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.

Question

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.

1. f(x)=x(x-3)^2
2. g(x) = x(x+2)(x-3)
3. h(x)=(x-3)^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.

Answer 1

1) Whether each graph crosses or only touches the x-axis at x = 3 depends on the exponent, or multiplicity, of the factor containing the zero.

a) f(x) = x(x - 3)²

The zero x = 3 comes from the factor (x - 3)². Its exponent is 2, which is even. Therefore, the graph touches the x-axis at x = 3 but does not cross it.

b) g(x) = x(x + 2)(x - 3)

The zero x = 3 comes from the factor (x - 3). Its exponent is 1, which is odd. Therefore, the graph crosses the x-axis at x = 3.

c) h(x) = (x - 3)³

The zero x = 3 has an exponent of 3, which is odd. Therefore, the graph crosses the x-axis at x = 3. Since the multiplicity is 3, the graph may flatten slightly as it crosses.

2) Let's cover some additional examples and the pattern. Consider this function:

p(x) = (x - 2)²(x + 1)

Its zeros are:

x = 2, with multiplicity 2

x = -1, with multiplicity 1

1. At x = 2, the graph only touches the x-axis because the multiplicity is even.
2. At x = -1, the graph crosses the x-axis because the multiplicity is odd.

Now lets consider:

q(x) = (x + 4)³(x - 1)²

1. At x = -4, the graph crosses because the multiplicity is 3, which is odd.
2. At x = 1, the graph only touches because the multiplicity is 2, which is even.

The general pattern is:

1. If a zero has an even multiplicity, the graph touches the x-axis and turns around.
2. If a zero has an odd multiplicity, the graph crosses the x-axis.

3) First, lets expand the original functions:

f(x) = x(x - 3)²

= x(x² - 6x + 9)

= x³ - 6x² + 9x

The degree is 3, and the leading coefficient is 1. Since it has an odd degree and a positive leading coefficient, the graph begins below the x-axis on the left and ends above the x-axis on the right.

g(x) = x(x + 2)(x - 3)

= x(x² - x - 6)

= x³ - x² - 6x

The degree is 3, and the leading coefficient is 1. Therefore, the graph begins below the x-axis and ends above the x-axis.

h(x) = (x - 3)³

= (x - 3)(x - 3)(x - 3)

= (x² - 3x - 3x + 9)(x - 3)

= (x² - 6x + 9)(x - 3)

= x³ - 9x² + 27x - 27

The degree is 3, and the leading coefficient is positive 1. Therefore, the graph begins below the x-axis and ends above the x-axis.

These are the end-behavior patterns to look at:

1. Even degree and positive leading coefficient: both ends are above.
2. Even degree and negative leading coefficient: both ends are below.
3. Odd degree and positive leading coefficient: begins below and ends above.
4. Odd degree and negative leading coefficient: begins above and ends below.

4) To make the graph touch the x-axis at x = -3 and x = 1, both zeros need even multiplicities. We can use:

(x + 3)² and (x - 1)²

To make the graph begin above the x-axis and end below it, the function must have an odd degree and a negative leading coefficient. Therefore, we need to include another linear factor:

F(x) = -(x + 3)²(x - 1)²(x - 4)

This function has the following properties:

1. It touches the x-axis at a x = -3 because the multiplicity is 2.
2. It touches the x-axis at x = 1 because the multiplicity is 2.
3. It has degree 5, which is odd.
4. Its leading coefficient is negative.
5. Therefore, it begins above x-axis and ends below the x-axis.

It also crosses the x-axis at x = 4. This additional zero is necessary because a polynomial that begins above and ends below must have an odd degree and must cross the x-axis somewhere.

Answer 2

The general concept is that if the factor has an EVEN exponent, it will touch the x-axis and bounce back.

Otherwise it has an odd exponent and will cross.

EX. y = (x-3)^2 (x+5)^3

will touch the x-axis at x=3 but cross at x=-5