Whenever a function has a zero, its graph touches the x-axis. In those cases, one of two things can happen. Either the graph can cross the x-axis, or it can touch the x-axis and not cross over it.
1) Lets look at some examples of different functions. Each of the following functions has a zero located at x = 3. For each function, determine if the function’s graph crosses or only touches the x-axis at x = 3.
- g(x) = x(x+2)(x-3)
2) . Do some additional examples for yourself using functions you create yourself by multiplying two or three linear factors so that there are clear zeros. Feel free to use squared or cubed linear factors like f(x) and h(x) above. Is there a pattern to when the graph of the function only touches the x-axis and when it actually crosses it?
3)The degree of a polynomial can be found by fully expanding the polynomial. Once this is done, look at the variable in the polynomial with the largest exponent. This exponent is the degree of the polynomial. The leading coefficient of a polynomial is the coefficient of the term with the largest exponent. Look at the graphs of each of the examples you have worked with so far and expand them using multiplication. Do you notice a pattern of when a graph starts below or above the x-axis? Do you notice a pattern when a graph ends below or above the x-axis?
4)Create a function that does the following: • It touches the x-axis at two points, x = -3 and x = 1 • It begins above the x-axis • It ends below the x-axis.