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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? In this Concept, you'll learn about horizontal and vertical lines so that you'll be able to create a graph for this type of scenario.

Guidance

Not all graphs are slanted or oblique. Some are horizontal or vertical. Read through the next situation to see why.

Example A

“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*}.

Solution: 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.

Example B

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

Solution:

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*}

Example C

Graph the following lines.

(a) \begin{align*}y=4\end{align*}

(b) \begin{align*}y=-4\end{align*}

(c) \begin{align*}x=4\end{align*}

(d) \begin{align*}x=-4\end{align*}

Solution:

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

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

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

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

Guided Practice

Graph the following:

1. \begin{align*}y=-3\end{align*}

2. \begin{align*} x=5 \end{align*}

Solutions:

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:

Practice

  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*}.

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