Problem 95 - College Algebra

Question

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)-x^{3}-3 $$

Step-by-Step Solution

Answer

The function \(g(x)=x^{3}-3\) is the graph of the original cubic function \(f(x)=x^{3}\), but shifted downwards by 3 units.

Step 1: Plot the original function

First, plot the original cubic function \(f(x)=x^{3}\). This will serve as the base from which transformations will be applied.

Step 2: Identify the transformation

Based on the given function to be graphed, \(g(x)=x^{3}-3\), there seems to be vertical shift happening. The '-3' indicates this function will be shifted downwards by 3 units in comparison to the original function \(f(x)=x^{3}\).

Step 3: Apply the transformation

Now, apply the transformation to the original function to obtain the graph for \(g(x)=x^{3}-3\). Take each point on the graph of \(f(x)=x^{3}\) and shift it downwards by 3 units.

Step 4: Verify the result

After plotting the graphs of the original function and the transformed function, check to see if every point on the cubic function has been moved downwards by 3 units. This will validate the transformation operation.

Key Concepts

Transformations of Graphs

Transformations are a fundamental concept in graphing functions. They allow us to modify basic graphs to create new ones. There are several types of transformations, such as translations, reflections, and dilations.

• Translations: These involve shifting the graph along the x or y-axis. This can be further broken down into horizontal and vertical shifts.
• Reflections: These are flips over a specific axis. For instance, reflecting a function over the x-axis will invert it upside down.
• Dilations: These changes involve stretching or compressing the graph, either vertically or horizontally.

For cubic functions, transformations help us understand how changes in the function's equation alter its shape and position. Recognizing these transformations can simplify the process of graphing and analyzing more complex functions.

Vertical Shifts

Vertical shifts are a specific type of graph transformation. They occur when a constant is added or subtracted from a function, moving the entire graph up or down.An equation of the form\[g(x) = f(x) + c\]where \(c > 0\), shifts the graph of \(f(x)\) upward by \(c\) units. Conversely, if \(c < 0\), the graph shifts downward by \(|c|\) units.In this example, the function \(g(x) = x^3 - 3\) experiences a vertical shift. The graph of \(f(x) = x^3\) is moved down by 3 units since \(c\) is -3. This adjustment alters the position of the graph on the y-axis without changing its overall shape.

Cubic Function Properties

Cubic functions are polynomial functions of degree three, and they exhibit specific properties that influence their graphs.

• Cubic functions have the general form \(f(x) = ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are coefficients with \(a eq 0\).
• The graph of a standard cubic function \(f(x) = x^3\) has a characteristic 'S' shaped curve, with symmetry about the origin.
• Cubic functions can have different slopes and intercepts, depending on their coefficients, which influence both the direction and steepness of the graph.
• They can exhibit one real root or three real roots, affecting the number of intersections with the x-axis.

Understanding these properties provides insight into how cubic functions behave and aids in predicting changes due to transformations such as shifts, stretches, or rotations.