The Effect of the Coefficient on the Graph

Oct 13, 2025 by recoveryshake.com 133 views

The parent function $f(x) = |x|$ is a well-known absolute value function. Its graph is a V-shaped graph with its vertex at the origin (0, 0). The graph is symmetric about the y-axis, and it has a slope of 1 on the right side and a slope of -1 on the left side.

The Effect of the Coefficient on the Graph

When we introduce a coefficient 'a' into the function, it affects the graph's width and height. In this case, we want to find the value of 'a' that makes the graph of $f(x) = a|x|$ wider than that of the parent function $f(x) = |x|$ .

The Relationship Between the Coefficient and the Graph's Width

To understand how the coefficient affects the graph's width, let's consider the graph of $f(x) = |x|$ . The graph has a slope of 1 on the right side and a slope of -1 on the left side. When we multiply the function by a coefficient 'a', the slope of the graph changes. If 'a' is greater than 1, the slope of the graph increases, and the graph becomes steeper. If 'a' is less than 1, the slope of the graph decreases, and the graph becomes less steep.

Determining the Value of 'a'

To make the graph of $f(x) = a|x|$ wider than that of the parent function $f(x) = |x|$ , we need to find the value of 'a' that makes the graph less steep. This means that we need to find the value of 'a' that is less than 1.

Analyzing the Options

Let's analyze the options given:

A. $-1$ : This value of 'a' would make the graph of $f(x) = a|x|$ steeper, not wider.
B. $\frac{1}{4}$ : This value of 'a$ is less than 1, which means that the graph of $f(x) = a|x|$ would be less steep and wider than that of the parent function $f(x) = |x|$ .
C. $1$ : This value of 'a$ would make the graph of $f(x) = a|x|$ the same as that of the parent function $f(x) = |x|$ .
D. $4$ : This value of 'a$ would make the graph of $f(x) = a|x|$ steeper, not wider.

Conclusion

Based on our analysis, the value of 'a' that makes the graph of $f(x) = a|x|$ wider than that of the parent function $f(x) = |x|$ is $\frac{1}{4}$ .

Final Answer

$The final answer is $\boxed{\frac{1}{4}}$.<br/> **Frequently Asked Questions (FAQs)**$

Q: What is the parent function $f(x) = |x|$ ?

A: The parent function $f(x) = |x|$ is a well-known absolute value function. Its graph is a V-shaped graph with its vertex at the origin (0, 0). The graph is symmetric about the y-axis, and it has a slope of 1 on the right side and a slope of -1 on the left side.

Q: How does the coefficient 'a' affect the graph of $f(x) = a|x|$ ?

A: The coefficient 'a' affects the graph's width and height. If 'a' is greater than 1, the slope of the graph increases, and the graph becomes steeper. If 'a' is less than 1, the slope of the graph decreases, and the graph becomes less steep.

Q: What is the relationship between the coefficient 'a' and the graph's width?

A: To make the graph of $f(x) = a|x|$ wider than that of the parent function $f(x) = |x|$ , we need to find the value of 'a' that makes the graph less steep. This means that we need to find the value of 'a' that is less than 1.

Q: How do I determine the value of 'a' that makes the graph of $f(x) = a|x|$ wider than that of the parent function $f(x) = |x|$ ?

A: To determine the value of 'a', we need to analyze the options given. We need to find the value of 'a' that is less than 1, as this will make the graph less steep and wider than that of the parent function.

Q: What are the options for the value of 'a'?

A: The options for the value of 'a' are:

A. $-1$
B. $\frac{1}{4}$
C. $1$
D. $4$

Q: Which option makes the graph of $f(x) = a|x|$ wider than that of the parent function $f(x) = |x|$ ?

A: Based on our analysis, the value of 'a' that makes the graph of $f(x) = a|x|$ wider than that of the parent function $f(x) = |x|$ is $\frac{1}{4}$ .

Q: What is the final answer?

A: The final answer is $\boxed{\frac{1}{4}}$ .

Additional Tips and Resources

To understand the graph of $f(x) = a|x|$ , it's essential to visualize the graph and analyze its properties.
The coefficient 'a' affects the graph's width and height, so it's crucial to understand how 'a' impacts the graph.
To determine the value of 'a', we need to analyze the options given and find the value that makes the graph less steep and wider than that of the parent function.

Conclusion

In conclusion, the value of 'a' that makes the graph of $f(x) = a|x|$ wider than that of the parent function $f(x) = |x|$ is $\frac{1}{4}$ . We hope this article has provided you with a clear understanding of the parent function, the effect of the coefficient 'a', and how to determine the value of 'a'.