MA007 Study Guide
Unit 5: Graphs of Linear Equations and Inequalities
5a. Graph points with given coordinates on the rectangular coordinate plane
- How do you graph (x,y) points on the coordinate plane?
The coordinate plane is made up of a horizontal axis, the x-axis, and a vertical axis, the y-axis. When plotting an (x,y) point on the coordinate plane, you begin at the origin, (0,0), and move left or right based on the value of x. From there, move vertically based on the value of y. That is the location of your point.
For example, let's plot the point (3, 7)
As you can see, we went 3 to the right because the x value is positive three, and then up 7 since the y value is positive 7. P is located on the point (3,7).
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5b. Determine the coordinates of a point on the rectangular coordinate system
- How do you determine the coordinates of a point on the coordinate plane?
If we see a point on the coordinate plane, we can identify its coordinates in the reverse way from how we plotted the point.
Let's find the coordinates of the point D. First, consider the x-coordinate of the point. Since D is 3 to the left, it has an x-coordinate of -3. Similarly, D has a y-coordinate of -3. Therefore, the coordinates of D are (-3,-3).
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5c. Determine whether a given ordered pair is a solution of the equation with two variable
- How can you check if a certain point is the solution to an equation?
When we graph an equation, every point on the graph is a solution to the equation that was graphed. Because of that, we can check if a certain point is a solution to the equation by simply checking if that point is on the graph.
For example, consider the equation \(y=x^2+4x+4\). Is the point (3,1) a solution to the equation?
The graph is:
Since we have been given the graph, all we need to do is check if the point (3,1) is on the graph. If we move to 3 on the positive x-axis, and then up 1, we find a point that is on the graph. Therefore, (3,1) is a solution to the equation \(y=x^2+4x+4\).
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5d. Find and graph solutions of the equation in two variables
- How do you find and graph the solution to an equation?
When you have an equation you want to graph the solution of, you should start by finding some specific solutions using an x-y table. Then, plot those points on the coordinate plane, and finally, connect the points to draw the graph.
For example, to graph the solutions to the equation y=3x-1, we will make an x-y table and select some x-values, which we will substitute into the equation to find the corresponding y-values. First, let's set up the x-y table. You can select any x-values you want, but values near the middle of your graph are generally good.
|
X |
Y |
|---|---|
|
-1 |
|
|
0 |
|
|
1 |
|
|
2 |
Now, we will substitute those x-values for x in the equation to find the y-values.
|
X |
Y |
|---|---|
|
-1 |
3(-1)-1= -4 |
|
0 |
3(0)-1= -1 |
|
1 |
3(1)-1= 2 |
|
2 |
3(2)-1= 5 |
Now we have 4 points on our graph. Plot those points, then connect them to graph the equation.
We now have the graph of the solutions to the equation.
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5e. Graph a line given either its equation or a slope and y-intercept
- What is slope?
- How do you find and use slope when graphing?
When graphing a linear equation, a key point to focus on is the slope. The slope is the change in y divided by the change in x. We often use the letter "m" to represent slope. The slope formula is:
\(m=\dfrac{y_2-y_1}{x_2-x_1}\)
When graphing, the slope of a line can be seen and calculated visually as well.
To calculate the slope visually, simply identify two points on the line, then count the change in y and change in x between those points, sometimes called "rise over run". Be sure to be careful to consider if the points are changing positively (up/right) or negatively (down/left) to accurately calculate the slope.
For example, we will calculate the slope of the following line:
If we focus on the points (-5,1) and (0,3), we can see that between these points, the y went up 2, and the x went to the right 5. Therefore our slope is \(\dfrac{2}{5}\).
When graphing a line, you can use any point along with the slope to make your graph.
For example, let's graph a line passing through the point (-3, 1) with a slope of ⅔.
First, we will plot a point at (-3,1). Then, from that point, we will move according to the slope, ⅔. We will move up 2 and to the right 3, and arrive at another point on the line, the point (0,3). Finally, connect these points, and you will have the graph of your line.
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5f. Find the slope and intercepts of a line given its equation or its graph
- How do you find the y-intercept of a line?
- How do you find the x-intercept of a line?
- How do you graph a line in slope-intercept form?
When graphing a line, one easy way to find some important points is to find the x-intercept and y-intercept. When viewing a graph, the intercepts can be found by simply looking at where the line crosses the x-axis and y-axis.
For example, the linear function above has a y-intercept of (0,-3) and an x-intercept of (2,0).
If you have the equation of a line, finding the intercepts is quite simple. To find the y-intercept, you simply let the \(x=0\) and solve for \(y\). Similarly, to find the x-intercept, let \(y=0\) and solve for \(x\).
For example, to find the intercepts of \(3x-2y=6\), we will first let \(x=0\) to find the y-intercept. \(3(0)-2y=6\), \(-2y=6\), \(y=-3\). Therefore, the y-intercept is (0,-3). Similarly, we can find the x-intercept by letting \(y=0\). \(3x-2(0)=6\), \(3x=6, x=2\). Therefore, the x-intercept is (2,0).
One way the equation of a line can be written is called slope-intercept form. Slope-intercept form is \(y=mx+b\), where the m is the slope, and b is the y-intercept. In slope-intercept form, we can start by plotting the y-intercept, then use the slope to find another point and graph the line.
For example, the line \(y=2x-3\), has a y-intercept of (0,-3) and a slope of 2. To graph, we begin by plotting the y-intercept, then from that point, graphing a slope of 2 to find another point and draw the graph.
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5g. Write the equation of a line passing through two given points
- How do you write the equation of a line passing through two points?
Another way to write the equation of a line is called point-slope form. Point-slope form is \(y-y_1=m(x-x_1)\). In point-slope form, \((x_1,y_1)\) is a point on the line, and m is still the slope. This form is ideal for problems where you are asked to write the equation of a line, as you only need any point on the line and the slope to write the equation.
For example, to find the equation of the line passing through (-2,3) and (-1,-2), first we must find the slope. Using the slope equation, the slope is \(m=\dfrac{-2-3}{-1-(-2)}=\dfrac{-5}{1}=-5\). Now that we have the slope, we can use either point to write the equation of the line. If we use the point (-2,3), then the equation of the line is \(y-3=-5(x+2)\).
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5h. Write the equation of a line with a given slope passing through a given point
- How do you write the equation of a line given a slope and a point?
When a slope and a point are given, rather than two points, writing the equation of a line is even simpler in point-slope form. Since a point and the slope are all that is needed to write the equation, you simply need to plug in the information given.
For example, to find the equation of the line passing through (-2,5) with a slope of ⅓, simply substitute into the point-slope equation, \(y-5=\dfrac{1}{3}(x+2)\).
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5i. Locate on a coordinate plane all solutions of a given inequality in two variables
- How do you graph the solutions to a linear inequality?
Linear inequalities are very similar to linear equations, except instead of just finding solutions on the line, we will be finding an entire area of the graph that has solutions to our inequality.
To graph a linear inequality like \(y<3x-1\), start by graphing the equivalent equation \(y=3x-1\). The y-intercept is (0,-1) and the slope is 3. When graphing, draw a dashed line instead of a solid line. For inequalities with the < or > symbols, always use a dashed line. For inequalities with the ≤ or ≥ symbols, you can use a solid line.
Now, pick any point on one side of the line. We will test that point in our inequality to see if it satisfies the inequality. If it does, then we will shade that side. If it doesn't, then we will shade the other side. For example, we will test the point (0,0), which is on the left/upper side of the line. 0<3(0)-1 simplifies to 0<-1. This is not true. Therefore, we must shade the other side.
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5j. Represent relationships between quantities as an equation or inequality in two variables
How do you represent the relationship between quantities in an inequality?
Inequalities are used every day in our lives. For example, if you want to buy gas and snacks but only have $20, you have solved an inequality. For example, if gas is $3 per gallon, and snacks are $4 each, you can create an inequality such as \(3g+4s \leq 20\). You can test values for g and s to find possible solutions, such as \(g=2\), \(s=2\). Since \(3(2)+4(2)=14\), and \(14 \leq 20\), then \(g=2\), \(s=2\) is a possible solution.
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5k. Interpret the meaning of slope and intercepts of the graph of a relationship between quantities
- How can you identify parallel and perpendicular slopes from their equations?
Parallel lines are two lines that never intersect. Think of parallel lines like the lines on a highway, they never intersect. When looking at the equations of two lines, the key to determining if the lines are parallel is to examine their slopes. Parallel lines must have the same slope. For example, the lines \(y = 2x + 1\) and \(y = 2x - 3\) are parallel because they both have a slope of 2.
Perpendicular lines are two lines that intersect at a 90 degree angle. The product of the slopes of two perpendicular lines is always equal to -1. Another way to identify perpendicular lines is that the slope of one line is the opposite reciprocal of the other line. The opposite means change the sign, and reciprocal means to flip the number, making the numerator the denominator, and vice versa. For example, the lines \(y=2x-1\) and \(y=-\dfrac{1}{2}x+5\) are perpendicular since the opposite reciprocal of 2 is -½.
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Unit 5 Vocabulary
This vocabulary list includes terms you will need to know to successfully complete the final exam.
- coordinate plane
- coordinates
- equivalent equation
- linear inequality
- opposite reciprocal
- parallel line
- perpendicular line
- point
- point-slope form
- product
- slope
- slope-intercept form
- x-axis
- x-intercept
- x-y table
- y-axis
- y-intercept