• We will review and explore the behaviour of the graphs of various polynomial functions ranging from degree \(0\) to degree \(6\).
• The focus will be on determining the possible number of turning points, the possible number of \( x \)-intercepts, the end behaviour of the function's graph, and how the end behaviour is influenced by the leading coefficient of the function (the coefficient of the highest degree term).

Let's first discuss some polynomial functions that are familiar to us.

Polynomial functions of degree \(0\) are constant functions of the form \( y = a, a \in \mathbb{R} \).

Their graphs are horizontal lines with a \(y\)-intercept at \( (0,a) \).

The end behaviours of these graphs can be summarized with the statement, “as \( x \rightarrow \pm \infty, y = a \).”

Constant functions have no turning points and no zeros, except for the case of \(y=0\), which is the equation of the \(x\)-axis and therefore has an infinite number of zeros.

Polynomial functions of degree \(1\) are linear functions of the form \( y=ax+b, a \neq 0 \).

We know that \(a\) provides the slope of the line and \(b\) is the \(y\)-intercept.

We also know that lines do not have turning points.

If we consider the graphs of the lines \(y=2x-3\) and \(y=-3x\), we see that a line will have one \(x\)-intercept if the line is not horizontal (as seen previously).

The end behaviours of a linear function are dependent on the slope of the line, which is given by the leading coefficient.

When the slope is positive, the end behaviour can be described in the following way:

• \(y\) approaches a large negative value as \(x\) approaches a large negative value.
• \(y\) will approach a large positive value as \(x\) approaches a large positive value.

The end behaviours of a linear function are dependent on the slope of the line, which is given by the leading coefficient.

When \( a > 0 \),

\( y \rightarrow -\infty \text{ as } x \rightarrow -\infty \)

\( y \rightarrow \infty \text{ as } x \rightarrow \infty \)

When \( a < 0 \),

\( y \rightarrow \infty \text{ as } x \rightarrow -\infty \)

\( y \rightarrow -\infty \text{ as } x \rightarrow \infty \)

The quadratic function, \(y=ax^2+bx+c\), is a polynomial function of degree \(2\).

The graph of a quadratic function (a parabola) has one turning point, which is an absolute maximum or minimum point on the curve.

We can see from the graphs of the quadratic functions shown that a parabola may have no \(x\)-intercept, \(1\) \(x\)-intercept, or \(2\) \(x\)-intercepts depending on where the vertex is and the direction in which the parabola opens.

If the leading coefficient is positive (\(a > 0\)), then the parabola will open upward.

When \( a > 0 \),

If the leading coefficient is negative (\(a < 0\)), then the parabola will open downward.

When \( a < 0 \),

The range of a quadratic function is dependent on the maximum or minimum value of the function and the direction of opening.

We will now investigate the graphs of higher degree polynomial functions.

We will focus on key features including: the number of turning points, the number of \(x\)-intercepts, and the end behaviours of the function.