Determining the Slope of a Line

In the lyrics of We Will Graph You we can find some information regarding the line we want to draw. The words a line is $4$ when $x$ is nil we read as $y=4$ when $x=0.$ Therefore, the graph has its $y$-intercept at $y=4.$ We have now found one point on the graph.

The next line, and decrease to $2$ when $x$ is $3$, gives us another point on the line, $(3,2).$ Let's mark this point in the diagram as well.

Next Brian tells us to draw the graph. We will, we will graph it by connecting the points with a line.

Brian May next puts us under pressure to find the slope of this line. But we are champions so we go ahead and present a formula that will help us calculate the slope . $m=\dfrac{\Delta y}{\Delta x}$ The points we used when we drew the line can be helpful in finding $\Delta y$ and $\Delta x.$

We will now substitute $\text{-} 2$ and $3$ for $\Delta y$ and $\Delta x$ in the formula and, like a kind of magic, we find the slope. $m=\dfrac{\Delta y}{\Delta x} \quad \Rightarrow \quad m=\dfrac{\text{-} 2}{3}=\text{-} \dfrac{2}{3}$ The final line before the chorus tells us to find its $x$ and $y$-intercepts. We already know the $y$-intercept, $(0, 4).$ The $x\text{-}$intercept we can identify by looking at the graph.

The line intercepts the $x\text{-}$axis at $(6,0).$ Therefore, the $x\text{-}$intercept is $6.$ Let's summarize what we have found about this graph. $\begin{aligned} \text{Slope: } & m=\text{-} \dfrac{2}{3} \\ x\text{-intercept: } & (6,0) \\ y\text{-intercept: } & (0,4) \end{aligned}$