• We will continue to concentrate on knowledge that will assist us in sketching the graph of a polynomial function.
• We will focus on the zeros (or \(x\)-intercepts) of the function.

The zeros of a polynomial function can easily be identified when the function is in factored form.

Consider the quadratic function

In factored form,

Set \( y = 0 \) to find the zeros of the function.

Therefore, \( 2x - 3 = 0 \) or \( x + 5 = 0 \); hence \( x = \dfrac{3}{2}\), and \(-5 \) are the roots.

Once the zeros of the function are identified, a quick sketch of the graph can be made.

The leading coefficient for \( y = 2x^2 + 7x - 15 \) is positive, so the parabola opens upward with end behaviour \( y \rightarrow +\infty \text{ as } x \rightarrow \pm \infty \).

The \(y\)-intercept of \(-15\) (the constant term) is found quickly by setting \( x = 0 \).

Here, the vertex will need to be determined to obtain a more accurate picture of the parabola.

To express \(y = 2x^2 + 7x - 15\) in this form, the leading coefficient, \(2\), can be divided out of the first factor, leaving a fractional value as a term in the factor.

Normally, we do not try to divide out a value that is not a factor of all the terms in the expression we are factoring, but it can be done.

Consider the family of cubic functions defined by

The zeros are at \( x = -2, 1\), and \(3\).

By varying the value of the parameter \( k \), the shape of the function can be changed.

A vertical stretch or compression is applied to the graph.

When \( k \lt 0 \), the function is reflected in the \( x \)-axis and the end behaviour changes.

The zeros do not change.

The quintic function \( f(x) = k(x - s)(x - t)(x - u)(x - v)(x - w) \) has zeros \( x = s, t, u, v\), and \( w \).

These zeros can be any real number and may sometimes be equal in value.

For example,

\( f(x) = k(x - s)(x - s)(x - t)(x - t)(x - t) = k\) \((x - s)^2\) \((x - t)^3\)

The zero \(x = s\) has multiplicity \(2\) and is called a double zero or zero of order \(2\).

The zero \(x = t\) has multiplicity \(3\) and is called a triple zero or zero of order \(3\).

How does the behaviour of the graph of a polynomial function change at its zero as the multiplicity of the zero changes?

Use the worksheet provided to investigate the behaviour of a polynomial function at its zeros.

In particular, note the behaviour of the graph of the function at the zero when the order (or multiplicity) of the zero changes.