Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity with the concepts. Understanding of certain skills is assumed, such as basic operations with integers, fractions and decimals, comparisons of numbers, and converting percents to decimals. Throughout the course, you will have access to a calculator, so you should make sure you are familiar with the way your calculator works. See Appendix A for more information on calculators. We ll start the prerequisite review with some reminders about graphing in the coordinate plane. The coordinate plane consists of two number lines, drawn perpendicular to one another. This divides the plane into four sections, which we call quadrants. The horizontal number line is called the x axis; the vertical number line is called the y axis. The axes intersect at a point that is called the origin. The coordinate plane allows us to assign a unique name to each point in the plane. We write that name using an ordered pair ( x, y ), where x tells how far to move to the left or right from the origin and y tells how far to move up or down. The origin 4,3 will be four units to is named by the ordered pair ( 0,0 ). The ordered pair ( ) the left of the origin and then up three units, while the ordered pair ( 3, 4) will be three units to the right of the origin and then down four units. Math 1314 Page 1 of 11 Section 1.1
You will need to be able to graph lines in the coordinate plane. A line is determined by two points, so you will need to know two points that lie on a line in order to graph it. We establish a relationship between the x and y coordinates of the points that lie on a line using an equation that contains both x and y. The general form for this equation is Ax + By = C. When you are given an equation in this form, you can graph the line by Choosing some values for x Computing the corresponding values for y Forming ordered pairs ( x, y ) and graphing them Drawing a line that passes through the ordered pairs You must find a minimum of two points; if you find more points, you may find it easier to sketch the graph. Example 1: Graph the line: 2x y = 3 Solution: Start by selecting some values for x. You may want to organize these in a table of values. Keep the numbers small. For this equation, we ll choose 1, 0, and 1. Next, compute the values for y. x -1 0 1 y Math 1314 Page 2 of 11 Section 1.1
If x = 1, then we have 2( 1) y = 3 2 y = 3 y = 5 y = 5 If x = 0, then we have 2(0) y = 3 0 y = 3 y = 3 y = 3 If x = 1, then we have 2(1) y = 3 2 y = 3 y = 1 y = 1 Now fill these values into the y column of the table of values. x y -1-5 0-3 1-1 We now have three ordered pairs, ( 1, 5 ), ( 0, 3 ) and ( 1, 1). Graph these in the coordinate plane. Math 1314 Page 3 of 11 Section 1.1
You can see that the three points lie along a line. Finally, draw a line through the three points and label the graph. Since you only need two points to graph a line, you may find that it s easy to find the two points where the graph crosses the axes and then use those to graph the line. These two points are called the intercepts. The x intercept is the point ( x,0) where the graph crosses the x axis, and the y intercept is the point ( 0, y ) where the graph crosses the y axis. To find the x intercept, let y = 0and solve for x. To find the y intercept, let x = 0and solve for y. Math 1314 Page 4 of 11 Section 1.1
Example 2: Find the x and y intercepts and use them to graph the line: 2x + 3y = 6. Solution: To find the x intercept, let y = 0and solve for x. 2x + 3(0) = 6 2x = 6 x = 3 To find the y intercept, let x = 0and solve for y. 2(0) + 3y = 6 3y = 6 y = 2 This gives you two points, ( 3,0) and ( 0,2 ). Graph them and draw the line. You may be familiar with the form y = mx + b. This is called the slope-intercept form, and it can be useful in graphing. In this equation, m represents the slope and b gives the y coordinate of the y intercept. The slope is the rate at which the y values are changing with respect to x. A slope of 1, then, states that, for each unit 2 change in x, the change in y is 1 rise unit. More often, we state that slope is 2 run, where the numerator tells how many units to move up or down and the Math 1314 Page 5 of 11 Section 1.1
denominator tells how many units to move left or right. So if the slope is 1 2, then you d rise 1 unit and run 2 units. Before the slope can help you graph a line, you ll need a starting point, and with the slope-intercept form, that point is the y intercept. It is the point ( 0,b ), so you can find the y intercept just by looking at the equation. Start by graphing the y intercept, then use the rise and run to obtain one or two more points. Once you can see the pattern for the line, draw in the line. Example 3: Graph the line 2 y = x 1. 3 Solution: Start by finding the information from the equation. The slope is 2 3, so rise is 2 and run is 3. The y intercept is the point ( 0, 1). Plot the point in the coordinate plane. Then use rise and run to find some more points. Now draw in the line and label it. Math 1314 Page 6 of 11 Section 1.1
If the slope that you are given is negative, then you will make either the rise or the run negative, but not both. (Remember that a negative number divided by a negative number is a positive number, so if you make both negative, the resulting slope is positive!) Also, if the slope you are given is an integer, you can write it as a fraction with a denominator of 1, so that you can determine both rise and run. Example 4: Graph y = 3x + 2. Solution: The slope is -3 and the y intercept is ( 0,2 ). Think of the slope as 3 m =, so rise is -3 and run is 1. Since the rise is negative, you ll move down 1 rather than up. Graph the y intercept, then use the rise and run to graph some more points. Now draw the line and label the graph. Math 1314 Page 7 of 11 Section 1.1
You will need to be able to write the equation of the line if you are given information about the line. You will usually be asked to write equations in slope-intercept form. To accomplish this, you will need to know the slope and one point that lies on the graph of the line. Example 5: Write an equation of the line that has slope -3 and passes through the point ( 0, 7). Solution: In this problem, you are given the slope, and the point that you are given is the y intercept. So you know both m and b. The equation of the line is y = 3x 7. Example 6: Write an equation of the line that has slope 2 and passes through the point ( 1, 4). Solution: In this problem, you are given the slope and a point that is not the y intercept. You will need to find b. From the ordered pair, you know that x = 1and y = 4. You are given the slope, so you know that m = 2. Substitute these numbers into y = mx + b and then solve for b. So you ll have 4 = 2(1) + b and b = 6. You now know both m and b, so you can write the equation: y = 2x 6. Math 1314 Page 8 of 11 Section 1.1
You may be given two points and asked to find the equation of the line. As noted earlier, to write an equation of the line, you need the slope and a point to write an equation of the line. You can find the slope of a line if you are given two points ( ) and (, ) x y x y that 1, 1 2 2 y2 y1 lie on the line. The formula for slope is m =. Once you know the slope, x x 2 1 you can use the slope and either one of the given points to write the equation of the line. Example 7: Write an equation of the line that passes through the points ( 2,5 ) and ( 1,4 ). Solution: Start by finding the slope. Two points are given, so you know that x = 2, y = 5, x = 1, and y = 4. Substitute these numbers into the slope formula. 1 1 2 2 4 5 1 1 m = = = 1 2 3 3 Now find b using the slope and either of the two given points. We ll use ( 2,5 ). 1 5 = ( 2) + b 3 2 5 = + b 3 2 5 = b 3 15 2 = b 3 3 13 = b 3 Now that you know both m and b, you can write the equation: 1 13 y = x +. 3 3 Math 1314 Page 9 of 11 Section 1.1
The equation of a horizontal line is y Example 8: Graph y = 2. Solution: The equation is written in the form y = k ; the equation of a vertical line is x = j. = k, so the graph will be a horizontal line. A horizontal line will cross the y axis. In this case, the line will cross the y axis where 2 0, 2. Plot the point and draw the line. y =, which is the point ( ) Example 9: Graph x = 1. Solution: The equation is written in the form x = j, so the graph will be a vertical line. A vertical line will cross the x axis. In this case, the line will cross the x axis where 1 1,0. Plot the point and draw the line. x =, which is the point ( ) Math 1314 Page 10 of 11 Section 1.1
Example 10: Write an equation of the horizontal line that passes through the point ( 1,3 ). Solution: The equation of a horizontal line has the form y = k. Find the y coordinate of the point that is given. The y value of every point on the horizontal line through this point will be the same, 3. The equation is 3 y =. Example 11: Write an equation of the vertical line that passes through the point ( 4,7). Solution: The equation of a vertical line has the form x = j. Find the x coordinate of the point that is given. The x value of every point on the vertical line through this point will be the same, -4. The equation is 4 x =. Math 1314 Page 11 of 11 Section 1.1