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
- What is the equation for the \begin{align*}x-\end{align*}axis?
- What is the equation for the \begin{align*}y-\end{align*}axis?
Write the equations for the graphed lines pictured below.
- \begin{align*}E\end{align*}
- \begin{align*}B\end{align*}
- \begin{align*}C\end{align*}
- \begin{align*}A\end{align*}
- \begin{align*}D\end{align*}
- Graph \begin{align*}x=-7\end{align*}.
- Graph \begin{align*}y=100\end{align*}.
- Graph \begin{align*}y=1/2\end{align*}.