add in grids…    Graphing and Slope: Workbook Examples/Practice

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7 add in grids… Graphing and Slope: Workbook Examples/Practice

Plotting and Labeling Points

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.

Note: Review the textbook chapter Skills Refresher: Graphing and Slope (3.1 – 3.4) — Intermediate Algebra for details on the skill of plotting points.

Examples/Practice

1.) What are the coordinates of each point on the graph below?

2.) Plot and label the following points on a graph.

$\begin{array}{lll} \text{A }(-5,5)&\text{B }(1,0)&\text{C }(-3,4) \\ \text{D }(-3,0)&\text{E }(-4, 2)&\text{F }(4,-2) \\ \text{G }(-2,-2)&\text{H }(3,-2)&\text{I }(0,3) \end{array}$

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:

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

Examples/Practice

Find the distance between the points.

1.) (−6, −1) and (6, 4)

2.) (1, −4) and (5, −1)

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 $y_{\text{mid}}=\dfrac{y_2+y_1}{2}$

Examples/Practice

Find the midpoint between the points.

1.) (−5, −1) and (3, 5)

2.) (6, −4) and (12, 4)

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.

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

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:

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

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:

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

Examples/Practice

Plot the points and connect. Then find the slope of the line that would connect each pair of points.

1.) (2, 10), (−2, 15)

2.) (1, 2), (−6, −12)

3.) (−5, 10), (0, 0)

4.) (2, −2), (7, 8)

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.