Problem 85 - Precalculus Mathematics for Calculus

Question

Graph the functions \(y=x^{2}\) \(y=x^{3}, y=x^{4},\) and \(y=x^{5},\) for \(-1 \leq x \leq 1,\) on the same coordinate axes. What do you think the graph of \(y=x^{100}\) would look like on this same interval? What about \(y=x^{101} ?\) Make a table of values to confirm your answers.

Step-by-Step Solution

Answer

The graph of \(y = x^{100}\) will be very flat around zero and shoot up past -1 and 1. The graph of \(y = x^{101}\) will be similar but switch signs at zero.

Step 1: Understanding the Functions

First, let's look at the functions that we need to graph: \(y = x^2\), \(y = x^3\), \(y = x^4\), and \(y = x^5\). Each function is a power function with increasing exponents. Our task is to graph them on the interval \(-1 \leq x \leq 1\) and observe their behavior to predict what \(y = x^{100}\) and \(y = x^{101}\) would look like.

Step 2: Calculating Values

To understand the behavior of these functions, calculate the values of \(y\) for specific \(x\) values: \(-1, -0.5, 0, 0.5, 1\). This will help us see the pattern as the power increases. **Example**: \(y = x^2\) calculations: \[\begin{align*} y(-1) &= 1, \ y(-0.5) &= 0.25, \ y(0) &= 0, \ y(0.5) &= 0.25, \ y(1) &= 1. \end{align*}\] Repeat this for each function.

Step 3: Analyzing Patterns in Table

Create a table of these values for each function. Notice that as the exponent increases, the graph's curvature becomes steeper in the middle (close to \(x = 0\)) and less steep at the ends (near \(x = \pm1\)). For even exponents, the graph is symmetric around the y-axis, while for odd exponents, you have similar behavior but with a slight twist due to the odd nature.

Step 4: Predicting the Graphs of y=x^{100} and y=x^{101}

With this pattern in mind, visualize that for \(y = x^{100}\), the graph will be very steep near \(x = 0\), hugging the x-axis between \(-1\) and \(1\) and then shooting upwards sharply past \(-1\) and \(1\). For \(y = x^{101}\), expect a similar steep characteristic, but with the odd behavior of the curve being over the x-axis in positive x and under for negative x, switching quickly at 0.

Key Concepts

Graphing Functions

Graphing power functions with different exponents helps us visualize how their shapes differ. When you plot functions like \( y = x^2 \), \( y = x^3 \), \( y = x^4 \), and \( y = x^5 \) over the same interval, you notice key characteristics:

• As exponents rise, the graphs become steeper.
• For even powers like \( y = x^2 \) and \( y = x^4 \), the graphs are symmetrical around the y-axis, resembling a U-shape.
• Odd powers, such as \( y = x^3 \) and \( y = x^5 \), create slightly twists in symmetry, appearing somewhat S-shaped.

Each graph has distinct curvature, increasing its steepness near \( x = 0 \) and becoming flater towards the ends of \( x = \pm1 \). Noticeable differences in curvature allow us to predict how higher power functions like \( y = x^{100} \) and \( y = x^{101} \) will appear.

Even and Odd Functions

The concepts of even and odd functions play a vital role in understanding symmetry in function graphs. A function \( f(x) \) is even if for every \( x \), \( f(x) = f(-x) \). This creates graphs that are symmetric with respect to the y-axis. Examples include \( y = x^2 \) and \( y = x^4 \). These functions maintain a mirror-like symmetry, resulting in a U-shaped curve for power functions.
Conversely, a function is odd if \( f(-x) = -f(x) \) for every \( x \). Odd functions show symmetry about the origin, giving them an S-shaped curve. Functions like \( y = x^3 \) and \( y = x^5 \) demonstrate this odd property, creating graphs that appear to twist or cross the y-axis. Notably, knowing these characteristics helps predict behaviors and shapes of functions with higher powers.

Predicting Function Behavior

Predicting the behavior of functions like \( y = x^{100} \) or \( y = x^{101} \) relies on examining patterns found in lower power graphs. When we consider very high even powers such as \( y = x^{100} \), the graph will:

• Be highly steep, especially as it nears \( x = 0 \).
• Tightly hug the x-axis between \(-1\) and \(1\).
• Rise sharply beyond the x-axis limits at \( x = \pm1 \), maintaining symmetry around the y-axis.

For very high odd powers like \( y = x^{101} \), expect the graph to exhibit:

• Similar steepness close to \( x = 0 \),
• A quick shift over and under the x-axis around \( 0 \), reflecting its odd nature.
• Symmetry about the origin, resulting in a steep S-shaped curve.

Understanding the basic behaviors of power functions allows us to extend these patterns, accurately predicting higher-power graph shapes and characteristics.