AI multiplication learning helper

It seems there might be some confusion in your request. The slope-intercept form is typically used in linear equations and is expressed as ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept. This form is not directly related to multiplication, but it does involve multiplication when calculating the value of ( y ) for a given ( x ).

If you’re looking to understand how to use tables to represent or solve problems involving the slope-intercept form, I can certainly help with that. Here’s an example:

### Example: Using a Table to Represent a Linear Equation

Suppose we have the linear equation ( y = 2x + 3 ). We can create a table to show different values of ( x ) and the corresponding values of ( y ):

| ( x ) | ( y = 2x + 3 ) |
|——–|——————|
| -1 | ( 2(-1) + 3 = 1 ) |
| 0 | ( 2(0) + 3 = 3 ) |
| 1 | ( 2(1) + 3 = 5 ) |
| 2 | ( 2(2) + 3 = 7 ) |
| 3 | ( 2(3) + 3 = 9 ) |

In this table, each value of ( y ) is calculated by multiplying the value of ( x ) by 2 (the slope) and then adding 3 (the y-intercept).

If you have a specific question or need further clarification on how to use tables with the slope-intercept form, feel free to ask!

I will only address the multiplication-related parts of your questions.

### Multiplication-Related Questions:

**II. Given that (sin(x+y) = sin x cos y + sin y cos x), determine (sin 2x).**

To determine (sin 2x), we can use the given identity by setting (y = x):

[
sin(x + x) = sin x cos x + sin x cos x
]

This simplifies to:

[
sin 2x = 2 sin x cos x
]

So, the value of (sin 2x) is (2 sin x cos x).

**V. Consider the function (f: mathbb{R} to mathbb{R}) defined by (f(x) = ax^2 + 3x – 2). Determine the derivative (f'(x)) of (f(x)). Deduce (f'(1)). Given that (f'(1) = 2a + 3), determine the real number (a). For which value(s) of (a) is (f'(1) = 0)?**

1. **Determine the derivative (f'(x)):**

The function is (f(x) = ax^2 + 3x – 2). To find the derivative, we apply the power rule:

[
f'(x) = 2ax + 3
]

2. **Deduce (f'(1)):**

Substitute (x = 1) into the derivative:

[
f'(1) = 2a(1) + 3 = 2a + 3
]

3. **Given that (f'(1) = 2a + 3), determine the real number (a):**

We are given that (f'(1) = 2a + 3). This is already in the form we derived, so no further action is needed for this part.

4. **For which value(s) of (a) is (f'(1) = 0)?**

Set (f'(1) = 0):

[
2a + 3 = 0
]

Solve for (a):

[
2a = -3
] [
a = -frac{3}{2}
]

So, the value of (a) for which (f'(1) = 0) is (-frac{3}{2}).

If you have any more multiplication-related questions, feel free to ask!

Sure! Here is the answer to your multiplication question:

1 × 1 = 1

If you have any other multiplication questions, feel free to ask!