Problem 38 - Algebra and Trigonometry

Question

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=-x^{3} $$

Step-by-Step Solution

Answer

Reflect the standard cubic graph \( f(x) = x^3 \) over the x-axis for \( f(x) = -x^3 \).

Step 1: Identify the standard function

The function given is a cubic function, and the standard form of the cubic function is \[ f(x) = x^3 \]. This function has a basic shape with a point of symmetry at the origin. It increases from negative infinity to zero and then increases from zero to positive infinity.

Step 2: Apply reflections

The function given is \( f(x) = -x^3 \). The negative sign in front of the cubic term indicates a reflection over the x-axis. This means that for each positive value of the standard function, the transformed function will have a corresponding negative value, and vice versa. The graph will be a horizontal reflection of the standard cubic function \( f(x) = x^3 \).

Step 3: Sketch the graph based on transformations

Start with the standard cubic function \( f(x) = x^3 \), which passes through the origin and shows increasing behavior on both sides of the y-axis, forming an S-like curve. After reflecting over the x-axis, the graph of \( f(x) = -x^3 \) will also pass through the origin. Instead of rising from left to right, it will fall from left to right, creating an inverted S-like shape. The curve will start from the third quadrant, cross the origin, and continue into the fourth quadrant.

Key Concepts

Cubic Function

A cubic function is a polynomial function where the highest degree is three, typically in the form

• \( f(x) = ax^3 + bx^2 + cx + d \)

In its simplest form, it is represented as \( f(x) = x^3 \). Cubic functions have interesting characteristics:

• They can change direction, featuring a turning point that creates an S-shaped curve.
• They are symmetric with respect to the origin, meaning that if you rotate the graph 180 degrees about the origin, it will look the same.

These properties make cubic functions unique compared to linear or quadratic functions. Knowing the form and symmetry helps in sketching the graph and understanding transformations applied to such functions.

Reflection

Reflection is a transformation that 'flips' a graph over a line, creating a mirror image. Specifically, when a function includes a negative sign in front of its main term, this indicates a reflection:

• For our cubic function \( f(x) = -x^3 \), reflection occurs over the x-axis.
• Reflection does not alter the shape of the graph but reverses its orientation.

For example, a positive value at \( f(x) = x^3 \) becomes a negative value at \( f(x) = -x^3 \). Similarly, a negative value becomes positive, effectively flipping the graph upside down. Understanding reflection helps in predicting how a function's graph will change without plotting multiple points.

X-axis

The x-axis is the horizontal number line in a coordinate plane. It is crucial in graphing as it serves as an axis of symmetry or reflection. In our transformed function \( f(x) = -x^3 \), the reflection flips the original curve of \( \pm x^3 \) over the x-axis by:

• Reversing the signs of all y-values.
• Maintaining the x-values, hence transforming rising curves into falling ones.

Reflections over the x-axis are common in transformations and are key to understanding how different signs affect the visual presentation of a function. Recognizing the role of the x-axis aids in executing and visualizing various graphical transformations.

Graph Sketching

Graph sketching involves creating a visual representation of a function based on transformations rather than individual plotting. With transformations like reflection over the x-axis, sketching becomes intuitive. For \( f(x) = -x^3 \):

• Begin with the basic cubic graph \( f(x) = x^3 \), which exhibits the characteristic S-shape.
• By applying the reflection, the graph will now move from the third quadrant to the fourth.

The essence of sketching is to capture the behavior, direction, and symmetry of a function. Understanding transformation techniques, such as reflecting over the x-axis, allows for quick and effective graph sketching, illustrating the influence of different function modifications.