Skills Refresher:  Graphing and Slope (3.1 – 3.4) — Intermediate Algebra

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10 Skills Refresher: Graphing and Slope (3.1 – 3.4) — Intermediate Algebra

3.1 Points and Coordinates

Often, to get an idea of the behavior of an equation or some function, a visual representation that displays the solutions to the equation or function in the form of a graph will be made. Before exploring this, it is necessary to review the foundations of a graph. The following is an example of what is called the coordinate plane of a graph.

The plane is divided into four sections by a horizontal number line ( $x$ -axis) and a vertical number line ( $y$ -axis). Where the two lines meet in the centre is called the origin. This centre origin is where $x = 0$ and $y = 0$ and is represented by the ordered pair $(0, 0)$ .

$x$ -axis

For the $x$ -axis, moving to the right from the centre 0, the numbers count up, and $x = 1, 2, 3, 4, 5.$ To the left of the centre 0, the numbers count down, and $x = -1, -2, -3, -4, -5.$

$y$ -axis

Similarly, for the $y$ -axis, moving up from the centre 0, the numbers count up, and $y = 1, 2, 3, 4, 5.$ Moving down from the centre 0, the numbers count down, and $y = -1, -2, -3, -4, -5.$

When identifying points on a graph, a dot is generally used with a set of parentheses following that gives the $x$ -value followed by the $y$ -value. This will look like $(x\text{-value}, y\text{-value})$ or $(x, y)$ and is given the formal name of an ordered pair.

This coordinate system is universally used, with the simplest example being the kind of treasure map that is usually encountered in childhood, or the longitude and latitude system used to identify any position on the Earth. For this system, the $x$ -axis (which represents latitude) is the equator and the $y$ -axis (which represents longitude) or the prime meridian is the line that passes though Greenwich, England. The origin of the Earth’s latitude and longitude (0°, 0°) is a fictional island called “Null Island.”

Representations of the globe that demonstrate latitude and longitude. Long description available.

Latitude and longitude. [Long Description]

A hand-drawn treasure map of an island.

The treasure map of Robert Louis Stevenson made popular by his work Treasure Island. From Cordingly, David (1995). Under the Black Flag: The romance and the reality of life among the pirates. Times Warner, 1996.

Example 3.1.1

Identify the coordinates of the following data points.

$\textbf{A.}$ For the $x$ -coordinate, move 4 to the right from the origin. For the $y$ -coordinate, move 4 up. This gives the final coordinates of (4, 4).

$\textbf{B.}$ For the $x$ -coordinate, stay at the origin. For the $y$ -coordinate, move 2 up. This gives the final coordinates of (0, 2).

$\textbf{C.}$ For the $x$ -coordinate, move 3 to the left from the origin. For the $y$ -coordinate, move 2 up. This gives the final coordinates of (−3, 2).

$\textbf{D.}$ For the $x$ -coordinate, move 2 to the left from the origin. For the $y$ -coordinate, move 4 down. This gives the final coordinates of (−2, −4).

$\textbf{E.}$ For the $x$ -coordinate, move 3 to the right from the origin. For the $y$ -coordinate, move 2 down. This gives the final coordinate of (3, −2).

Example 3.1.2

Graph the points A(3, 2), B(−2, 1), C(3, −4), and D(−2, −3).

The first point, A, is at (3, 2). This means $x = 3$ (3 to the right) and $y = 2$ (up 2). Following these instructions, starting from the origin, results in the correct point.

The second point, B(−2, 1), is left 2 for the $x$ -coordinate and up 1 for the $y$ -coordinate.

The third point, C(3 ,−4), is right 3, down 4.

The fourth point, D(−2, −3), is left 2, down 3.

Long Descriptions

Latitude and longitude long description: Two views of the globe that show the landmark points of the latitude and longitude system.

The first globe demonstrates the lines of latitude. The centre line of latitude is called the equator and represents 0° latitude. It wraps around the centre of the Earth from west to east. The globe shows North and South America, and the equator runs through the northern part of South America. The North Pole is at 90° latitude and the South Pole is at −90° latitude. Positive latitude is above the equator, and negative latitude is below it.

The second globe demonstrates the lines of longitude. The centre line of longitude is called the prime meridian and represents 0° longitude. It wraps around the centre of the Earth from north to south. It passes through Greenwich, England, by convention, as well as parts of France, Spain, and western Africa. Positive longitude is east of the prime meridian, and negative longitude is west of it. [Return to Latitude and longitude]

3.2 Midpoint and Distance Between Points

Finding the Distance Between Two Points

The logic used to find the distance between two data points on a graph involves the construction of a right triangle using the two data points and the Pythagorean theorem $(a^2 + b^2 = c^2)$ to find the distance.

To do this for the two data points $(x_1, y_1)$ and $(x_2, y_2)$ , the distance between these two points $(d)$ will be found using $\Delta x = x_2 - x_1$ and $\Delta y = y_2 - y_1.$

Using the Pythagorean theorem, this will end up looking like:

$d^2 = \Delta x^2 + \Delta y^2$

or, in expanded form:

$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$

On graph paper, this looks like the following. For this illustration, both $\Delta x$ and $\Delta y$ are 7 units long, making the distance $d^2 = 7^2 + 7^2$ or $d^2 = 98$ .

The square root of 98 is approximately 9.899 units long.

Example 3.2.1

Find the distance between the points $(-6,-4)$ and $(6, 5)$ .

Start by identifying which are the two data points $(x_1, y_1)$ and $(x_2, y_2)$ . Let $(x_1, y_1)$ be $(-6,-4)$ and $(x_2, y_2)$ be $(6, 5)$ .

Now:

$\Delta x^2 = (x_2 - x_1)^2$ or $[6 - (-6)]^2$ and $\Delta y^2 = (y_2 - y_1)^2$ or $[5 - (-4)]^2$ .

This means that

$d^2 = [6 - (-6)]^2 + [5 - (-4)]^2$

or

$d^2 = [12]^2 + [9]^2$

which reduces to

$d^2 = 144 + 81$

or

$d^2 = 225$

Taking the square root, the result is $d = 15$ .

Finding the Midway Between Two Points (Midpoint)

The logic used to find the midpoint between two data points $(x_1, y_1)$ and $(x_2, y_2)$ on a graph involves finding the average values of the $x$ data points $(x_1, x_2)$ and the of the $y$ data points $(y_1, y_2)$ . The averages are found by adding both data points together and dividing them by $2$ .

In an equation, this looks like:

$x_{\text{mid}}=\dfrac{x_2+x_1}{2}$ and $y_{\text{mid}}=\dfrac{y_2+y_1}{2}$

Example 3.2.2

Find the midpoint between the points $(-2, 3)$ and $(6, 9)$ .

We start by adding the two $x$ data points $(x_1 + x_2)$ and then dividing this result by 2.

$x_{\text{mid}} = \dfrac{(-2 + 6)}{2}$

or

$\dfrac{4}{2} = 2$

The midpoint’s $y$ -coordinate is found by adding the two $y$ data points $(y_1 + y_2)$ and then dividing this result by 2.

$y_{\text{mid}} = \dfrac{(9 + 3)}{2}$

or

$\dfrac{12}{2} = 6$

The midpoint between the points $(-2, 3)$ and $(6, 9)$ is at the data point $(2, 6)$ .

3.3 Slopes and Their Graphs

Another important property of any line or linear function is slope. Slope is a measure of steepness and indicates in some situations how fast something is changing—specifically, its rate of change. A line with a large slope, such as 10, is very steep. A line with a small slope, such as $\dfrac{1}{10},$ is very flat or nearly level. Lines that rise from left to right are called positive slopes and lines that sink are called negative slopes. Slope can also be used to describe the direction of a line. A line that goes up as it moves from from left to right is described as having a positive slope whereas a line that goes downward has a negative slope. Slope, therefore, will define a line as rising or falling.

Slopes in real life have significance. For instance, roads with slopes that are potentially dangerous often carry warning signs. For steep slopes that are rising, extra slow moving lanes are generally provided for large trucks. For roads that have steep down slopes, runaway lanes are often provided for vehicles that lose their ability to brake.

Caution signs for steep upward and downward slopes.

When quantifying slope, use the measure of the rise of the line divided by its run. The symbol that represents slope is the letter $m,$ which has unknown origins. Its first recorded usage is in an 1844 text by Matthew O’Brian, “A Treatise on Plane Co-Ordinate Geometry,”^[1] which was quickly followed by George Salmon’s “A Treatise on Conic Sections” (1848), in which he used $m$ in the equation $y = mx + b.$

$\text{slope }=\dfrac{\text{rise of the line}}{\text{run of the line}}$

Since the rise of a line is shown by the change in the $y$ -value and the run is shown by the change in the $x$ -value, this equation is shortened to:

$m =\dfrac{\Delta y}{\Delta x},\text{ where }\Delta\text{ is the symbol for change and means final value } - \text{ initial value}$

This equation is often expanded to:

$m=\dfrac{y_2-y_1}{x_2-x_1}$

Example 3.3.1

Find the slope of the following line.

First, choose two points on the line on this graph. Any points can be chosen, but they should fall on one of the corner grids. These are labelled $(x_1, y_1)$ and $(x_2, y_2).$

To find the slope of this line, consider the rise, or vertical change, and the run, or horizontal change. Observe in this example that the $\Delta y$ -value (the rise) goes from 4 to −2.

Therefore, $\Delta y = y_2 - y_1$ , or (4 − −2), which equals (4 + 2), or 6.

The $\Delta x$ -value (the run) goes from −2 to 4.

Therefore, $\Delta x = x_2 - x_1$ , or (−2 − 4), which equals (−2 + −4), or −6.

This means the slope of this line is $m=\dfrac{\Delta y}{\Delta x}$ , or $\dfrac{6}{-6}$ , or −1.

$m = -1$

Example 3.3.2

Find the slope of the following line.

First, choose two points on the line on this graph. Any points can be chosen, but to fall on a corner grid, they should be on opposite sides of the graph. These are $(x_1, y_1)$ and $(x_2, y_2).$

To find the slope of this line, consider the rise, or vertical change, and the run, or horizontal change. Observe in this example that the $\Delta y$ -value (the rise) goes from −4 to 1.

Therefore, $\Delta y = y_2 - y_1$ , or (1 − −4), which equals 5.

The $\Delta x$ -value (the run) goes from −6 to 6.

Therefore, $\Delta x = x_2 - x_1$ or (6 − −6), which equals 12.

This means the slope of this line is $m=\dfrac{\Delta y}{\Delta x}$ , or $\dfrac{5}{12}$ , which cannot be further simplified.

$m = \dfrac{5}{12}$

There are two special lines that have unique slopes that must be noted: lines with slopes equal to zero and slopes that are undefined.

Undefined slopes arise when the line on the graph is vertical, going straight up and down. In this case, $\Delta x=0,$ which means that zero is divided by while calculating the slope, which makes it undefined.

Zero slopes are flat, horizontal lines that do not rise or fall; therefore, $\Delta y=0.$ In this case, the slope is simply 0.

Most often, the slope of the line must be found using data points rather than graphs. In this case, two data points are generally given, and the slope $m$ is found by dividing $\Delta y$ by $\Delta x.$ This is usually done using the expanded slope equation of:

$m=\dfrac{y_2-y_1}{x_2-x_1}$

Example 3.3.3

Find the slope of a line that would connect the data points $(-4, 3)$ and $(2, -9)$ .

Choose Point 1 to be $(-4, 3)$ and Point 2 to be $(2, -9)$ .

$\begin{array}{l} m=\dfrac{y_2-y_1}{x_2-x_1} \\ \\ m=\dfrac{-9-3}{2-4} \\ \\ m=\dfrac{-12}{6} \text{ or } -2 \end{array}$

Example 3.3.4

Find the slope of a line that would connect the data points $(-5, 3)$ and $(2, 3)$ .

Choose Point 1 to be $(-5, 3)$ and Point 2 to be $(2, 3)$ .

$\begin{array}{l} m=\dfrac{y_2-y_1}{x_2-x_1} \\ \\ m=\dfrac{3-3}{2-5} \\ \\ m=\dfrac{0}{7} \text{ or } 0 \end{array}$

This is an example of a flat, horizontal line.

Example 3.3.5

Find the slope of a line that would connect the data points $(4, 3)$ and $(4, -5)$ .

Choose Point 1 to be $(4, 3)$ and Point 2 to be $(4, -5)$ .

$\begin{array}{l} m=\dfrac{y_2-y_1}{x_2-x_1} \\ \\ m=\dfrac{-5-3}{4-4} \\ \\ m=\dfrac{-8}{0} \text{ or undefined} \end{array}$

This is a vertical line.

Example 3.3.6

Find the slope of a line that would connect the data points $(-4, -3)$ and $(2, 6)$ .

Choose Point 1 to be $(-4, -3)$ and Point 2 to be $(2, 6)$ .

$\begin{array}{l} m=\dfrac{y_2-y_1}{x_2-x_1} \\ \\ m=\dfrac{6-3}{2-4} \\ \\ m=\dfrac{9}{6} \text{ or } \dfrac{3}{2} \end{array}$

Derivation of Slope: https://services.math.duke.edu//education/webfeats/Slope/Slopederiv.html ↵

3.4 Graphing Linear Equations

There are two common procedures that are used to draw the line represented by a linear equation. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation.

If the equation is given in the form $y = mx + b$ , then $m$ gives the rise over run value and the value $b$ gives the point where the line crosses the $y$ -axis, also known as the $y$ -intercept.

Example 3.4.1

Given the following equations, identify the slope and the $y$ -intercept.

$\begin{array}{lll} y = 2x - 3\hspace{0.14in} & \text{Slope }(m)=2\hspace{0.1in}&y\text{-intercept } (b)=-3 \end{array}$
$\begin{array}{lll} y = \dfrac{1}{2}x - 1\hspace{0.08in} & \text{Slope }(m)=\dfrac{1}{2}\hspace{0.1in}&y\text{-intercept } (b)=-1 \end{array}$
$\begin{array}{lll} y = -3x + 4 & \text{Slope }(m)=-3 &y\text{-intercept } (b)=4 \end{array}$
$\begin{array}{lll} y = \dfrac{2}{3}x\hspace{0.34in} & \text{Slope }(m)=\dfrac{2}{3}\hspace{0.1in} &y\text{-intercept } (b)=0 \end{array}$

When graphing a linear equation using the slope-intercept method, start by using the value given for the $y$ -intercept. After this point is marked, then identify other points using the slope.

This is shown in the following example.

Example 3.4.2

Graph the equation $y = 2x - 3$ .

First, place a dot on the $y$ -intercept, $y = -3$ , which is placed on the coordinate $(0, -3).$

Now, place the next dot using the slope of 2.

A slope of 2 means that the line rises 2 for every 1 across.

Simply, $m = 2$ is the same as $m = \dfrac{2}{1}$ , where $\Delta y = 2$ and $\Delta x = 1$ .

Placing these points on the graph becomes a simple counting exercise, which is done as follows:

For m = 2, go up 2 and forward 1 from each point.

Once several dots have been drawn, draw a line through them, like so:

Note that dots can also be drawn in the reverse of what has been drawn here.

Slope is 2 when rise over run is $\dfrac{2}{1}$ or $\dfrac{-2}{-1}$ , which would be drawn as follows:

For m = 2, go down 2 and back 1 from each point.

Example 3.4.3

Graph the equation $y = \dfrac{2}{3}x$ .

First, place a dot on the $y$ -intercept, $(0, 0)$ .

Now, place the dots according to the slope, $\dfrac{2}{3}$ .

When m = 2 over 3, go up 2 and forward 3 to get the next point.

This will generate the following set of dots on the graph. All that remains is to draw a line through the dots.

Line with slope 2 over 3. Passes through (−3, −2), (0, 0), (3, 2), and (6, 4).

The second method of drawing lines represented by linear equations and functions is to identify the two intercepts of the linear equation. Specifically, find $x$ when $y = 0$ and find $y$ when $x = 0$ .

Example 3.4.4

Graph the equation $2x + y = 6$ .