is a solution set to linear equality; we must evaluate the inequality at the ordered pair and see if it is a true statement. If it is a true statement, then the ordered pair is a solution, and if it is false then it is not a solution. The solutions are even more infinite than those solutions for a linear equation in 2 variables. Example: a) Determine if the following ordered pairs are solutions to 7y + 2 -2x (-1, 0) b) (-1,-1) c) (-1,1) Graphing a Linear Inequality in Two Variables Step 1: Solve the equation for y (don't forget that the sense of the inequality will reverse if multiplying or dividing by a negative.) Step 2: Step 3: Graph the line y = ax + b, using at least 2 labeled ordered pairs a) If or then line is solid b) If or the line is dotted When in slope-intercept form, shade the region as indicated by the inequality a) Choose a checkpoint in the region that you shade and put it into the original equation to make sure it creates a true statement (this is really to check that you correctly flipped your inequality if you had to mult./divide by a negative in put the equation into slope intercept form) Step 4: Label the solution. Example: Graph y 2x + 4 y x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 30 Example: Graph 2x + y -1 y x Example: Graph 3x 4y 12 y x Y. Butterworth Ch. 2 Notes – Martin-Gay/Greene 31