Graph the polynomial $f(x)=x^4-3x^3-4x^2+3x-2$ using a graphing calculator

Question

Graph the polynomial $f(x)=x^4-3x^3-4x^2+3x-2$ using a graphing calculator?

a) The Range
b) The Real Zeros
c) The y-intercept
d) relative minimum and Relative Maximum
e) The interval where $f(x)\le0$

Answer 1

Just using Wolfram, you obtain:

At the link, you'll also find a graph using a larger scale.

The point of the exercise is to "get to know" you graphing calculator: zeroing in (zoom in) to see behavior near the origin, otherwise missing maxima, minima, e.g., with a different scale, you might obtain the following, which doesn't look any different than a distorted parabola:

Indeed, you may need to "zoom in" to clarify the behavior even more.

And, the point of the exercise is to learn how approximate (zero in on) the key points (which graphing calculators can do): where, approximately, are there relative minima? A relative (and global) maximum? Where does it (approximately) intersect the $x-axis$ (where is y zero)?, the $y$-axis?, etc.

Note that the interval on which $f(x) < 0 \approx (-1.5163, 3.8681)$. At approximately $x_1 \approx -1.5163, x_2 \approx 3.8681,\;\;y = 0$.

So what's stopping you from using your graphing calculator and/or graphing technology? These are meant to help you! But you must be clear about what it means to be an minimum, maximum, "zero", etc.

Answer 2

If you didn't mistype it, then André is probably correct, and you're just supposed to approximate these things. Your graphing calculater should have tools that allow you to find approximations for the zeroes and the local extrema. You should see that the interval where $f(x)\le0$ is the closed interval with those two zeroes as endpoints. The lesser of the two local minima will be the global minimum of your function, so it should be clear that the range is all numbers $\ge$ the global minimum.

Answer 3

If your calculator is similar to a TI-84+, then perhaps this YouTube video will help. I haven't watched it, but the comments look good.

Answer 4

With the help of the calculator is easy to establish: $$ \begin{array}{c|ccccc} a & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline \\ f(a) & 115 & 16 & -5 & -2 &-5 & -20 & -29 & 10\end{array} $$ This suggests the minimum is close to $-29$. In fact it is $-29.4136$. For very large $|x|$, the fourth power will dominate other terms, therefore the polynomial grows boundlessly for large $|x|$. Thus the range is $(-29.4136, \infty)$.

The tabulation indicates that real zeros are in the intervals $(-2,-1)$ and $(3,4)$. In fact, using calculator (or W|A) you could establish that $$ x^4 -3 x^3 - 4x^2+3x-2 = \left(x-\frac{3}{4} \right)^4 - \frac{59}{8} \left(x-\frac{75}{236} \right)^2 - \frac{23737}{15104} $$ which shows that these are the only real roots, and the polynomial is negative in between them. Approximate values of these roots are $-1.5163$ and $3.86814$