Problem 73 - Algebra and Trigonometry

Question

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions that you can make from your graphs. \(f(X)=X^{c}\) (a) \(c=\frac{1}{2}, \frac{1}{4}, \frac{1}{6} ; \quad[-1,4]\) by \([-1,3]\) (b) \(c=1, \frac{1}{3}, \frac{1}{5} ; \quad[-3,3]\) by \([-2,2]\) (c) How does the value of \(c\) affect the graph?

Step-by-Step Solution

Answer

As \( c \) decreases, graphs stretch horizontally and flatten; steeper for higher \( c \).

Step 1: Understand the Problem

The task requires graphing different functions of the form \( f(x) = x^c \) for various values of \( c \). We need to use specified intervals for both x-values and y-values (viewing rectangles) and then analyze how changing \( c \) affects the graphs.

Step 2: Graph Functions for Part (a)

For part (a), graph the functions \( f(x) = x^{1/2}, x^{1/4}, x^{1/6} \). Use the viewing interval \([-1, 4]\) for x and \([-1, 3]\) for y.- Plot \( f(x) = x^{1/2} \); this is the square root function, which is defined for \( x \geq 0 \) and increases as \( x \) increases.- Plot \( f(x) = x^{1/4} \); this graph is more stretched compared to the square root function and also only defined for \( x \geq 0 \).- Plot \( f(x) = x^{1/6} \); it stretches even more horizontally.Observing these functions, notice they are only defined for non-negative x and increase more slowly as c gets smaller.

Step 3: Graph Functions for Part (b)

For part (b), graph the functions \( f(x) = x^{1}, x^{1/3}, x^{1/5} \) using the viewing rectangle \([-3, 3]\) for x and \([-2, 2]\) for y.- Plot \( f(x) = x^{1} \); this is a straight line with a 45-degree angle.- Plot \( f(x) = x^{1/3} \); this graph is S-shaped and symmetrical about the origin.- Plot \( f(x) = x^{1/5} \); this graph is similar to the cube root but flatter, and also symmetrical about the origin.As \( c \) decreases, the graphs resemble stretched S-shapes, showing less steepness and flattening as \( c \) becomes smaller.

Step 4: Analyze and State Conclusions for Part (c)

From the graphs, observe how the value of \( c \) affects the function:- For \( c > 0 \), the graph is either defined or undefined based on positive or negative x-values.- As \( c \) becomes smaller (fractions decrease), the graph stretches horizontally for positive x or becomes less steep.- A fractional exponent \( c \) means the graph includes roots and pulls the graph lower as the power gets smaller.Values greater than 1 will increase the steepness, while values between 0 and 1 decrease the steepness and stretch the graphs.

Key Concepts

Family of Functions

A family of functions refers to a set of functions that share a similar form or characteristic. In our context, we are exploring the family of functions given by the equation \( f(x) = x^c \), where the variable \( c \) can vary to produce different functions within this family.

When we talk about graphing a family of functions, we are looking at how these functions behave both individually and collectively. By changing the parameter \( c \), we can analyze how each function compares to others in the family. This concept is important because it helps us understand the underlying patterns and characteristics of mathematical functions.

For example, in this exercise, the family \( f(x) = x^c \) includes functions such as square roots \( (c = \frac{1}{2}) \), cube roots \( (c = \frac{1}{3}) \), and so forth. Each member of this family shares a similar root structure but visually manifests different graph shapes depending on the value of \( c \). This allows us to observe different behaviors and properties shared among the functions within this family.

Effects of Exponent on Graphs

The exponent \( c \) in the function \( f(x) = x^c \) plays a significant role in determining the shape of its graph. As you change \( c \), the resulting graphs will exhibit different characteristics.

• When \( c \) is 1, the function becomes a linear one, forming a straight line. This is the simplest form where the graph has a 45-degree angle from the horizontal, indicating a constant rate of change.
• For values of \( c \) between 0 and 1, such as \( c = \frac{1}{2} \) or \( c = \frac{1/3} \), the graphs take the form of root functions. The smaller the fraction, the more the graph stretches horizontally, indicating a slower rate of increase compared to a linear graph. This is especially noticeable with fractional exponents.
• As \( c \) decreases, these graphs become flatter and wider, indicating a decrease in steepness. This flattening illustrates that the graph is becoming less steep or steeper as per the increase or decrease of \( c \).
• When \( c \) is greater than 1, the graphs become steeper, showing exponential growth as the value of \( x \) increases. This makes the shoot upwards more quickly compared to a linear graph.

By experimenting with different values of \( c \), one can easily observe these visual changes, which are crucial to understanding how mathematical functions behave differently based on their exponents.

Graphing Root Functions

Graphing root functions involves understanding the behavior of functions where the exponent is a fraction, such as \( x^{1/2} \) or \( x^{1/3} \).

• The function \( f(x) = x^{1/2} \) is commonly known as the square root function. The graph of this function is only defined for positive values of \( x \), starting at the origin \((0,0)\) and increases steadily as \( x \) increases. Its growth is slower than a linear function because it involves taking the square root.
• In contrast, the function \( f(x) = x^{1/3} \), known as the cube root function, differs because it is defined for both positive and negative \( x \) values. This symmetry about the origin results in an S-shaped graph, indicating a different growth dynamic where negative inputs yield negative results.
• As we see with these fractional exponents, the graph becomes more stretched horizontally, and the increase slows down. The smaller the fraction, the more the graph stretches, indicating the root-taking nature of these functions.

Thus, graphing root functions helps in visualizing the gradual increase compared to the rapid rise seen with whole number exponents, allowing us to comprehend and compare different growth rates and behaviors in mathematical graphs.