# How to determine the shape and sketch curve by hand for products of polynomials and exponentials

How to:

• Determine the shape of,
• and sketch the curve (Only x/y axis intercept points are required for labeling)

for equations in the format of:

$$f(x) = a_1x^{b_1}\boldsymbol{e}^{c_1 x} +\ ...\ + a_nx^{b_n}\boldsymbol{e}^{c_n x}$$

where $a_i \neq 0$, $b_i, c_i \in \mathbb{R}$.

Some examples of such equation:

$$y = 20\ \boldsymbol{e}^{-{x\over2}} -20\ \boldsymbol{e}^{2x\over 5}+2 x\ \boldsymbol{e}^{-{x\over2}}$$

$$y = \boldsymbol{e}^{-3x} + 3x\ \boldsymbol{e}^{-3x} + 25x^2\ \boldsymbol{e}^{-3x}$$

• the 1st example you give is "out-of-shape" vs, the question you pose (why $e^{2x/5}$ ?) – G Cab Oct 29 '16 at 14:32
• @GCab The coefficient $c_i$ of $x$ in $e^{c_ix}$ may vary in each term. – Eana Hufwe Oct 29 '16 at 23:11