Asked by sayletyler
Fill in the blank
function (in factored form)
red circle=___(x+____)(x-____)
function in standard form
=-2(x^3-___x+___
end behavior
as x goes to -infinity f(x) goes to ___
as x goes to infinity f(x) goes to __
roots with mutiplicity
-2,1 (mutiplicity 2)
value of leading coeffecient -2
degree 3
function (in factored form)
red circle=___(x+____)(x-____)
function in standard form
=-2(x^3-___x+___
end behavior
as x goes to -infinity f(x) goes to ___
as x goes to infinity f(x) goes to __
roots with mutiplicity
-2,1 (mutiplicity 2)
value of leading coeffecient -2
degree 3
Answers
Answered by
sayletyler
Just the ansswers to the blanks all I need
Answered by
sayletyler
Oh and also graph this too pls just tell me where to plot the points
Answered by
sayletyler
Like where is 0 on a square graph
Answered by
GPT-4o mini
Based on the details you provided, here's how we can fill in the blanks step by step:
### Factored Form
1. The red circle function in factored form:
- Starting with the roots you provided:
- Root at \( x = -2 \) gives a factor of \( (x + 2) \)
- Root at \( x = 1 \) with multiplicity 2 gives a factor of \( (x - 1)^2 \)
Thus, the function in factored form can be expressed as:
\[ \text{red circle} = -2(x + 2)(x - 1)^2 \]
Filling the blanks:
- \( = -2(x + 2)(x - 1) \)
- \( \text{First blank: } 2 \)
- \( \text{Second blank: } 1 \)
- \( \text{Third blank: } 1 \)
### Standard Form
- The function in standard form is derived from multiplying the factors:
- \( = -2(x + 2)(x - 1)(x - 1) \)
When expanded, it results in:
\[ -2 \left( (x^2 - 2x + 1)(x + 2) \right) = -2(x^3 - 2x^2 + x + 2x^2 - 4x - 2) = -2(x^3 - 1x - 2) \]
So, in standard form:
\[ = -2(x^3 - 1x - 2) \]
Filling the blanks:
- \( = -2(x^3 - 1x + 2) \)
- \( \text{Fourth blank: } 1 \)
- \( \text{Fifth blank: } 2 \)
### End Behavior
- End behavior:
- As \( x \to -\infty \), the leading coefficient is negative, thus \( f(x) \to +\infty \).
- As \( x \to \infty \), \( f(x) \to -\infty \).
Filling the blanks:
- \( \text{End behavior as } x \to -\infty: f(x) \to +\infty \)
- \( \text{End behavior as } x \to \infty: f(x) \to -\infty \)
### Summary of the Blanks
1. **Factored Form:**
- Red circle = \(-2(x+2)(x-1)(x-1)\)
- (Blanks: **2, 1**)
2. **Standard Form:**
- \(-2(x^3 - 1x + 2)\)
- (Blanks: **1, 2**)
3. **End Behavior:**
- As \( x \to -\infty, f(x) \to +\infty \)
- As \( x \to \infty, f(x) \to -\infty \)
4. **Roots with Multiplicity:**
- \( -2, 1 \) (multiplicity \(2\))
5. **Leading Coefficient:**
- \( -2 \)
6. **Degree:**
- \( 3 \)
This completes your function description!
### Factored Form
1. The red circle function in factored form:
- Starting with the roots you provided:
- Root at \( x = -2 \) gives a factor of \( (x + 2) \)
- Root at \( x = 1 \) with multiplicity 2 gives a factor of \( (x - 1)^2 \)
Thus, the function in factored form can be expressed as:
\[ \text{red circle} = -2(x + 2)(x - 1)^2 \]
Filling the blanks:
- \( = -2(x + 2)(x - 1) \)
- \( \text{First blank: } 2 \)
- \( \text{Second blank: } 1 \)
- \( \text{Third blank: } 1 \)
### Standard Form
- The function in standard form is derived from multiplying the factors:
- \( = -2(x + 2)(x - 1)(x - 1) \)
When expanded, it results in:
\[ -2 \left( (x^2 - 2x + 1)(x + 2) \right) = -2(x^3 - 2x^2 + x + 2x^2 - 4x - 2) = -2(x^3 - 1x - 2) \]
So, in standard form:
\[ = -2(x^3 - 1x - 2) \]
Filling the blanks:
- \( = -2(x^3 - 1x + 2) \)
- \( \text{Fourth blank: } 1 \)
- \( \text{Fifth blank: } 2 \)
### End Behavior
- End behavior:
- As \( x \to -\infty \), the leading coefficient is negative, thus \( f(x) \to +\infty \).
- As \( x \to \infty \), \( f(x) \to -\infty \).
Filling the blanks:
- \( \text{End behavior as } x \to -\infty: f(x) \to +\infty \)
- \( \text{End behavior as } x \to \infty: f(x) \to -\infty \)
### Summary of the Blanks
1. **Factored Form:**
- Red circle = \(-2(x+2)(x-1)(x-1)\)
- (Blanks: **2, 1**)
2. **Standard Form:**
- \(-2(x^3 - 1x + 2)\)
- (Blanks: **1, 2**)
3. **End Behavior:**
- As \( x \to -\infty, f(x) \to +\infty \)
- As \( x \to \infty, f(x) \to -\infty \)
4. **Roots with Multiplicity:**
- \( -2, 1 \) (multiplicity \(2\))
5. **Leading Coefficient:**
- \( -2 \)
6. **Degree:**
- \( 3 \)
This completes your function description!
Answered by
GPT-4o mini
Sure! Here are the answers to the blanks:
1. **Factored Form:**
- Red circle = \(-2(x + 2)(x - 1)(x - 1)\)
- First blank: **2**
- Second blank: **1**
- Third blank: **1**
2. **Standard Form:**
- \(-2(x^3 - 1x + 2)\)
- Fourth blank: **1**
- Fifth blank: **2**
3. **End Behavior:**
- As \( x \to -\infty, f(x) \to **+\infty**
- As \( x \to \infty, f(x) \to **-\infty**
So the complete answers are:
- **2**
- **1**
- **1**
- **1**
- **2**
- **+\infty**
- **-\infty**
1. **Factored Form:**
- Red circle = \(-2(x + 2)(x - 1)(x - 1)\)
- First blank: **2**
- Second blank: **1**
- Third blank: **1**
2. **Standard Form:**
- \(-2(x^3 - 1x + 2)\)
- Fourth blank: **1**
- Fifth blank: **2**
3. **End Behavior:**
- As \( x \to -\infty, f(x) \to **+\infty**
- As \( x \to \infty, f(x) \to **-\infty**
So the complete answers are:
- **2**
- **1**
- **1**
- **1**
- **2**
- **+\infty**
- **-\infty**
Answered by
GPT-4o mini
To plot the function \(-2(x + 2)(x - 1)^2\), we can identify the important points such as the roots, vertex, and key points. Here are the steps:
### Key Points to Plot
1. **Roots (x-intercepts)**:
- From the factorization, the roots are:
- \(x = -2\) (multiplicity 1)
- \(x = 1\) (multiplicity 2, meaning the graph will touch the x-axis at this point but not cross it)
**Points to plot:**
- \((-2, 0)\)
- \( (1, 0)\) (only plot for touch but note it's a single point)
2. **Y-intercept (where x = 0)**:
- Calculate \(f(0)\):
\[
f(0) = -2(0+2)(0-1)(0-1) = -2(2)(-1)(-1) = -2(2) = -4
\]
**Point to plot:**
- \((0, -4)\)
3. **End Behavior**:
- As \(x \to -\infty, f(x) \to +\infty\) (the graph will rise)
- As \(x \to +\infty, f(x) \to -\infty\) (the graph will fall)
4. **Additional Points**:
You may want to calculate a few more points to assist in plotting the curve:
- For example, calculate:
- \(f(-1)\)
\[
f(-1) = -2(-1 + 2)(-1 - 1)(-1 - 1) = -2(1)(-2)(-2) = -2 \times 1 \times 4 = -8
\]
- This gives you another point: \((-1, -8)\)
- You can also calculate \(f(2)\):
\[
f(2) = -2(2 + 2)(2 - 1)^2 = -2(4)(1) = -8
\]
- This gives you another point: \((2, -8)\)
### Summary of Points to Plot
1. \((-2, 0)\) (root)
2. \((1, 0)\) (root, touches)
3. \((0, -4)\) (y-intercept)
4. \((-1, -8)\) (additional point)
5. \((2, -8)\) (additional point)
### Overview of the Graph
- Your graph will start high on the left (due to end behavior).
- It will come down to \( -2 \) at \( (-2, 0) \), touch the x-axis at \( (1,0) \), then drop to the y-intercept at \( (0, -4) \), go down further to \((-1, -8)\) and \((2, -8)\), and continue downward on the right side.
Plot these points on a Cartesian plane to visualize the curve. The graph will show a "W" shape due to the presence of a double root at \(x = 1\). The behavior is important to indicate how the graph behaves at the extremes and around the roots. Feel free to connect the points smoothly to illustrate the cubic shape!
### Key Points to Plot
1. **Roots (x-intercepts)**:
- From the factorization, the roots are:
- \(x = -2\) (multiplicity 1)
- \(x = 1\) (multiplicity 2, meaning the graph will touch the x-axis at this point but not cross it)
**Points to plot:**
- \((-2, 0)\)
- \( (1, 0)\) (only plot for touch but note it's a single point)
2. **Y-intercept (where x = 0)**:
- Calculate \(f(0)\):
\[
f(0) = -2(0+2)(0-1)(0-1) = -2(2)(-1)(-1) = -2(2) = -4
\]
**Point to plot:**
- \((0, -4)\)
3. **End Behavior**:
- As \(x \to -\infty, f(x) \to +\infty\) (the graph will rise)
- As \(x \to +\infty, f(x) \to -\infty\) (the graph will fall)
4. **Additional Points**:
You may want to calculate a few more points to assist in plotting the curve:
- For example, calculate:
- \(f(-1)\)
\[
f(-1) = -2(-1 + 2)(-1 - 1)(-1 - 1) = -2(1)(-2)(-2) = -2 \times 1 \times 4 = -8
\]
- This gives you another point: \((-1, -8)\)
- You can also calculate \(f(2)\):
\[
f(2) = -2(2 + 2)(2 - 1)^2 = -2(4)(1) = -8
\]
- This gives you another point: \((2, -8)\)
### Summary of Points to Plot
1. \((-2, 0)\) (root)
2. \((1, 0)\) (root, touches)
3. \((0, -4)\) (y-intercept)
4. \((-1, -8)\) (additional point)
5. \((2, -8)\) (additional point)
### Overview of the Graph
- Your graph will start high on the left (due to end behavior).
- It will come down to \( -2 \) at \( (-2, 0) \), touch the x-axis at \( (1,0) \), then drop to the y-intercept at \( (0, -4) \), go down further to \((-1, -8)\) and \((2, -8)\), and continue downward on the right side.
Plot these points on a Cartesian plane to visualize the curve. The graph will show a "W" shape due to the presence of a double root at \(x = 1\). The behavior is important to indicate how the graph behaves at the extremes and around the roots. Feel free to connect the points smoothly to illustrate the cubic shape!
Answered by
GPT-4o mini
On a Cartesian coordinate system (which is typically represented as a square graph), the point where both the x-coordinate and y-coordinate are zero is called the **origin**.
### Coordinate System Overview:
1. **Axes**:
- The horizontal line is the **x-axis**.
- The vertical line is the **y-axis**.
- The point where these two axes intersect is the **origin** (0, 0).
2. **Quadrants**:
The Cartesian plane is divided into four quadrants:
- **Quadrant I**: Top right (both x and y are positive).
- **Quadrant II**: Top left (x is negative, y is positive).
- **Quadrant III**: Bottom left (both x and y are negative).
- **Quadrant IV**: Bottom right (x is positive, y is negative).
### Origin:
- The **origin** is located at:
- **Coordinates**: \((0, 0)\)
- It is in the center of the graph where the values of x and y are both zero.
### How to Locate Points:
To graph the points mentioned earlier:
- **\((-2, 0)\)**: Move 2 units to the left on the x-axis and stay on the x-axis (y = 0).
- **\((1, 0)\)**: Move 1 unit to the right on the x-axis and stay on the x-axis (y = 0).
- **\((0, -4)\)**: Stay at the origin, then move down 4 units on the y-axis.
- **\((-1, -8)\)**: Move left to -1 on the x-axis and down 8 units on the y-axis.
- **\((2, -8)\)**: Move right to 2 on the x-axis and down 8 units on the y-axis.
### Visual Representation:
Here’s a simple way to imagine the points on the graph:
```
y
|
5 |
4 |
3 |
2 |
1 |
----------|---------- x
-4 | (0, -4)
-5 |
-6 |
-7 |
-8 | (2, -8)
| (1,0)
|
| (0,0) Origin
-2 | (-2,0)
|
-3 |
```
This illustrates where you would place the key points on the Cartesian graph relative to the origin (0, 0). Simply correspond the coordinates to the described movements from the origin!
### Coordinate System Overview:
1. **Axes**:
- The horizontal line is the **x-axis**.
- The vertical line is the **y-axis**.
- The point where these two axes intersect is the **origin** (0, 0).
2. **Quadrants**:
The Cartesian plane is divided into four quadrants:
- **Quadrant I**: Top right (both x and y are positive).
- **Quadrant II**: Top left (x is negative, y is positive).
- **Quadrant III**: Bottom left (both x and y are negative).
- **Quadrant IV**: Bottom right (x is positive, y is negative).
### Origin:
- The **origin** is located at:
- **Coordinates**: \((0, 0)\)
- It is in the center of the graph where the values of x and y are both zero.
### How to Locate Points:
To graph the points mentioned earlier:
- **\((-2, 0)\)**: Move 2 units to the left on the x-axis and stay on the x-axis (y = 0).
- **\((1, 0)\)**: Move 1 unit to the right on the x-axis and stay on the x-axis (y = 0).
- **\((0, -4)\)**: Stay at the origin, then move down 4 units on the y-axis.
- **\((-1, -8)\)**: Move left to -1 on the x-axis and down 8 units on the y-axis.
- **\((2, -8)\)**: Move right to 2 on the x-axis and down 8 units on the y-axis.
### Visual Representation:
Here’s a simple way to imagine the points on the graph:
```
y
|
5 |
4 |
3 |
2 |
1 |
----------|---------- x
-4 | (0, -4)
-5 |
-6 |
-7 |
-8 | (2, -8)
| (1,0)
|
| (0,0) Origin
-2 | (-2,0)
|
-3 |
```
This illustrates where you would place the key points on the Cartesian graph relative to the origin (0, 0). Simply correspond the coordinates to the described movements from the origin!
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