Sectioning Off by Quadrants
Another description for a point is the quadrant that the point lies in. The quadrants are referred to in many applications because of the common characteristics of points that lie in the same quadrant. The quadrants are numbered one through four, usually with Roman numerals. Check out Figure 21-1 to see how the quadrants are identified.
Figure 21-1: Categorizing points.
Q. Referring to Figure 21-1, describe which coordinates are positive or negative in the different quadrants.
A. In quadrant I, both the x and y coordinates are positive numbers. In quadrant II, the x coordinate is negative, and the y coordinate is positive. In quadrant III, both the x and y coordinates are negative. In quadrant IV, the x coordinate is positive, and the y coordinate is negative.
3. Referring to Figure 21-1, in which quadrant do the points (–3, 2) and (–4, 11) lie?
Solve It
4. Referring to Figure 21-1, in which quadrant do the points (–4, –1) and (–2, –2) lie?
Solve It
Using Points to Lay Out Lines
One of the most basic graphs you can construct by using the coordinate system is the graph of a straight line. You may remember from geometry that only two points are required to determine a particular line. When graphing lines using points, though, plot three points to be sure that you’ve graphed the points correctly and put them in the correct positions. You can think of the third point as a sort of a check (like the check digit in a UPC Code). The third point can be anywhere, but try to spread out the three points and not have them clumped together. If the three points aren’t in a straight line, you know that at least one of them is wrong.
Q. Use the following to graph the line represented by the equation 2x + 3y = 10.
A. To graph the line, first find three sets of coordinates that satisfy the equation. Three points that work for this line are (5, 0), (2, 2), and (–1, 4). Plot the three points and then draw a line through them.
These aren’t the only three points you could have chosen. I’m just demonstrating how to spread the points out so that you can draw a better line.
5. Find three points that lie on the line 2x – y = 3, plot them, and draw the line through them.
Solve It
6. Find three points that lie on the line x + 3y + 6 = 0, plot them, and draw the line through them.
Solve It
7. Find three points that lie on the line x = 4, plot them, and draw the line through them.
Solve It
8. Find three points that lie on the line y = –2, plot them, and draw the line through them.
Solve It
Graphing Lines with Intercepts
An intercept is a point that a figure shares with one of the axes. Vertical lines (but not the y-axis) have one intercept; it’s a point somewhere on the x-axis. Horizontal lines (but not the x-axis) have one intercept; and that’s a point on the y-axis. And then there’s the group of lines such as y = x, y = 2x, y = –4x, and so on that have just one intercept — at the origin. Other lines, with a constant added or subtracted in the equation, cross both of the axes and have two intercepts.
Intercepts are easy to find when you have the equation of a line. To find the x-intercept, you let y be equal to 0 in the equation and solve for x. To find the y-intercept, you let x be equal 0 and solve for y.
Q. Find the intercepts of the line 9x – 4y = 18.
A. (2,0) and . First, to find the x intercept, let y = 0 in the equation of the line to get 9x = 18. Solving that, x = 2, and the intercept is (2, 0). Next, for the y intercept, let x = 0 to get –4y = 18. Solving for y, . Intercepts are especially helpful when graphing the line, too. Use the two intercepts you found to graph the line, and then you can check with one more point. For instance, in the following figure, you can check to see whether the point is on the line.
9. Use the intercepts to graph the line 3x + 4y = 12.
Solve It
10. Use the intercepts to graph the line x – 2y = 4.
Solve It
Computing Slopes of Lines
The slope of a line is simply a number that describes the steepness of the line and whether it’s rising or falling, as the line moves from left to right in a graph. When referring to how steep a line is, when you’re given its slope, the general rule is that the farther the number is from 0, the steeper the line. A line with a slope of 7 is much steeper than a line with a slope of 2. And a line whose slope is –6 is steeper than a line whose slope is –3.
To find the slope of a line, you can use two points on the graph of the line and apply the formula . The letter m is the traditional symbol for slope; the (x1, y1) and (x2, y2) are the coordinates of any two points on the line. The point you choose to go first in the formula doesn’t really matter. Just be sure to keep the order the same — from the same point — because you can’t mix and match.
A horizontal line has a slope of 0, and a vertical line has no slope. To help you remember, picture the sun coming up on the horizon — that 0 is just peeking out at you.
Q. Find the slope of the line that goes through the two points (–3, 4) and (1, –8) and use the following figure to graph it.
A. –3. To find the slope, use . The following figure shows a graph of that line. It’s fairly steep — any slope greater than 1 or less than –1 is steep. The negative part indicates that the line’s falling as you go from left to right. Another description of slope is that the bottom number is the change in x, and the top is the change in y. Here’s how you read a slope of –3: For every 1 unit you move to the right parallel to the x-axis, you drop down 3 units parallel to the y-axis.
11. Find the slope of the line through the points (3, 2) and (–4, –5) and graph the line.
Solve It
12. Find the slope of the line through the points (–1, 7) and (1, 3) and graph the line.
Solve It
13. Find the slope of the line through (3, –4) and (5, –4) and graph it.
Solve It
14. Find the slope of the line through (2, 3) and (2, –8) and graph it.
Solve It
Graphing with the Slope-Intercept Form
Equations of lines can take many forms, but one of the most useful is called the slope-intercept form. The numbers for the slope and y-intercept are part of the equation. When you use this form to graph a line, you just plot the y-intercept and use the slope to find another point from there.
The slope intercept form is y = mx + b. The m represents the slope of the line. The b is the y-coordinate of the intercept where the line crosses the y-axis. A line with the equation y = –3x + 2 has a slope of –3 and a y-intercept of (0, 2).
Having the equation of a line in the slope-intercept form makes graphing the line an easy chore. Follow these steps:
1. Plot the y-intercept on the y-axis.
2. Write the slope as a fraction.
Using the equation y = –3x + 2, the fraction would be . (If the slope is negative, you put the negative part in the numerator.) The slope has the change in y in the numerator and the change in x in the denominator.
3. Starting with the y-intercept, count the amount of the change in x (the number in the denominator) to the right of the intercept, and then count up or down from that point (depending on whether the slope is positive or negative), using the number in the numerator.
Wherever you end up is another point on the line.
4. Mark that point and draw a line through the new point and the y-intercept.
Q. Graph y = –3x + 2, using the method in the previous steps.
A.
Q. Graph .
A.
15. Graph the line , using the y-intercept and slope.
Solve It
16. Graph the line y = 5x – 2, using the y-intercept and slope.
Solve It
Changing to the Slope-Intercept Form
Graphing lines by using the slope-intercept form is a piece of cake. But what if the equation you’re g
iven isn’t in that form? Are you stuck with substituting in values and finding coordinates of points that work? Not necessarily. Changing the form of the equation using algebraic manipulations — and then graphing using the new form — is often easier.
To change the equation of a line to the slope-intercept form, y = mx + b, first isolate the term with y in it on one side of the equation and then divide each side by any coefficient of y. You can rearrange the terms so the x term, with the slope multiplier, comes first.
Q. Change the equation 3x – 4y = 8 to the slope-intercept form.
A. . First, subtract 3x from each side: –4y = –3x + 8. Then divide each term by –4:
This line has a slope of and a y-intercept at (0, –2).
Q. Change the equation y – 3 = 0 to the slope-intercept form.
A. y = 3. There’s no x term, so the slope must be 0. If you want a complete slope-intercept form, you could write the equation as y = 0x + 3 to show a slope of 0 and an intercept of (0, 3).
17. Change the equation 8x + 2y = 3 to the slope-intercept form.
Solve It
18. Change the equation 4x – y – 3 = 0 to the slope-intercept form.
Solve It
Writing Equations of Lines
Up until now, you’ve been given the equation of a line and been told to graph that line using either two points, the intercepts, or the slope and y-intercept. But how do you re-create the line’s equation if you’re given either two points (which could be the two intercepts) or the slope and some other point?
Use the point-slope form, to write the equation of a line. The letter m represents the slope of the line, and is any point on the line. After filling in the information, simplify the form.
Q. Find the equation of the line that has a slope of 3 and goes through the point (–4, 2).
A. y = 3x + 14. Using the point-slope form, you write y – 2 = 3(x –(–4)). Simplifying, you get .
Q. Find the equation of the line that goes through the points (5, –2) and (–4, 7).
A. y = –x + 3. First, find the slope of the line. . Now use the point-slope form with the slope of –1 and with the coordinates of one of the points. It doesn’t matter which one, so I choose (5, –2). Filling in the values, y –(–2) = –1(x – 5). Simplifying, you get y + 2 = –x + 5 or y = –x + 3.
19. Find the equation of the line with a slope of that goes through the point (0, 7).
Solve It
20. Find the equation of the line that goes through the points (–3, –1) and (–2, 5).
Solve It
Picking on Parallel and Perpendicular Lines
When two lines are parallel to one another, they never touch, and their slopes are exactly the same number. When two lines are perpendicular to one another, they cross in exactly one place creating a 90 degree angle, and their slopes are related in a special way. If two perpendicular lines aren’t vertical and horizontal (parallel to the two axes), then their slopes are opposite in sign and the numerical parts are reciprocals. (The reciprocal of a number is its flip, what you get when you reverse the numerator and denominator of a fraction.) In other words, if you multiply the values of the two slopes of perpendicular lines together, you always get an answer of –1.
Q. Determine how these lines are related:
, , and .
A. They’re parallel. They’re written in slope-intercept form, and all have the same slope. The y-intercepts are the only differences between these three lines.
Q. Determine how these lines are related:
and .
A. They’re perpendicular to one another. Their slopes are negative reciprocals of one another. It doesn’t matter what the y-intercepts are. They can be different or the same.
21. What is the slope of a line that’s parallel to the line 2x – 3y = 4?
Solve It
22. What is the slope of a line that’s perpendicular to the line 4x + 2y + 7 = 0?
Solve It
Finding Distances between Points
A segment can be drawn between two points that are plotted on the coordinate axes. You can determine the distance between those two points by using a formula that actually incorporates the Pythagorean theorem — it’s like finding the length of a hypotenuse of a right triangle. (Check Chapter 18 for more practice with the Pythagorean theorem.) If you want to find the distance between the two points (x1, y1) and (x2, y2), use the formula .
Q. Find the distance between the points (–8, 2) and (4, 7).
A. 13 units. Use the distance formula and plug in the coordinates of the points:
Of course, not all the distances come out nicely with a perfect square under the radical. When it isn’t a perfect square, either simplify the expression or give a decimal approximation (refer to Chapter 5).
Q. Find the distance between the points (4, –3) and (2, 11).
A. units. Using the distance formula, you get
If you want to estimate the distance, just replace the with 1.4 and multiply by 10. The distance is about 14 units.
23. Find the distance between (3, –9) and (–9, 7).
Solve It
24. Find the distance between (4, 1) and (–2, 2). Round the decimal equivalent of the answer to two decimal places.
Solve It
Finding the Intersections of Lines
Two lines will intersect in exactly one point — unless they’re parallel to one another. You can find the intersection of lines by careful graphing of the lines or by using simple algebra. Graphing is quick and easy, but it’s hard to tell the exact answer if there’s a fraction in one or both of the coordinates of the point of intersection.
To use algebra to solve for the intersection, you either add the two equations together (or some multiples of the equations), or you use substitution. I show you the substitution method here, because you can use the slope-intercept (see the earlier section “Changing to the Slope-Intercept Form”) forms of the equations to accomplish the job.
To find the intersection of two lines, use their slope-intercept forms and set their mx + b portions equal to one another. Solve for x and then find y by putting the x value you found into one of the equations.
Q. Find the intersection of the lines y = 3x – 2 and y = –2x – 7.
A. (–1, –5). Set 3x – 2 = –2x – 7 and solve for x. Adding 2x to each side and adding 2 to each side, you get 5x = –5. Dividing by 5 gives you x = –1. Now substitute –1 for x in either of the original equations. You get y = –5. It’s really a good idea to do that substitution back into both of the equations as a check.
Q. Find the intersection of the lines x + y = 6 and 2x – y = 6.
A. (4, 2). First, write each equation in the slope-intercept form. The line x + y = 6 becomes y = –x + 6, and the line 2x – y = 6 becomes y = 2x – 6. Now, setting –x + 6 = 2x – 6, you get –3x = – 12 or x = 4. Substituting 4 for x in either equation, you get y = 2.
25. Find the intersection of the lines y = –4x + 7 and y = 5x – 2.
Solve It
26. Find the intersection of the lines 3x – y = 1 and x + 2y + 9 = 0
Solve It
Graphing Parabolas and Circles
A parabola is a sort of U-shaped curve. It’s one of the conic sections. A circle is the most easily recognized conic (the other conics are hyperbolas and ellipses). The equations and graphs of parabolas are used to describe all sorts of natural phenomena. For instance, headlight reflectors are formed from parabolic shapes. Circles — well, circles are just circles: handy, easy to deal with, and so symmetric.
An equation for parabolas that open upward or downward is y = ax2 + bx + c, where a isn’t 0. If a is a positive number, then the parabola opens upward; a negative a gives you a downward parabola.
Algebra I Workbook For Dummies Page 21