2.2. Shape of the Graph

• The graph is S-shaped or inverted S-shaped.

Cubic functions exhibit a distinctive S-shape when graphed. This means that the curve rises and falls in a way that resembles the letter 'S'. If the leading coefficient (the value of 'a' in the cubic function) is positive, the graph appears as a regular S. However, if 'a' is negative, the graph is shaped like an inverted S, which means it starts high, dips down, and then rises again.

Imagine a roller coaster ride that goes up slowly, then drops down steeply, and finally rises again at the end. This is similar to how the graph behaves if it is S-shaped. In contrast, if the roller coaster started high and then dipped deeply before rising, it represents an inverted S-shape.

• May have one or two turning points (local maximum and minimum).

Turning points are places on the graph where the curve changes direction. A local maximum is the highest point in a nearby region, and a local minimum is the lowest point. A cubic function can have either one or two of these turning points. This is important because they help determine the overall shape of the graph and represent crucial values where the function's behavior changes.

Think about a mountain range. Each peak represents a local maximum, where the landscape is at its highest point before descending again. Each valley represents a local minimum, where the landscape is at its lowest point before rising again. These points are essential for understanding the terrain, just as they are essential for interpreting the cubic graph.

• Can cross the x-axis up to three times (up to three real roots).

The x-intercepts of a cubic function are the points where the graph crosses the x-axis. This corresponds to the values of 'x' for which the cubic function equals zero (i.e., f(x) = 0). A cubic function can cross the x-axis up to three times, meaning it can have up to three real roots. Depending on how the graph is shaped, some roots might be repeated or complex (meaning non-real). This feature provides insights into the solutions of the corresponding cubic equations.

Visualize a car travelling along a road. The points where the car goes above or below a specific line on the road can be seen as the intersections with that line. If the road curves like an S-shape, it can intersect with the line at multiple points, representing different moments when the car might be at the same height as the line. These intersections are analogous to the roots of the cubic function.

• As 𝑥 → ∞, 𝑓(𝑥) → ∞ if 𝑎 > 0. • As 𝑥 → −∞, 𝑓(𝑥) → −∞ if 𝑎 > 0. • Reversed if 𝑎 < 0.

End behavior describes how the graph behaves as the input values ('x') become extremely large or small. For a cubic function with a positive leading coefficient (a > 0), as x approaches positive infinity, the function goes towards positive infinity, which means the graph rises. Conversely, as x approaches negative infinity, the function goes down towards negative infinity, indicating the graph falls. If the leading coefficient is negative (a < 0), these behaviors are reversed. Understanding end behavior helps predict how the graph will look far away from the y-axis.

Consider a mountain road traveling up a hill. As you go further up the hill (positive infinity), you can see that the road keeps rising. When driving down the other side (negative infinity), it continues downward. If the hill were reversed, where it started low and went high in the first part, navigating that road would mimic how the cubic graph behaves depending on the leading coefficient.

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