Join IDNLearn.com today and start getting the answers you've been searching for. Get the information you need from our community of experts who provide accurate and thorough answers to all your questions.
Sagot :
To determine the correct equation for the absolute value function with the given vertex and x-intercepts, let's break down the problem step by step.
### Step 1: Identify the Vertex of the Absolute Value Function
The vertex of the absolute value function is given as [tex]\((2, 3)\)[/tex]. The general form of an absolute value function is:
[tex]\[ y = a |x - h| + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex. For our problem, [tex]\(h = 2\)[/tex] and [tex]\(k = 3\)[/tex]. Therefore, the equation can start as:
[tex]\[ y = a |x - 2| + 3 \][/tex]
### Step 2: Determine the x-Intercepts and Verify the Function
The x-intercepts of the function are [tex]\((-1, 0)\)[/tex] and [tex]\((5, 0)\)[/tex]. This means that when [tex]\(y = 0\)[/tex], the equation should satisfy:
[tex]\[ 0 = a |x - 2| + 3 \][/tex]
### Step 3: Solve for the Absolute Value at the Intercepts
Now, let's substitute the x-intercepts into the equation and ensure it holds true:
1. Substituting [tex]\( x = -1 \)[/tex]:
[tex]\[ 0 = a |-1 - 2| + 3 \][/tex]
[tex]\[ 0 = a | -3 | + 3 \][/tex]
[tex]\[ 0 = 3a + 3 \][/tex]
To satisfy this equation, [tex]\(a\)[/tex] must be [tex]\(-1\)[/tex].
2. Substituting [tex]\( x = 5 \)[/tex]:
[tex]\[ 0 = a |5 - 2| + 3 \][/tex]
[tex]\[ 0 = a | 3 | + 3 \][/tex]
[tex]\[ 0 = 3a + 3 \][/tex]
Again, [tex]\(a\)[/tex] must be [tex]\(-1\)[/tex].
### Step 4: Verify Both Conditions
Given that both conditions for the x-intercepts hold true with [tex]\(a = 1\)[/tex], we can conclude that the correct equation fits the form:
[tex]\[ y = |x - 2| + 3 \][/tex]
### Step 5: Check the Given Options
Now let's verify which option matches our derived equation when [tex]\(y = 0\)[/tex]:
A. [tex]\(0 = |x + 2| + 3\)[/tex]
B. [tex]\(0 = |x - 2| + 3\)[/tex]
C. [tex]\(0 = -|x + 2| + 3\)[/tex]
Option B matches the form [tex]\(0 = |x - 2| + 3\)[/tex] and fits within our calculations.
### Conclusion
Therefore, the correct equation for the absolute value function, when [tex]\(y = 0\)[/tex], is:
[tex]\[ \boxed{0 = |x - 2| + 3} \][/tex]
### Step 1: Identify the Vertex of the Absolute Value Function
The vertex of the absolute value function is given as [tex]\((2, 3)\)[/tex]. The general form of an absolute value function is:
[tex]\[ y = a |x - h| + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex. For our problem, [tex]\(h = 2\)[/tex] and [tex]\(k = 3\)[/tex]. Therefore, the equation can start as:
[tex]\[ y = a |x - 2| + 3 \][/tex]
### Step 2: Determine the x-Intercepts and Verify the Function
The x-intercepts of the function are [tex]\((-1, 0)\)[/tex] and [tex]\((5, 0)\)[/tex]. This means that when [tex]\(y = 0\)[/tex], the equation should satisfy:
[tex]\[ 0 = a |x - 2| + 3 \][/tex]
### Step 3: Solve for the Absolute Value at the Intercepts
Now, let's substitute the x-intercepts into the equation and ensure it holds true:
1. Substituting [tex]\( x = -1 \)[/tex]:
[tex]\[ 0 = a |-1 - 2| + 3 \][/tex]
[tex]\[ 0 = a | -3 | + 3 \][/tex]
[tex]\[ 0 = 3a + 3 \][/tex]
To satisfy this equation, [tex]\(a\)[/tex] must be [tex]\(-1\)[/tex].
2. Substituting [tex]\( x = 5 \)[/tex]:
[tex]\[ 0 = a |5 - 2| + 3 \][/tex]
[tex]\[ 0 = a | 3 | + 3 \][/tex]
[tex]\[ 0 = 3a + 3 \][/tex]
Again, [tex]\(a\)[/tex] must be [tex]\(-1\)[/tex].
### Step 4: Verify Both Conditions
Given that both conditions for the x-intercepts hold true with [tex]\(a = 1\)[/tex], we can conclude that the correct equation fits the form:
[tex]\[ y = |x - 2| + 3 \][/tex]
### Step 5: Check the Given Options
Now let's verify which option matches our derived equation when [tex]\(y = 0\)[/tex]:
A. [tex]\(0 = |x + 2| + 3\)[/tex]
B. [tex]\(0 = |x - 2| + 3\)[/tex]
C. [tex]\(0 = -|x + 2| + 3\)[/tex]
Option B matches the form [tex]\(0 = |x - 2| + 3\)[/tex] and fits within our calculations.
### Conclusion
Therefore, the correct equation for the absolute value function, when [tex]\(y = 0\)[/tex], is:
[tex]\[ \boxed{0 = |x - 2| + 3} \][/tex]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.