A question asks to compute $$\int_{0}^{1} f(x) \; dx$$ given a piecewise defined function: $$\begin{equation} f(x)= \begin{cases} x, & \text{for}\;\ 0\leq x \leq c \\ c\dfrac{x-1}{c-1}, & \text{for}\; \ c \leq x \leq 1 \end{cases} \end{equation}$$
I know how to integrate it, but then the question asks to draw the graph of $f(x)$, I am confused regarding this part.. how do I graph this exactly? The variable $c$ is what confuses me. It would be very helpful if someone shows me how to graph this.
On
$c$ is a parameter that tells you where the function changes behaviour. Since it's variable, when you draw the graph you pick an arbitrary value of $c$ - the graph still looks quite similar for many valid values of $c$, so just pick one somewhere near $c = 0.5$ and draw that. If the graph did change significantly, you might want to draw a few different graphs demonstrating the different behaviours.
Since $x\mapsto x$ is linear, the left part of the graph is just the line segment that goes from $(0,0)$ to $(c,c)$. The map $x\mapsto c\frac{x-1}{c-1}$ is an affine map, So, its graph is a line segment too. Since $c\mapsto c$ and since $1\mapsto 0$, its graph is the line segment from $(c,c)$ to $(1,0)$.
So, you get the broken line that goes from $(0,0)$ to $(c,c)$ and then from $(c,c)$ to $(1,0)$.