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

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.
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!
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!

Related Questions