How to calcuate the slope or gradeint usign function only with one point.

Question

Hello I am studying functions deeply and I came up with a question.

I have this function.

$f(x)=3x+2$;

I am reading a book. It says- The graph of $f$ is a single line, passing through the point $(0,2)$ with slope $3$.

I know if $x=0$, then it gives $2$ as the output, then I got my first point $(0,2)$. But without calcuating the second point how can the author say the slope is $3$? Is there any other method?

I know if I calculate the second point by substituting $x=1$ gives $5$ as the output. By using the slope formula, i.e $\frac {y_2-y_1}{x_2-x_1}=3$, I can get the same result. But is there any other way to get the slope value?

Answer 1

I shall show a more general result:

If $f(x) = mx + c$, then the graph passes through $(0,c) $ and has slope $m$.

Evaluating at $ x =0 $ shows that the graph passes through $(0,c) $ .

Now let $ x_1, x_2 $ be any two distinct points, then - by the definition of the slope that you give in your question - we have:

$slope = \frac {(m x_1 +c) -(m x_2 +c) }{x_1 - x_2} = \frac{m x_1 - m x_2}{x_1 - x_2} =m $