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.

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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.