The table of values represents a linear function g(x), where x is the number of days that have passed and g(x) is the balance in the bank account:

xg(x)

0

$325

5

$400

10

$475

**Part A:** Find and interpret the slope of the function. (3 points)

**Part B:** Write the equation of the line in point-slope, slope-intercept, and standard forms. (3 points)

**Part C:** Write the equation of the line using function notation. (2 points)

**Part D:** What is the balance in the bank account after 12 days? (2 points)

Let's break this down step by step:

### **Part A: Find and interpret the slope of the function. (3 points)**

The slope of a linear function represents the rate of change of the dependent variable (in this case, the bank balance $ g(x) $) with respect to the independent variable $ x $ (the number of days).

The formula for the slope $ m $ is:

\[
m = \frac{\Delta y}{\Delta x} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\]

Using the two points from the table:

- $ (x_1, g(x_1)) = (0, 325) $
- $ (x_2, g(x_2)) = (5, 400) $

Substitute into the slope formula:

\[
m = \frac{400 - 325}{5 - 0} = \frac{75}{5} = 15
\]

**Interpretation:**  
The slope is 15, which means the balance in the bank account increases by $15 per day.

### **Part B: Write the equation of the line in point-slope, slope-intercept, and standard forms. (3 points)**

1. **Point-slope form:**  
   The point-slope form of a line is:

   \[
   y - y_1 = m(x - x_1)
   \]

   Using the slope $ m = 15 $ and the point $ (0, 325) $:

   \[
   g(x) - 325 = 15(x - 0)
   \]

   Simplified:

   \[
   g(x) - 325 = 15x
   \]

2. **Slope-intercept form:**  
   The slope-intercept form of a line is:

   \[
   g(x) = mx + b
   \]

   From the slope $ m = 15 $ and the initial value $ g(0) = 325 $, the equation is:

   \[
   g(x) = 15x + 325
   \]

3. **Standard form:**  
   The standard form of a line is:

   \[
   Ax + By = C
   \]

   Rearranging the slope-intercept form $ g(x) = 15x + 325 $:

   \[
   15x - g(x) = -325
   \]

   Multiplying through by -1:

   \[
   -15x + g(x) = 325
   \]

   Or:

   \[
   15x - g(x) = -325
   \]

### **Part C: Write the equation of the line using function notation. (2 points)**

In function notation, the equation is simply:

\[
g(x) = 15x + 325
\]

This represents the same linear equation as in Part B but written explicitly as a function.

### **Part D: What is the balance in the bank account after 12 days? (2 points)**

To find the balance after 12 days, substitute $ x = 12 $ into the equation:

\[
g(12) = 15(12) + 325 = 180 + 325 = 505
\]

So, the balance after 12 days is **$505**.

---

Would you like any further details or clarification on any part?

### 5 Related Questions:
1. How would the slope change if the bank balance decreased over time instead of increasing?
2. What is the significance of the y-intercept in this problem?
3. How can you use this model to predict the balance after 20 days?
4. How do you convert between point-slope form and slope-intercept form?
5. What happens to the equation if there was an initial fee, reducing the balance at the start?

**Tip:** The slope in a linear function always represents the rate of change, whether it's an increase or decrease.

The table of values represents a linear function g(x), where x is the number of days that have passed and g(x) is the balance in the bank account:

xg(x)

0
$325

5
$400

10
$475

**Part A:** Find and interpret the slope of the function. (3 points)

**Part B:** Write the equation of the line in point-slope, slope-intercept, and standard forms. (3 points)

**Part C:** Write the equation of the line using function notation. (2 points)

**Part D:** What is the balance in the bank account after 12 days? (2 points)

Learn how to find and interpret the slope of a linear function based on bank account balance over time. This example covers writing equations in point-slope, slope-intercept, and standard forms, using function notation, and calculating future balances. Suitable for Grades 8-10.

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