Question
Fill in the blank pls
Function:
(x+___) (x-__)
end behavior
as x goes to -infinity f(x) goes to __
as x goes to positive infinity f(x) goes to ___
roots (with mutiplicity)
x=__ mutiplicity of ___
x=___ mutiplicity of ___
value of leading coefeecient 1
DOmain (____,)_____
Range (____,) ____
Other: f(0)=16
Function:
(x+___) (x-__)
end behavior
as x goes to -infinity f(x) goes to __
as x goes to positive infinity f(x) goes to ___
roots (with mutiplicity)
x=__ mutiplicity of ___
x=___ mutiplicity of ___
value of leading coefeecient 1
DOmain (____,)_____
Range (____,) ____
Other: f(0)=16
Answers
GPT-4o mini
To fill in the blanks for the given function format and characteristics, let's analyze the provided information. We can express the function in the form:
\[ f(x) = (x + a)(x - b) \]
From the data you provided, we need to determine the appropriate values for \( a \) and \( b \) that meet the conditions specified.
### 1. Function:
Therefore, let's fill in the values:
- Function:
\[ (x + a)(x - b) \]
### 2. End Behavior:
For a quadratic function, the leading term will dictate the end behavior:
- As \( x \) goes to \(-\infty\), \( f(x) \) goes to \(\infty\) (if leading coefficient is positive).
- As \( x \) goes to \(+\infty\), \( f(x) \) goes to \(\infty\) (if leading coefficient is positive).
- So we would write:
- As \( x \) goes to \(-\infty\), \( f(x) \) goes to \( \infty \).
- As \( x \) goes to \(+\infty\), \( f(x) \) goes to \( \infty \).
### 3. Roots (with multiplicity):
The roots of the function are represented by \( x + a = 0 \) (root at \(-a\)) and \( x - b = 0 \) (root at \(b\)).
- Roots:
- \( x = -a \) multiplicity of 1
- \( x = b \) multiplicity of 1
### 4. Value of the Leading Coefficient:
You stated the "value of leading coeefecient" is 1, meaning:
- Leading coefficient: \( 1 \)
### 5. Domain:
The domain of any quadratic function is all real numbers:
- Domain: \((-\infty, \infty)\)
### 6. Range:
Since the leading coefficient is positive and the parabola opens upwards, the minimum value occurs at the vertex.
Since we also have the value \( f(0) = 16 \):
- This means likely the vertex h (in the vertex form \( f(x) = a(x-h)^2+k \)) is above the x-axis, indicating a minimum point of the parabola is above 0.
Thus:
- Range: \([k, \infty)\), where \( k = f(0) = 16 \). Therefore:
- Range: \([16, \infty)\)
### 7. Other:
We already know that \( f(0) = 16 \).
### Summary Filled Out:
Here’s the full filled-in version with what we've deduced:
Function:
\((x + a)(x - b)\)
End behavior:
- As \( x \to -\infty\), \( f(x) \to \infty\)
- As \( x \to +\infty\), \( f(x) \to \infty\)
Roots (with multiplicity):
- \( x = -a\), multiplicity of 1
- \( x = b\), multiplicity of 1
Value of leading coefficient: 1
Domain: \((-\infty, \infty)\)
Range: \([16, \infty)\)
Other: \( f(0) = 16\)
To further determine \( a \) and \( b \), more specific values or interactions would be needed.
\[ f(x) = (x + a)(x - b) \]
From the data you provided, we need to determine the appropriate values for \( a \) and \( b \) that meet the conditions specified.
### 1. Function:
Therefore, let's fill in the values:
- Function:
\[ (x + a)(x - b) \]
### 2. End Behavior:
For a quadratic function, the leading term will dictate the end behavior:
- As \( x \) goes to \(-\infty\), \( f(x) \) goes to \(\infty\) (if leading coefficient is positive).
- As \( x \) goes to \(+\infty\), \( f(x) \) goes to \(\infty\) (if leading coefficient is positive).
- So we would write:
- As \( x \) goes to \(-\infty\), \( f(x) \) goes to \( \infty \).
- As \( x \) goes to \(+\infty\), \( f(x) \) goes to \( \infty \).
### 3. Roots (with multiplicity):
The roots of the function are represented by \( x + a = 0 \) (root at \(-a\)) and \( x - b = 0 \) (root at \(b\)).
- Roots:
- \( x = -a \) multiplicity of 1
- \( x = b \) multiplicity of 1
### 4. Value of the Leading Coefficient:
You stated the "value of leading coeefecient" is 1, meaning:
- Leading coefficient: \( 1 \)
### 5. Domain:
The domain of any quadratic function is all real numbers:
- Domain: \((-\infty, \infty)\)
### 6. Range:
Since the leading coefficient is positive and the parabola opens upwards, the minimum value occurs at the vertex.
Since we also have the value \( f(0) = 16 \):
- This means likely the vertex h (in the vertex form \( f(x) = a(x-h)^2+k \)) is above the x-axis, indicating a minimum point of the parabola is above 0.
Thus:
- Range: \([k, \infty)\), where \( k = f(0) = 16 \). Therefore:
- Range: \([16, \infty)\)
### 7. Other:
We already know that \( f(0) = 16 \).
### Summary Filled Out:
Here’s the full filled-in version with what we've deduced:
Function:
\((x + a)(x - b)\)
End behavior:
- As \( x \to -\infty\), \( f(x) \to \infty\)
- As \( x \to +\infty\), \( f(x) \to \infty\)
Roots (with multiplicity):
- \( x = -a\), multiplicity of 1
- \( x = b\), multiplicity of 1
Value of leading coefficient: 1
Domain: \((-\infty, \infty)\)
Range: \([16, \infty)\)
Other: \( f(0) = 16\)
To further determine \( a \) and \( b \), more specific values or interactions would be needed.
sayletyler
For the function I need numbers
GPT-4o mini
Let's determine some specific numbers to fill in the blanks for the function and other characteristics. We know:
1. \( f(0) = 16 \)
2. The function is in the form \( f(x) = (x + a)(x - b) \) or \( f(x) = x^2 + (a - b)x - ab \).
Now let's find specific values for \( a \) and \( b\) based on the information provided.
### Step 1: Find \( a \) and \( b \)
Since we are looking for \( f(0) = 16 \):
\[
f(0) = (0 + a)(0 - b) = a(-b) = -ab = 16
\]
This means:
\[
ab = -16
\]
### Step 2: Choose specific values for \( a \) and \( b \)
Let's assume:
- Let \( a = 4 \)
- Let \( b = -4 \)
This satisfies:
\[
ab = 4 \times (-4) = -16
\]
### Step 3: Rebuild Function and Apply Values
1. **Function:**
\[
f(x) = (x + 4)(x + 4) = (x + 4)^2
\]
(This is a perfect square and hence has multiplicity 2.)
### Summary Filled in with Specific Numbers
- **Function:**
\[
(x + 4)(x-(-4)) = (x + 4)(x + 4)
\]
- **End behavior:**
- As \( x \) goes to \(-\infty\), \( f(x) \) goes to \(\infty\)
- As \( x \) goes to \(+\infty\), \( f(x) \) goes to \(\infty\)
- **Roots (with multiplicity):**
- \( x = -4\), multiplicity of 2
- **Value of leading coefficient:** \(1\)
- **Domain:** \((-\infty, \infty)\)
- **Range:** \([16, \infty)\)
- **Other:** \( f(0) = 16\)
### Final Summary:
Now filling in all values with specific numbers:
- Function: **\((x + 4)(x + 4)\)**
- End behavior:
- As \( x \to -\infty\), \( f(x) \to \infty\)
- As \( x \to +\infty\), \( f(x) \to \infty\)
- Roots (with multiplicity):
- \( x = -4\), multiplicity of 2
- Value of leading coefficient: **1**
- Domain: \((-\infty, \infty)\)
- Range: \([16, \infty)\)
- Other: \( f(0) = 16\)
If you have different numbers in mind for \( a \) and \( b \) or any additional restrictions or changes, please let me know!
1. \( f(0) = 16 \)
2. The function is in the form \( f(x) = (x + a)(x - b) \) or \( f(x) = x^2 + (a - b)x - ab \).
Now let's find specific values for \( a \) and \( b\) based on the information provided.
### Step 1: Find \( a \) and \( b \)
Since we are looking for \( f(0) = 16 \):
\[
f(0) = (0 + a)(0 - b) = a(-b) = -ab = 16
\]
This means:
\[
ab = -16
\]
### Step 2: Choose specific values for \( a \) and \( b \)
Let's assume:
- Let \( a = 4 \)
- Let \( b = -4 \)
This satisfies:
\[
ab = 4 \times (-4) = -16
\]
### Step 3: Rebuild Function and Apply Values
1. **Function:**
\[
f(x) = (x + 4)(x + 4) = (x + 4)^2
\]
(This is a perfect square and hence has multiplicity 2.)
### Summary Filled in with Specific Numbers
- **Function:**
\[
(x + 4)(x-(-4)) = (x + 4)(x + 4)
\]
- **End behavior:**
- As \( x \) goes to \(-\infty\), \( f(x) \) goes to \(\infty\)
- As \( x \) goes to \(+\infty\), \( f(x) \) goes to \(\infty\)
- **Roots (with multiplicity):**
- \( x = -4\), multiplicity of 2
- **Value of leading coefficient:** \(1\)
- **Domain:** \((-\infty, \infty)\)
- **Range:** \([16, \infty)\)
- **Other:** \( f(0) = 16\)
### Final Summary:
Now filling in all values with specific numbers:
- Function: **\((x + 4)(x + 4)\)**
- End behavior:
- As \( x \to -\infty\), \( f(x) \to \infty\)
- As \( x \to +\infty\), \( f(x) \to \infty\)
- Roots (with multiplicity):
- \( x = -4\), multiplicity of 2
- Value of leading coefficient: **1**
- Domain: \((-\infty, \infty)\)
- Range: \([16, \infty)\)
- Other: \( f(0) = 16\)
If you have different numbers in mind for \( a \) and \( b \) or any additional restrictions or changes, please let me know!
sayletyler
All I want is the answers to the blanks
GPT-4o mini
Apologies for the misunderstanding. Here are the answers to your blanks:
Function:
**(x + 4)(x - (-4))**
End behavior:
As \( x \) goes to \(-\infty\), \( f(x) \) goes to **\(\infty\)**
As \( x \) goes to \(+\infty\), \( f(x) \) goes to **\(\infty\)**
Roots (with multiplicity):
\( x = -4 \) multiplicity of **2**
\( x = 4 \) multiplicity of **1**
Value of leading coefficient: **1**
Domain: \((-∞, ∞)\)
Range: \([16, ∞)\)
Other: \( f(0) = 16 \)
If any of these values need adjusting based on specific criteria, please let me know!
Function:
**(x + 4)(x - (-4))**
End behavior:
As \( x \) goes to \(-\infty\), \( f(x) \) goes to **\(\infty\)**
As \( x \) goes to \(+\infty\), \( f(x) \) goes to **\(\infty\)**
Roots (with multiplicity):
\( x = -4 \) multiplicity of **2**
\( x = 4 \) multiplicity of **1**
Value of leading coefficient: **1**
Domain: \((-∞, ∞)\)
Range: \([16, ∞)\)
Other: \( f(0) = 16 \)
If any of these values need adjusting based on specific criteria, please let me know!