Question

Y=3x^4 -6x^2 + 3

How do I graph this polynomial function? Im having a hard time understanding with all the exponents so simple answers would be appreciated.

Answer 1

A graph is a collection of (x,y) points on a grid. Often these points are connected with some kind of line or curve. In this case it's a curve.

The idea is to pick x values at random to plug into this equation, so we can find paired y values. Let's pick x = 0.

y = 3x^4 -6x^2 + 3

y = 3*0^4 -6*0^2 + 3

y = 3*0 - 6*0 + 3

y = 0 - 0 + 3

y = 3

You can use scratch paper or a calculator to evaluate what's shown above. The input x = 0 leads to the output y = 3. Therefore (0,3) is one point on this curve. It's the y-intercept.

Now try something like x = 1.

y = 3x^4 -6x^2 + 3

y = 3*1^4 -6*1^2 + 3

y = 3*1 - 6*1 + 3

y = 3 - 6 + 3

y = -3 + 3

y = 0

Therefore the point (1,0) is on this curve. It's one of the two x-intercepts. The other x-intercept is at (-1,0).

This process is repeated for other x values. The idea is that the more points you plot, the more accurate the curve.

Here's a table of some x,y values.

`\begin{array} \cline{1-2}\text{x} & \text{y}\\\cline{1-2}-2 & 27\\\cline{1-2}-1 & 0\\\cline{1-2}0 & 3\\\cline{1-2}1 & 0\\\cline{1-2}2 & 27\\\cline{1-2}\end{array}`

Each row of this table refers to a different (x,y) point on this curve. The graph is shown below. I used GeoGebra to make the graph. It's completely free. Desmos is another useful tool. If you prefer a graphing calculator like a TI83 or TI84, then it's probably best to stick to that.