At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To determine which absolute value function has a graph that is wider than the parent function [tex]\( f(x) = |x| \)[/tex] and is translated to the right 2 units, we need to examine each given function step by step.
### Step 1: Understanding the Parent Function
The parent function is [tex]\( f(x) = |x| \)[/tex]. This function has a V-shaped graph with the vertex at the origin (0,0) and opens upward.
### Step 2: Identifying a Wider Graph
A graph of an absolute value function is wider than the parent function if the coefficient of the absolute value is between -1 and 1 (excluding 0). Coefficients greater than 1 or less than -1 make the graph narrower.
### Step 3: Translation to the Right
To translate the graph of an absolute value function to the right by 2 units, the function must include [tex]\(|x - 2|\)[/tex] as part of its argument. This shifts the vertex of the V-shaped graph from [tex]\((0,0)\)[/tex] to [tex]\((2,0)\)[/tex].
### Evaluating the Given Functions
Let's examine each provided function:
1. [tex]\( f(x) = 1.3|x| - 2 \)[/tex]
- The coefficient [tex]\(1.3\)[/tex] is greater than 1, meaning the graph is narrower, not wider.
- There is no shifting term [tex]\(x - 2\)[/tex], so it is not translated to the right.
- This function does not meet the criteria.
2. [tex]\( f(x) = 3|x - 2| \)[/tex]
- The coefficient 3 is greater than 1, making the graph narrower.
- The term [tex]\(x - 2\)[/tex] indicates the graph is translated to the right by 2 units.
- This function only partially meets the criteria (translated right but not wider).
3. [tex]\( f(x) = \frac{3}{4}|x - 2| \)[/tex]
- The coefficient [tex]\(\frac{3}{4}\)[/tex] is between 0 and 1, meaning the graph is wider.
- The term [tex]\(x - 2\)[/tex] indicates the graph is translated to the right by 2 units.
- This function meets both criteria: wider graph and translated right.
4. [tex]\( f(x) = \frac{4}{3}|x| + 2 \)[/tex]
- The coefficient [tex]\(\frac{4}{3}\)[/tex] is greater than 1, making the graph narrower.
- There is no shifting term [tex]\(x - 2\)[/tex], so it is not translated to the right.
- This function does not meet the criteria.
### Conclusion
The function that has a graph which is wider than the parent function [tex]\( f(x) = |x| \)[/tex] and translated to the right 2 units is:
[tex]\[ f(x) = \frac{3}{4}|x-2| \][/tex]
### Step 1: Understanding the Parent Function
The parent function is [tex]\( f(x) = |x| \)[/tex]. This function has a V-shaped graph with the vertex at the origin (0,0) and opens upward.
### Step 2: Identifying a Wider Graph
A graph of an absolute value function is wider than the parent function if the coefficient of the absolute value is between -1 and 1 (excluding 0). Coefficients greater than 1 or less than -1 make the graph narrower.
### Step 3: Translation to the Right
To translate the graph of an absolute value function to the right by 2 units, the function must include [tex]\(|x - 2|\)[/tex] as part of its argument. This shifts the vertex of the V-shaped graph from [tex]\((0,0)\)[/tex] to [tex]\((2,0)\)[/tex].
### Evaluating the Given Functions
Let's examine each provided function:
1. [tex]\( f(x) = 1.3|x| - 2 \)[/tex]
- The coefficient [tex]\(1.3\)[/tex] is greater than 1, meaning the graph is narrower, not wider.
- There is no shifting term [tex]\(x - 2\)[/tex], so it is not translated to the right.
- This function does not meet the criteria.
2. [tex]\( f(x) = 3|x - 2| \)[/tex]
- The coefficient 3 is greater than 1, making the graph narrower.
- The term [tex]\(x - 2\)[/tex] indicates the graph is translated to the right by 2 units.
- This function only partially meets the criteria (translated right but not wider).
3. [tex]\( f(x) = \frac{3}{4}|x - 2| \)[/tex]
- The coefficient [tex]\(\frac{3}{4}\)[/tex] is between 0 and 1, meaning the graph is wider.
- The term [tex]\(x - 2\)[/tex] indicates the graph is translated to the right by 2 units.
- This function meets both criteria: wider graph and translated right.
4. [tex]\( f(x) = \frac{4}{3}|x| + 2 \)[/tex]
- The coefficient [tex]\(\frac{4}{3}\)[/tex] is greater than 1, making the graph narrower.
- There is no shifting term [tex]\(x - 2\)[/tex], so it is not translated to the right.
- This function does not meet the criteria.
### Conclusion
The function that has a graph which is wider than the parent function [tex]\( f(x) = |x| \)[/tex] and translated to the right 2 units is:
[tex]\[ f(x) = \frac{3}{4}|x-2| \][/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
