Mathematics Study Guide for the TABE Test

Graphing

As an important aspect of algebra, graphing involves creating a visual representation of algebraic information. This section explains the basics of graphing and presents some of its uses.

Expressions and equations can be graphed on what is known as a coordinate plane (or coordinate grid). A coordinate plane is created using two number lines that cross at their zero points. We call those the axes (that’s “AX-ees,” not the tools you chop trees down with). The horizontal axis is called the \(x\)-axis, and the vertical axis is called the \(y\)-axis. This is an example:

To locate a point using the \(x\)- and \(y\)-coordinates, we use a pair of numbers inside parentheses called an ordered pair. The point where the axes cross is the ordered pair \((0, 0)\).

Look at point \(A\) plotted on the coordinate plane grid below:

Point \(A\) has the coordinates \((4, 5)\). That is the point’s ordered pair. The first number is called the \(x\)-coordinate, and it always comes first. In this example, it tells us that the point is four units to the right of \((0, 0)\) in the \(x\) direction. After moving four units to the right, the second number, the \(y\)-coordinate, tells you that point \(A\) is five units up in the \(y\) direction.

Notice that the numbers along the \(x\) and \(y\) axes get bigger going to the right and up from \((0,0)\), but they decrease into negative numbers as they go to the left and down.

Look at point \(B\). The directions to get there from \((0, 0)\) can be written this way: Go left four spaces and go up five spaces. We write those directions as \((-4,5)\) because \(-4\) means go left and \(+5\) means go up.

Finally, look at point \(C\). To get there from \((0, 0)\) we need to go left to \(-4\) and down to \(-5\). The \((x, y\)) coordinates of point \(C\) are \((-4, -5)\).

Four Quadrants

Graphs are split into four equal sections known as quadrants. These quadrants are numbered as shown below:

Notice how the axis markings are positive and negative. Based on the notation of a given point, you can tell which of the four quadrants it is in:

• Quadrant I: (\(+\),\(+\))
• Quadrant II: (\(-\),\(+\))
• Quadrant III: (\(-\),\(-\))
• Quadrant IV: (\(+\),\(-\))

Consider the graph below and write down the coordinates of each lettered point. Then look below the graph to see if you have the right values:

A \(= (2, 3)\)
B \(= (1,-4)\)
C \(= (-3, 6)\)
D \(= (-6, -3)\)

The Distance Between Two Points

Another important concept when working with graphs is finding the distance between two points. This helps us understand how far apart two locations are on the coordinate plane.

Let’s start with a simple case. Suppose we have the points \((-3, 4)\) and \((6, 4)\). Notice that both points lie on the same horizontal line because they have the same \(y\)-coordinate. Instead of thinking about two directions, we only need to find the horizontal distance between them. In this case, we compare the \(x\)-values:

\[\vert 6 - (-3) \vert = \vert 6 + 3 \vert = \vert 9 \vert = 9\]

It does not matter which order you subtract in, because distance is always positive:

\[\vert -3 - 6 \vert = \vert -9 \vert = 9\]

So, the distance between the two points is nine units.

Now consider a vertical example. Suppose we have the points \((2, -1)\) and \((2, 5)\). Since the \(x\)-values are the same, we find the vertical distance using the \(y\)-values:

\[\vert 5 - (-1) \vert = \vert 6 \vert = 6\]

In this case, the distance between these two points is six units.

In general:

• If two points share the same \(y\)-coordinate, you will find the horizontal distance.
• If two points share the same \(x\)-coordinate, you will find the vertical distance.

In both cases, we use absolute value to make sure the distance is positive.

Graphing a Line

Graphing a line connects algebra with visual understanding. Instead of just working with numbers, you can actually see how values change. It helps you visualize relationships between variables, identify patterns, and make predictions.

A line is made by plotting multiple points and connecting them. These points usually come from an equation, known as a linear equation.

For example, consider this equation:

\[y = 2x + 1\]

To graph this line, we create a table of values:

\[\begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 5 \\ \hline \end{array}\]

Now, we plot these points:

\[(0,1)\] \[(1,3)\] \[(2,5)\]

Once plotted, we’ll draw a straight line through them:

This illustrates an important idea:

A linear equation produces a straight line.

Each point on the line satisfies the equation, meaning if you plug the \(x\)-value into the equation, you will get the correct \(y\)-value.

Scaling Graphs and Data Displays

When making a graph or other form of data display, be sure to choose a reasonable data scale for each axis. If the scale of your intervals is off, the graph may be unreadable. For instance, if you’re measuring the growth of various plants in centimeters but the scale you use has intervals in feet, the difference between the plants might not be obvious on your graph.

Using the data in the table below, what scales would be good to use for time and distance?

Time (Hours)	Distance (Miles)
0	0
1	59
3	177
6	354
7	413
10	590
12	708

Let’s look at the time data first. It ranges from \(0\) to \(12\) hours. If you make each interval equal to \(1\), that will make the graph \(12\) intervals high. That’s reasonable, so let’s go with that. The vertical time scale will be:

\[0, \,1, \,2, \,3, \,4, \,5, \,6, \,7, \,8, \,9, \,10, \,11, \,12\]

If space is a concern, you could just label every other interval:

\[0, \,2, \,4, \,6, \,8, \,10, \,12\]

How about the distance scale? It has a bigger range, from \(0\) to \(708\) miles. We definitely can’t have one interval equal one mile. No graph paper in the world would be large enough for that. How many intervals would be reasonable? Somewhere between \(10\) and \(20\), depending on the graph paper. Let’s try \(15\).

If \(15\) squares have to go from \(0\) to \(708\), we need to divide \(708 \div 15=47.2\) for each interval. That would be an awkward number of miles for each interval, so let’s just round it off to \(50\). So, our horizontal scale will be:

\[0, \,50, \,100, \,150, \,200, \,250,\, 300, \,350,\, 400,\, 450, \,500, \,550, \,600, \,650, \,700, \,750, \,800\]

This scale covers the whole range of distances and it’s reasonably easy to do.

Scaling isn’t an exact science. It’s more a reasonable estimate that will not give you two many or too few intervals. Each situation will be different, so you’ll need to use your best judgement.

Distance and Time

There are many ways in which distance and time are related. In everyday life, we discuss distance in relation to time frequently, as when we talk about miles per hour or feet per second. When discussing these concepts, we can represent the data visually in three different ways.

Suppose a child on a tricycle is pedaling down the sidewalk, and every second he travels four feet. We could show his rate of motion by making a little table showing his total distance traveled in different amounts of time:

\[\begin {array}{|c|c|} \hline \text{Time (Seconds)} & \text{Distance (Feet)}\\ \hline 1 & 4\\ \hline 5 & 20\\ \hline 10 & 40\\ \hline \end{array}\]

A chart like this makes it easy to see how distance and time relate to each other. Does this chart look familiar? It should, because a distance-time chart is essentially just a chart of independent and dependent variables.

Another thing we could do is use the three pairs of numbers in the table to make a distance-time graph, which represents distance along the vertical axis and time along the horizontal axis:

A third way to represent the tricycle’s motion is to write an equation. If you look at the points in the table, you can see that the distance value is always four times the time value. That leads directly to this equation, where \(d\) is distance and \(t\) is time:

\[d=4\ t\]

Note: Later, you will learn more about the concept of speed and how it is measured by dividing distance by time. That builds naturally off of what you’ve just learned here.

Let’s try an example problem that uses distance and time.

Two cars are driving on a highway. Car 1’s journey is described by the chart below. Car 2’s journey is described by the formula \(d=32 \ t\), where \(d\) is measured in feet and \(t\) is measured in seconds. Assuming both cars maintain their rate of motion, which one will travel the farthest in a minute?

Car 1

\[\begin {array}{|c|c|} \hline \text{Time (Seconds)} & \text{Distance (feet)}\\ \hline 1 & 30\\ \hline 2 & 60\\ \hline 3 & 90\\ \hline 4 & 120\\ \hline \end{array}\]

Solution

The question is asking us to compare the rate of motion for two cars; however, that information was given using two different methods. It would be easier to make a comparison if we used the same method. Since we already have a chart for Car 1, let’s make a chart for Car 2. To make this chart, we replace the time (\(t\)) variable, one second at a time:

\[\begin {array}{|c|c|} \hline \text{Time (Seconds)} & \text{Distance (feet)}\\ \hline 1 & 32\\ \hline 2 & 64\\ \hline 3 & 96\\ \hline 4 & 128\\ \hline \end{array}\]

Clearly, Car 2 is traveling a little farther each second than Car 1. We could create a formula for Car 1 and insert 60 (for one minute) for the time variable, but there’s no need. Since we’re told the cars are traveling at a constant rate of motion, we know Car 2 will travel farthest.