Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Compare the graph of \(f(x)=8 / x^{3}\) with the graph of \(g\). $$g(x)=f(x)+5=\frac{8}{x^{3}}+5$$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is the same as the graph of \(f(x)\), but shifted 5 units upwards.

Step 1: Considering the function \(f(x)\)

Start with \(f(x) = \frac{8}{x^3}\). The graph of this function will be the cubic root function which is reflected over the x-axis since the reciprocal of a negative number is also negative and vice-versa. It will pass through points (1,8), (-1,-8) and it will tend to zero as x goes to \(\infty\) or \(-\infty\). The function also has a vertical asymptote located at x = 0.

Step 2: Considering the function \(g(x)\)

Now move to \(g(x) = f(x) + 5 = \frac{8}{x^3} + 5\). The function g(x) is obtained by adding 5 to the function \(f(x)\). This operation results in vertically shifting the graph of \(f(x)\) 5 units upwards. The vertical asymptote of f still remains an asymptote for g at x = 0, but the whole graph is now shifted 5 units upwards. Which means the graph will pass through points (1,13), (-1, -3) and it will tend to 5 as x goes to \(\infty\) or \(-\infty\).

Step 3: Comparing the two graphs

After plotting both the functions \(f(x)\) and \(g(x)\) on a graph it can be seen that, the graph of \(g(x)\) is just the graph of \(f(x)\) shifted upwards by 5 units. Apart from this vertical shift, the graphs of both functions will have the same shape.

Key Concepts

Vertical Shift

When we talk about a vertical shift in graph transformation, we mean moving a graph up or down on the coordinate plane without changing its shape. In our example, the initial function is \( f(x) = \frac{8}{x^3} \).
To generate \( g(x) = \frac{8}{x^3} + 5 \), we add 5 to each output value of \( f(x) \). Thus, every point on the graph is moved up by 5 units.
Here's how you can identify a vertical shift:

• If a constant is added to the function, like \( f(x) + 5 \), the graph shifts upward by that constant amount.
• If a constant is subtracted, like \( f(x) - 5 \), it shifts down by that amount.

This transformation doesn't alter the function's asymptotes, but it changes where the graph intersects the y-axis. For instance, the intersection at (1,8) for \( f(x) \) becomes (1,13) for \( g(x) \) due to the vertical shift.

Asymptote

Asymptotes are lines that a graph approaches but never actually touches. In the context of our example, both functions \( f(x) = \frac{8}{x^3} \) and \( g(x) = \frac{8}{x^3} + 5 \) have a vertical asymptote at \( x = 0 \). This is because the denominator \( x^3 \) becomes zero, creating a division by zero, which is undefined.
The presence of the vertical asymptote means that as \( x \) approaches 0, the function values explode towards infinity or negative infinity.

• The vertical shift in \( g(x) \) does not affect the position of the vertical asymptote.
• It remains at \( x = 0 \) for both functions.

It's essential to remember that vertical shifts do not alter the positions of asymptotes. Asymptotes are crucial in understanding the behavior of functions as \( x \) approaches certain critical values that make the function undefined.

Function Comparison

Function comparison involves analyzing the characteristics and behaviors of multiple functions. Comparing \( f(x) = \frac{8}{x^3} \) and \( g(x) = \frac{8}{x^3} + 5 \) involves evaluating their similarities and differences.
Here's how to approach the comparison:

• **Shape**: Both graphs have the same overall shape since \( g(x) \) is just \( f(x) \) vertically shifted.
• **Asymptotes**: The vertical asymptote remains at \( x = 0 \) for both functions.
• **Vertical Shift**: The primary distinction is that \( g(x) \) is 5 units higher on the y-axis compared to \( f(x) \).

When comparing these functions graphically, the key difference you will notice is the shift in the intersection points and the values the functions tend towards as \( x \) approaches infinity. For example, \( g(x) \) tends towards 5, while \( f(x) \) tends towards 0. By understanding these aspects, you can predict how modifications like adding or subtracting constants affect graphs.