Problem 50 - Calculus for Biology and Medicine

Question

Use a graphing calculator to sketch the graphs of the functions. $$ y=2 x^{-7 / 8}, x>0 $$

Step-by-Step Solution

Answer

Graph \( y = 2x^{-7/8} \) as a decreasing curve for \( x > 0 \).

Step 1: Understanding the Function

The function you are asked to graph is \( y = 2x^{-7/8} \). It is an algebraic function that includes a negative fractional exponent, which implies division and root operations. The domain specified is \( x > 0 \).

Step 2: Analysis before Sketching

Analyzing the function: \( y = 2x^{-7/8} \) can be rewritten as \( y = \frac{2}{x^{7/8}} \). As \( x \) increases, \( x^{7/8} \) increases, making \( \frac{2}{x^{7/8}} \) decrease. Hence, the function is decreasing for \( x > 0 \). As \( x \) approaches 0 from the right, \( y \) approaches infinity. As \( x \) approaches infinity, \( y \) approaches 0.

Step 3: Using a Graphing Calculator

Turn on the graphing calculator. Enter \( y = 2x^{-7/8} \) as the function to graph. Ensure your viewing window includes positive \( x \)-values because the domain is \( x > 0 \). Set the window to include small \( y \) values to observe the graph decreasing as \( x \) increases.

Step 4: Sketching the Graph

After graphing on the calculator, you notice the graph starts off high near the y-axis and decreases towards the x-axis without touching it as \( x \) becomes larger. This behavior confirms the analysis in Step 2. Draw this sketch on graph paper, ensuring that the curve does not cross the x-axis and remains above it at all times.

Key Concepts

Algebraic Function

An algebraic function is any mathematical function involving only algebraic operations, such as addition, subtraction, multiplication, division, and raising to a power. These functions are foundational to algebra and include a wide range of expressions, from simple polynomials to more complex rational functions.
In our example, the function \( y = 2x^{-7/8} \) is an algebraic function. Here, the operations involved are multiplication and exponentiation, albeit with a fractional and negative exponent. Algebraic functions often form curves rather than straight lines, making them an interesting subject for graphing, as their behavior can vary widely based on the function's components.
When graphing such a function, it is important to analyze and interpret the effects of each operation, as they dictate the shape and direction of the curve relative to the axes.

Negative Fractional Exponent

Negative fractional exponents are a fascinating aspect of algebra that involve both root and reciprocal operations. When you see an exponent like \(-7/8\) in the function \( y = 2x^{-7/8} \), it indicates that you will first take the 8th root of \( x \), and then take the reciprocal of the result due to the negative exponent.
This means:

• \( x^{-7/8} = \frac{1}{x^{7/8}} \)
• Take the 8th root of \( x \)
• Raise this result to the 7th power
• Finally, take the reciprocal

This sequence can alter the graph significantly:

• As \( x \) increases, the term \( x^{7/8} \) gets larger, making the entire expression \( \frac{2}{x^{7/8}} \) approach zero.
• Conversely, as \( x \) approaches zero from the right, \( x^{7/8} \) becomes very small, and \( \frac{2}{x^{7/8}} \) shoots up towards infinity.

Understanding these changes helps predict the graph’s behavior, which is critical for interpreting algebraic functions with negative fractional exponents.

Graphing Techniques

Graphing complex functions like \( y = 2x^{-7/8} \) can be simplified with graphing techniques and tools like a graphing calculator. Such tools help visualize how variables interact within a function.
When grapging a function:

• Input the function into the graphing calculator accurately. Here, enter \( y = 2x^{-7/8} \).
• Set the viewing window—focus on positive \( x \) values due to the domain \( x > 0 \), and adjust the \( y \)-range to capture the curve's downward trajectory.
• Observe the graph shape: it should start high and decrease, never touching the x-axis, as it approaches zero from a positive situation, due to the negative fractional exponent.

Drawing a function manually requires plotting several points to sketch the curve accurately. However, the calculator simplifies this by providing a complete graph quickly. Whether by calculator or hand-sketch, understanding these techniques ensures accurate depictions and interpretations of algebraic functions.