Extended Response Questions. 1. Graph $$ y=x^{3}+2 x^{2}-3 x-4 $$ a) Provide a t table with the $$ y $$ intercept over [-4, 1].

Question

UpStudy AI · Accepted Answer

Certainly! Below is a table of values for the function $$ y = x^{3} + 2x^{2} - 3x - 4 $$ over the interval $$[-4, 1]$$. This table includes the y-intercept, which occurs at $$ x = 0 $$.

### Table of Values

| $$ x $$ | $$ y = x^{3} + 2x^{2} - 3x - 4 $$ |
|:-------:|:---------------------------------:|
|  **-4** |             **-24**               |
|  **-3** |              **-4**               |
|  **-2** |               **2**               |
|  **-1** |               **0**               |
|   **0** |             **-4** _(y-intercept)_|
|   **1** |             **-4**               |

### Calculation Details

1. **For $$ x = -4 $$:**
   $$
   y = (-4)^3 + 2(-4)^2 - 3(-4) - 4 = -64 + 32 + 12 - 4 = -24
   $$

2. **For $$ x = -3 $$:**
   $$
   y = (-3)^3 + 2(-3)^2 - 3(-3) - 4 = -27 + 18 + 9 - 4 = -4
   $$

3. **For $$ x = -2 $$:**
   $$
   y = (-2)^3 + 2(-2)^2 - 3(-2) - 4 = -8 + 8 + 6 - 4 = 2
   $$

4. **For $$ x = -1 $$:**
   $$
   y = (-1)^3 + 2(-1)^2 - 3(-1) - 4 = -1 + 2 + 3 - 4 = 0
   $$

5. **For $$ x = 0 $$ (Y-Intercept):**
   $$
   y = 0^3 + 2(0)^2 - 3(0) - 4 = -4
   $$

6. **For $$ x = 1 $$:**
   $$
   y = 1^3 + 2(1)^2 - 3(1) - 4 = 1 + 2 - 3 - 4 = -4
   $$

### Graphical Representation

Using these points, you can plot the graph of the function $$ y = x^{3} + 2x^{2} - 3x - 4 $$ over the interval $$[-4, 1]$$. The y-intercept is clearly visible at the point $$(0, -4)$$.

![Graph of y = x^3 + 2x^2 - 3x - 4](https://i.imgur.com/yourgraphlink.png)

*(Please replace the image link with an actual graph if needed.)*

### Summary

- **Y-Intercept:** The graph intersects the y-axis at $$(0, -4)$$.
- **Behavior:**
  - As $$ x $$ decreases from 1 to -4, the function value decreases initially, reaches a minimum, and then increases.
  - The function crosses the x-axis at $$ x = -1 $$.

Feel free to plot these points on graph paper or use graphing software to visualize the behavior of the function more clearly.
Here is the table of values for the function $$ y = x^{3} + 2x^{2} - 3x - 4 $$ over the interval $$[-4, 1]$$: | $$ x $$ | $$ y $$ | |:-------:|:-------:| | -4 | -24 | | -3 | -4 | | -2 | 2 | | -1 | 0 | | 0 | -4 | | 1 | -4 | The y-intercept is at $$(0, -4)$$.

Extended Response Questions.

Graph
a) Provide a t table with the intercept over [-4, 1].

Upstudy AI Solution

Answer

Solution

Table of Values

Calculation Details

Graphical Representation

Summary

Beyond the Answer

Related Questions

Latest Algebra Questions


-4	-24
-3	-4
-2	2
-1	0
0	-4 (y-intercept)
1	-4

Extended Response Questions. Graph a) Provide a t table with the intercept over [-4, 1].