Question

How do you graph $$y=2x-9$$?

Answer 1

Hint: Suppose an equation of straight line to be $$y=ax+b$$. We can draw the graph of $$y=ax+b$$ from the simple graph $$y=x$$. We need to modify the $$y=x$$ graph by shifting and scaling methods. It is a better idea to modify the graph of $$y=x$$ in such a manner that we get the required graph by going from left side to right side of the equation $$y=2x-9$$.

As per the given question, we need to graph a straight line which is given by the equation $$y=2x-9$$.
A straight line can be traced out on the cartesian plane by just two points lying on it. We can also use a third point for sort of check. It is very simple to graph the $$y=x$$ line as it is symmetric to both x and y axes.

The graph of $$y=x$$ is as shown in below figure:

If we go from left hand side to right hand side of the equation $$y=2x-9$$, it is clear that we need to first scale the $$y=x$$ graph by a factor 2. Then we get, $$y=2x$$.
And the graph of $$y=2x$$ is as shown in the below figure:

Now, we need to shift the $$y=2x$$ graph right hand side by $${\displaystyle \frac{9}{2}}$$ units to get the required straight line $$y=2x-9$$. And the graph of $$y=2x-9$$ is shown in the below figure:

$$\therefore$$ We have to compress $$y=x$$ by 2 and then shift it to the right hand side by $${\displaystyle \frac{9}{2}}$$ units to get the desired line $$y=2x-9$$.

Note:
We can trace the graph of $$y=2x-9$$ by substitution by any two random values of x. We can also trace the graph by going from the right hand side to left hand side of the straight-line equation $$y=2x-9$$. So, by taking 2 common, we get $$y=2(x-{\displaystyle \frac{9}{2}})$$. That is, we have to shift the $$y=x$$ graph by $${\displaystyle \frac{9}{2}}$$ units and then compress it by a factor 2 to get the graph of $$y=2x-9$$.