Problem 2 - Precalculus: Functions and Graphs

Question

Sketch the graph of \(f\) for the indicated value of \(c\) or \(a\) $$f(x)=-2 x^{3}+c$$ (a) \(c=-2\) (b) \(c=2\)

Step-by-Step Solution

Answer

Shift the graph of \(-2x^3\) by -2 for (a) and by +2 for (b).

Step 1: Understand the Function

The function given is a cubic function of the form \(f(x) = -2x^3 + c\), where the term \(-2x^3\) is the cubic term and \(c\) is a constant. This type of function is symmetric about the y-axis since it has only one variable \(x\) raised to the cube (odd degree polynomial).

Step 2: Identify the Effect of 'c'

The constant \(c\) adds a vertical shift to the graph of the function. For \(c = -2\), the graph of \(f(x) = -2x^3 - 2\) will shift down by 2 units. For \(c = 2\), the graph of \(f(x) = -2x^3 + 2\) will shift up by 2 units relative to the cubic graph \(f(x) = -2x^3\).

Step 3: Sketch the Base Graph

Sketch the graph of the base function \(f(x) = -2x^3\). This graph passes through the origin and has a general cubic shape, but since the coefficient of \(x^3\) is negative, it will start from the upper left quadrant and go to the lower right quadrant (downwards).

Step 4: Apply Vertical Shift for c = -2

To find the graph for \(c = -2\), take the base cubic graph and shift it downward by 2 units. This means that each point on the graph will have its y-coordinate decreased by 2.

Step 5: Apply Vertical Shift for c = 2

Similarly, for \(c = 2\), take the base graph and shift it upwards by 2 units. Here, each point on the graph will have its y-coordinate increased by 2.

Step 6: Plot the Graphs

Finally, sketch both transformed graphs on the same coordinate system for comparison: the graph of \(f(x) = -2x^3 - 2\), which is shifted down, and the graph of \(f(x) = -2x^3 + 2\), which is shifted up.

Key Concepts

Graph Transformations

Graph transformations involve changes to the original graph of a function in order to create new graphs. These transformations can include shifts, reflections, stretches, and compressions. For cubic functions like the one given in the exercise, the primary transformation applied is a vertical shift influenced by the constant term.

• Shifts move the graph in a specific direction, either vertically or horizontally.
• Reflections flip the graph across an axis.
• Stretches and compressions change the shape of the graph, either making it taller or flatter.

In this problem, we are specifically looking at vertical transformations resulting from the constant term 'c' in the equation.

Vertical Shift

One of the simplest transformations you can apply to graphs of functions is the vertical shift. This transformation is done by adding or subtracting a constant from the function.
For the function given in the problem, the term '+ c' dictates the vertical shift:

• If you add a positive constant (like in case (b) when c=2), the entire graph moves up.
• If you subtract a constant (like in case (a) when c=-2), the graph shifts down.

This movement doesn’t change the shape of the graph but merely moves it along the y-axis. The graph of the function compared to the base function equals vertical shifts upwards or downwards, depending on the value of 'c'.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They have a general form of:
\[f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\]In the given problem, the polynomial function is a cubic function because it contains a term with an exponent of 3:

• The leading term \(-2x^3\) is the highest degree term, indicating this function is cubic.
• The constant term '+ c' allows modification of the graph's position vertically.

Polynomial functions can take various forms but understanding their degree helps predict the number of roots and the graph's shape. Cubic functions, with their degree of 3, can have up to three real roots.

Odd Degree Polynomials

Odd degree polynomials, such as cubic functions, have specific characteristics that distinguish them from even degree polynomials. The crucial feature of odd degree polynomials is their behavior at the ends of the graph, also known as their end behavior.

• An odd degree polynomial will rise towards positive infinity on one end and fall towards negative infinity on the other.
• The graph of an odd degree polynomial is symmetrical about its central point, though not necessarily about the y-axis.

In the example provided, we have a cubic function \(-2x^3\). The negative sign in front of the \(-2x^3\) term indicates that the graph falls from left to right, starting in the upper-left quadrant and ending at the lower-right quadrant. This is because the coefficient is negative, inverting the typical path of a cubic graph.