Did you ever think that a four hundred year-old spider may be why we study linear relationships today?

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Jodie Hutchinson
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1 Show Me: Determine if a Function is Linear M8221 Did you ever think that a four hundred year-old spider may be why we study linear relationships today? Supposedly, while lying in bed Rene Descartes noticed a spider walking on the ceiling. This inspired Descartes to invent coordinate geometry. Today, we learn about ordered pairs of numbers, equations, and graphs, all thanks to Descartes. So, let s take a look at Descartes s discoveries and see how these mathematical ideas are related. Let s start with two ordered pairs like zero, three and one, five. The coordinates of each ordered pair can be written in a table. These two ordered pairs can also be plotted in a coordinate plane. Since lines can also be drawn in a coordinate plane, how many different lines can we draw that would pass through both points? A line like this one is a function because A function is a set of ordered pairs in which each first coordinate X is paired with one and only one second coordinate Y.

2 Since all points on this line have a different value of x as their first coordinate, and it is the graph of a straight line, the points in this table, and this graph, determine a linear function. Notice that the word linear has the word line in it. Let s look at linear functions more closely. Here, the points negative one, two and negative one, negative one, are plotted in the coordinate plane. Exactly one line can be drawn through these two points. Click points on this line to see their coordinates. What do you notice about the coordinates of each point on this line? When you look at this table of ordered pairs, you can see that all of the X coordinates are equal to negative one. But the Y coordinates are different. Therefore, according to the definition of a function, this line, a vertical line, is not a function. So, even if these two points determine a line, they do not determine a linear function. Here s an example. Which table contains ordered pairs that determine a line but do not determine a linear function? Click here to hear the solution.

3 The answer is D because the X value, negative 2, has two Y values, negative 2 and 2. So, this is not a function. Remember, the definition of a function says that each x value may have one and only one y value. In all of the other tables, each x value has only one y value. Which table contains ordered pairs that determine a line but do not determine a linear function? The two points in this table determine a line. Click the line to see coordinates of the highlighted point. The coordinates of this point are one, zero. The line is not a vertical line because no two points have the same X coordinate. Therefore, it is a linear function. We can also say that this table that contains its ordered pairs determines a linear function. The equation of the line is Y equals X minus one and can be used to describe all the coordinates on the line and in the table. You can see this by substituting the value of X into the expression X minus one and then simplifying the result. Zero minus one is negative one. Two minus one is one. One minus one is zero. In each case, the result after simplifying the expression is the value of Y in the table.

4 These points determine a linear function because the coordinates of the points satisfy the equation of the line. So, they lie on the line. Substitute the points into the equation to determine which point does not lie on this line. Let s plot these three points in the coordinate plane. These points determine a function because no value of X in the table appears more than once. However, you can see that it is not possible to draw exactly one straight line through all three points. So, these points do not determine a linear function. So, a table, and its graph, can represent a function because each value of X is paired with exactly one value of Y, but not represent a linear function. Plot the three ordered pairs in this table. Click on the add button and then click on its location in the plane. Like this: Now, plot the other two points Click on each ordered pair in the table and then click the location of each point in the plane. Here s an example. Graph the points in this table to determine if they appear to form a linear function.

5 Click here to hear the solution. The graph shows that it is not possible to draw one line through all three of these points. So, the answer is B. The values in this table do not determine a linear function. Which statement best describes the set of points plotted in the coordinate plane? This time, let s start with the equation Y equals two X minus 4 and see how to determine whether it represents a linear function. We can use this equation to create a table of values. The first step is to start with a value for x, say negative one. To find the value of y when X equals negative one, substitute negative one for X in the expression two X minus four, like this. So, we get two times negative one minus four, which equals negative six. Now, find two more ordered pairs that satisfy this equation. What are the values of Y, when X equals zero and when X equals one? These three ordered pairs can be plotted in the coordinate plane. And the three points appear to lie on one non-vertical line. So, it appears that the equation Y equals two X minus four represents a linear function.

6 Is Y equals X squared a linear function? Click Solution to hear the answer. Y equals X squared is not a linear function. To determine if y equals X squared is linear, we generate a table of values. Completing the table of values using substitution, gives us three points that satisfy the equation. Here, the values in the table represent a function. After plotting the points, it is clear that there is no way to draw one straight line through these three points, so the equation is not a linear function. Is Y equals 1 minus X a linear function? Another way to determine if a function is linear is by using slopes. The slopes between different pairs of points on a linear function are always equal. The three points in the table and plotted in the coordinate plane appear to lie on the same line. But let s check the slopes to be sure. The slope between the points negative six, negative two and negative one, three is the ratio of the rise to the run between the two points. The rise is positive five, and the run is positive five.

7 So, the slope of the line between these two points is five over five, or one. Now let s find the slope between these two points. We can also calculate slope by using the slope formula. The slope formula is Y two minus Y one divided by X two minus X one. Here, Y two is five, Y one is three, X two is one and X one is negative one. The slope is five minus 3 divided by one minus negative one which equals two divided by two. So, the slope of the line between these two points is 1. Because both slopes are one, all three points in the table must fall on the same line. We can tell from the graph that it is not a vertical line, so this table represents a linear function. Complete this statement about a linear function and its slope. Sometimes when you plot the points in a table of values, it might look like the points lie on the same line, when in fact they do not. The slope between negative three, three and one, six, calculated using the slope formula is three-fourths. On the other hand, the slope between one, six and four eight is two-thirds.

8 So, the slopes between the two pairs of points are not the same. In fact, if we were to draw the line between the first pair of points and the line between the second pair of points on the same graph, we could see that they are not exactly the same line. True or False? This table determines a linear function. Click Solution for the answer. This table does not determine a linear function. The slope between the first two points is two. The slope between the second pair of points is three. These two slopes are not the same, so all three points cannot lie on the same line. Now, you try one. True or False? This table determines a linear function. In this lesson, you have learned how to tell if a table of values, a graph, or an equation determine a linear function. There is exactly one line that can be drawn through two points. If the line is not vertical, like this one, then the two points determine a linear function. If the two points have the same X value, then you can draw a line but the points do not determine a linear function. Every linear function has an equation.

9 You can decide if an equation is linear by looking at points that satisfy the equation. If the coordinates that satisfy an equation lie in exactly one non-vertical straight line, then the equation represents a linear function. If the coordinates that satisfy an equation do not lie on one non-vertical line, the equation does not represent a linear function. And finally, if the slopes between all of the pairs of points in a table are equal, the table of values determines a linear function. If you d like to review this activity again, click Review. If you re ready to exit, click Done.

Section Graphs and Lines

Section Graphs and Lines 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