Graphing Horizontal and Vertical Lines

We have been given the equation $$\begin{array}{c}y=3x-y=\frac{1}{4}\end{array}$$ and asked to graph a line to match the function. Let's first identify our $y$-intercept, the value for $y$ when $x=0\text{:}$ $$\begin{array}{c}y=3\cdot 0-\frac{1}{4}\Rightarrow y=\text{-}\frac{1}{4}\end{array}$$ Now we know that $(0,\text{-}\frac{1}{4})$ is part of our graphed line. Next, we should consider $3$ from the equation. What does $3$ tell us? It tells us that with each new $x$ value, the line increases by $3$ units. This is the slope of the line. We can also think of it like this: $$\begin{array}{c}m=\frac{3}{1}=\frac{\Delta y}{\Delta x},\end{array}$$ every time we move by $1$ unit to the right, the line increases by $3$ units. Let's graph the line.

Unlike Part A, there is no way to find the $y$-intercept for this line. We can choose for our function to start anywhere along the $y$-axis. What we do know is that, the line will decrease by $3$ units for every $9$ units we move to the right. Our slope can be written as: $$\begin{array}{c}\frac{\Delta y}{\Delta x}=\frac{\text{-}3}{9}\Rightarrow \text{-}\frac{1}{3}=m\end{array}$$ Let's assume that the $y$-intercept is $0$ and graph our line.

This function, $g(x)=x+3.5,$ will be graphed in a very similar way to the equation in Part A. This is because, although it is written with function notation, we can still find the $y$-intercept and the slope. Recall that the $y$-intercept it the value of the function when $x=0.$ Here we have: $$\begin{array}{c}g(0)=0+3.5\Rightarrow g(0)=3.5.\end{array}$$ From the function, we can also see that for each new value of $x,$ there is only an increase by $x.$ In other words: $$\begin{array}{c}m=1\Rightarrow \frac{1}{1}=\frac{\Delta y}{\Delta x},\end{array}$$ every time we move $1$ unit to the right, our function increases by $1$ unit. Let's graph this line.

We are not given information that would help us find the $y\text{-}$intercept for this line. We can, however, use the given $\Delta x$ and $\Delta y$ to find the slope of the line. $$\begin{array}{c}\text{slope}=\frac{\Delta y}{\Delta x}=\frac{5-1}{4-4}=\frac{4}{0}\end{array}$$ But wait! We can't divide by zero, that's just not possible. This means that our line is not a function. There is no change in $x,$ so we have a vertical line. Let's see what this looks like in a coordinate plane.