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How can you predict what the function

$$f(x) = \frac{(x - 5)(x + 4)(x - 3)^2(x)}{(x-5)(x)}$$

looks like before you plot it?

Amzoti
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user117520
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2 Answers2

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Observations:

$$f(x) = \frac{(x - 5)(x + 4)(x - 3)^2(x)}{(x-5)(x)} = (x+4)(x-3)^2,\;\;x\neq 5, x\neq 0$$The function is not defined at $x = 0, x= 5$. Why not? These are called, however, removable discontinuities, and do not impact the graph of the function $f(x) = (x + 4)(x-3)^2, $ save for there being "holes" at $x = 5,\text{ and } x = 0$

$f(x)$ intersects the x axis at $x = -4, x = 3$. Why? We call these values of $x$ "zeros" of the function.

The polynomial we are left with is a cubic polynomial.

We can also take the derivative to find critical points; this will help immensely when graphing: it allows us to determine where, if anywhere, there are local minimums, local maximums, etc. To find these, we compute $f'(x)$ and solve for $x$ when $f'(x) = 0$.

amWhy
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    How would taking the derivative help to find critical points? – user117520 Dec 24 '13 at 15:19
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    When we take the derivative and put $f'(x) = 0$, the solutions give us the critical points: the values of $x$ at which $f(x)$ reaches local maximums and minimums (the peaks and valleys of the curve.) Note that when $f'(x) = 0$, the tangent lines to the curve have slope $0$, which means that the function, at those points, is changes between increasing to decreasing, or decreasing to increasing: I.e., the tangent lines are horizontal, like lines balancing at the top of a peak, or at the bottom of a valley in the curve. – amWhy Dec 24 '13 at 15:22
  • Note, it's also possible that $f'(x)=0$ but $f$ does not attain a local max or min at $x$. To check this, one should first check $f''(x)$. If $f''(x)=0$ then one should check values close enough to $x$ that there are no other critical points near by. – Dan Rust Dec 24 '13 at 15:26
  • @amWhy: Needs another UV +1 – Amzoti Dec 25 '13 at 00:33
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Hint: When is the denominator $0$? When is the numerator $0$ and what does this mean for the graph (you should consider limits of $f$ as $x$ tends to these zeros)? What are the turning points of a function and how can you find them, and their characteristics (minimum, maximum, saddle point)? Before you do any of this though, you should make sure nothing cancels (removable discontinuities).

Dan Rust
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