 Describe the graph of yequalsx Superscript 12. Select the correct choice below and fill in any answer boxes to complete your choice.
​(Type your answer in interval​ notation.)
A.
Of the graphs that are symmetric about the​ y-axis, it is the one that is less steep for StartAbsoluteValue x EndAbsoluteValue greater than 1. It decreases and increases on the open intervals 
  
enter your response here and 
  
enter your response here​, respectively.
B.
It is symmetric about the origin and increases on the open interval 
  
enter your response here.
C.
Of the graphs that are symmetric about the​ y-axis, it is the one that is more steep for StartAbsoluteValue x EndAbsoluteValue greater than 1. It decreases and increases on the open intervals 
  
enter your response here and 
  
enter your response here​, respectively.

The function \( y = x^{12} \) is an even function because the exponent is even. This means the graph is symmetric about the **y-axis**.

Let's analyze the behavior:

1. **Shape**: The graph of \( y = x^{12} \) resembles a very steep parabola. It is flat near \( x = 0 \) and becomes much steeper as \( |x| \) grows larger than 1. 

2. **Intervals of Increase and Decrease**:
   - The function **decreases** for negative \( x \), i.e., on the interval \( (-\infty, 0) \).
   - The function **increases** for positive \( x \), i.e., on the interval \( (0, \infty) \).

3. **Steepness**: The graph is **more steep** for \( |x| > 1 \) compared to lower powers of even exponents (like \( x^2 \), \( x^4 \), etc.).

So, the correct choice is **C**:
- Of the graphs that are symmetric about the y-axis, it is the one that is more steep for \( |x| > 1 \).
- It decreases on the open interval \( (-\infty, 0) \) and increases on the open interval \( (0, \infty) \).

Do you want further details or have any questions? Here are 5 questions to expand on this concept:

1. How does the steepness of a graph change with higher even powers of \( x \)?
2. What would the graph of \( y = x^{11} \) look like compared to \( y = x^{12} \)?
3. Why is \( y = x^{12} \) symmetric about the y-axis?
4. How do the intervals of increase and decrease change for odd-powered functions?
5. What is the behavior of the graph near \( x = 0 \)?

**Tip**: The symmetry of the graph of a function depends on whether the function is even or odd. Even functions like \( y = x^{12} \) are symmetric about the y-axis!

Describe the graph of y equals x Superscript 12. Select the correct choice below and fill in any answer boxes to complete your choice. (Type your answer in interval notation.) A. Of the graphs that are symmetric about the y-axis, it is the one that is less steep for |x| greater than 1. It decreases and increases on the open intervals enter your response here and enter your response here, respectively. B. It is symmetric about the origin and increases on the open interval enter your response here. C. Of the graphs that are symmetric about the y-axis, it is the one that is more steep for |x| greater than 1. It decreases and increases on the open intervals enter your response here and enter your response here, respectively.

Explore the graph of y = x^12, an even polynomial function. Learn about its symmetry about the y-axis, steepness for |x| > 1, and how it decreases on the interval (-∞, 0) and increases on (0, ∞). Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation