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Horizontal and Vertical Line Graphs

Line y = # is parallel to x-axis and line x = # is parallel to y-axis

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Horizontal and Vertical Line Graphs

Suppose you're at an all-you-can-eat pancake house where you can pay $8.99 and have all the pancakes you want. What if you graphed the number of pancakes you ate along the \begin{align*}x-\end{align*}axis and the amount of money you have to pay along the \begin{align*}y-\end{align*}axis. Would the line representing this situation be horizontal or vertical? 

Horizontal and Vertical Line Graphs

Not all graphs are slanted or oblique. Some are horizontal or vertical. 

Let's graph the following situations and equations:

  1. “Mad-cabs” have an unusual offer going on. They are charging $7.50 for a taxi ride of any length within the city limits. Graph the function that relates the cost of hiring the taxi \begin{align*}(y)\end{align*} to the length of the journey in miles \begin{align*}(x)\end{align*}.

No matter the mileage, your cab fare will be $7.50. To see this visually, create a graph. You can also create a table to visualize the situation.

# of miles \begin{align*}(x)\end{align*} Cost \begin{align*}(y)\end{align*}
0 7.50
10 7.50
15 7.50
25 7.50
60 7.50

Because the mileage can be anything, the equation should relate only to the restricted value, in this case, \begin{align*}y\end{align*}. The equation that represents this situation is:

\begin{align*}y=7.50\end{align*}

Whenever there is an equation of the form \begin{align*}y= \text{constant}\end{align*}, the graph is a horizontal line that intercepts the \begin{align*}y-\end{align*}axis at the value of the constant.

Similarly, if there is an equation of the form \begin{align*}x=\text{constant}\end{align*}, the graph is a vertical line that intercepts the \begin{align*}x-\end{align*}axis at the value of the constant. Notice that this is a relation but not a function because each \begin{align*}x\end{align*} value (there’s only one in this case) corresponds to many (actually an infinite number of) \begin{align*}y\end{align*} values.

  1. Graph \begin{align*}y=-2\end{align*} by making a table and plotting the points.

Notice that there is no \begin{align*}x\end{align*} in this equation. So, no matter what the value of \begin{align*}x\end{align*} is, \begin{align*}y\end{align*} will always be -2. A table of points on this line will look like the following:

\begin{align*}(x)\end{align*} \begin{align*}(y)\end{align*}
\begin{align*}-2\end{align*} \begin{align*}-2\end{align*}
\begin{align*}-1\end{align*} \begin{align*}-2\end{align*}
\begin{align*}0\end{align*} \begin{align*}-2\end{align*}
\begin{align*}1\end{align*} \begin{align*}-2\end{align*}
\begin{align*}2\end{align*} \begin{align*}-2\end{align*}

  1. \begin{align*}y=4\end{align*}

\begin{align*}y=4\end{align*} is a horizontal line that crosses the \begin{align*}y-\end{align*}axis at 4.

  1. \begin{align*}y=-4\end{align*}

\begin{align*}y=-4\end{align*} is a horizontal line that crosses the \begin{align*}y-\end{align*}axis at –4.

  1. \begin{align*}x=4\end{align*}

\begin{align*}x=4\end{align*} is a vertical line that crosses the \begin{align*}x-\end{align*}axis at 4.

  1. \begin{align*}x=-4\end{align*}

\begin{align*}x=-4\end{align*} is a vertical line that crosses the \begin{align*}x-\end{align*}axis at –4.

   

 

Examples

Example 1

Earlier, you were told that you're at an all-you-can-eat pancake house where you can pay $8.99 and have all the pancakes you want. You were asked if the line representing how much the number of pancakes you ate along the \begin{align*}x-\end{align*}axis and the amount of money you have to pay along the \begin{align*}y-\end{align*}axis would be horizontal or vertical.

Similar to problem 1 about the taxi from above, the line that would represent this situation would be a horizontal line. No matter what value you are looking at on the \begin{align*}x-\end{align*}axis (the number of pancakes you eat), the \begin{align*}y-\end{align*}value would be the same, $8.99. 

Example 2

Graph \begin{align*}y=-3\end{align*} and \begin{align*} x=5 \end{align*} on the same plot.

The graph of \begin{align*}y=-3\end{align*} is a horizontal line where \begin{align*}y\end{align*} is always equal to 3 no matter what \begin{align*}x\end{align*} is, and the graph of \begin{align*}x=5\end{align*} is a vertical line where \begin{align*}x\end{align*} is always equal to 5 no matter what \begin{align*}y\end{align*} is:

Review

  1. What is the equation for the \begin{align*}x-\end{align*}axis?
  2. What is the equation for the \begin{align*}y-\end{align*}axis?

Write the equations for the graphed lines pictured below.

  1. \begin{align*}E\end{align*}
  2. \begin{align*}B\end{align*}
  3. \begin{align*}C\end{align*}
  4. \begin{align*}A\end{align*}
  5. \begin{align*}D\end{align*}
  1. Graph \begin{align*}x=-7\end{align*}.
  2. Graph \begin{align*}y=100\end{align*}.
  3. Graph \begin{align*}y=1/2\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.3. 

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Vocabulary

oblique graph

A graph is oblique if it is slanted, rather than horizontal or vertical.

Horizontally

Horizontally means written across in rows.

Vertically

Vertically means written up and down in columns.

Image Attributions

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