Problem 13 - Algebra 2 and Trigonometry

Question

a. Graph the functions \(y=x^{4}\) and \(y=4^{x}\) on a graphing calculator using the following viewing windows: (1) \(\operatorname{Xmin}=0, \operatorname{Xmax}=3,\) Ymin \(=0,\) Ymax \(=50\) (2) \(X \min =0, \operatorname{Xmax}=5, Y \min =0,\) Ymax \(=500\) (3) \(X \min =0, \operatorname{Xmax}=5, Y \min =0, Y \max =1,000\) b. How many points of intersection can you find? Find the coordinates of these intersection points to the nearest tenth. c. Which function grows more rapidly for increasing values of \(x ?\)

Step-by-Step Solution

Answer

The functions intersect at approximately (2, 16). The function \( y = 4^x \) grows more rapidly as \( x \) increases.

Step 1: Understand the problem

You are asked to graph two functions, \( y = x^4 \) and \( y = 4^x \), using specific viewing windows in a graphing calculator. Then you need to find the points where they intersect and determine which function grows faster as \( x \) increases.

Step 2: Graph the functions using the first window

Set the graphing calculator to the first window settings: \( \text{Xmin} = 0, \text{Xmax} = 3, \text{Ymin} = 0, \text{Ymax} = 50 \). Graph both functions and observe their behavior and points of intersection within this range.

Step 3: Graph the functions using the second window

Change the calculator settings to the second window: \( \text{Xmin} = 0, \text{Xmax} = 5, \text{Ymin} = 0, \text{Ymax} = 500 \). Graph the functions again and check for points where they intersect.

Step 4: Graph the functions using the third window

Adjust the settings to the third window options: \( \text{Xmin} = 0, \text{Xmax} = 5, \text{Ymin} = 0, \text{Ymax} = 1000 \). Graph the functions one more time, noting intersections and growth trends.

Step 5: Identify and calculate intersection points

In each window, look for points where the two graphs meet. You may use the calculator's intersect feature or estimate manually. Verify and record these coordinates, rounding to the nearest tenth.

Step 6: Compare growth rates

Compare the shapes and trends of the graphs as \( x \) increases. The function that rises more sharply for larger \( x \) values is the one that grows faster.

Key Concepts

Exponential Growth

Exponential growth is a powerful mathematical concept where quantities increase rapidly. In the case of the function \( y = 4^x \), we observe that this represents exponential growth. Exponential functions are unique because they grow at a consistent rate, multiplying by a constant. This leads to very quick increases as \( x \) becomes larger.
To visualize this, think of how doubling something repeatedly can lead to massive numbers in just a few steps, like the example of a single mold spore doubling. If you plot an exponential function on a graph, you will notice its characteristic upward curve.
In our exercise, \( y = 4^x \) grows faster than the polynomial function \( y = x^4 \) because every increase in \( x \) results in the function multiplying by 4, not just a gradual increase.

Graphing Calculator

A graphing calculator is an essential tool for plotting and analyzing functions. It allows you to visualize complex equations easily, providing a clear picture of how functions interact and change. In the context of our exercise, a graphing calculator enables you to experiment with different viewing windows.
When using one, simply enter the equations \( y = x^4 \) and \( y = 4^x \), and adjust the window settings to explore different ranges. For example:

• First Window: \( \text{Xmin} = 0, \text{Xmax} = 3, \text{Ymin} = 0, \text{Ymax} = 50 \)
• Second Window: \( \text{Xmin} = 0, \text{Xmax} = 5, \text{Ymin} = 0, \text{Ymax} = 500 \)
• Third Window: \( \text{Xmin} = 0, \text{Xmax} = 5, \text{Ymin} = 0, \text{Ymax} = 1000 \)

By using the graphing calculator, you can clearly see how different functions behave within these ranges, making it easier to analyze their growth and intersection points.

Intersection Points

Intersection points are where two graphs meet on a coordinate plane. These points are important because they represent solutions or common values of the equations involved.
When graphing \( y = x^4 \) and \( y = 4^x \), look for places where the graphs cross. These intersection points tell you the specific \( x \) and \( y \) values where both functions have the same output.
To find these points accurately, utilize the intersect feature of a graphing calculator, which calculates the exact coordinates. In the exercise window, three Xmin-Xmax ranges help to visually inspect or calculate that both functions intersect at specific points. Rounding these coordinates to the nearest tenth often makes them easier to interpret and use in further mathematical problems.

Polynomial Functions

Polynomial functions, like \( y = x^4 \), include terms where variables are raised to whole number exponents. These functions can have various shapes and are fundamental to calculus and algebra.
The polynomial function in our exercise, \( y = x^4 \), describes a curve that starts at zero, increases gradually, and reflects symmetry about the y-axis. As a degree four polynomial, it grows slower than its exponential counterpart, particularly for larger \( x \) values.
Polynomials are versatile and can model diverse data sets. Their growth rate depends on the coefficient and degree of the highest term. Unlike exponential functions, polynomial growth is more predictable and often displays symmetry or consistent patterns within their graphs.