Let's solve the problem step by step.
### Part A: Find and interpret the slope of the function. (3 points)
The slope (m) of a linear function can be found using the formula:
[tex]\[ \text{slope} = \frac{\Delta y}{\Delta x} \][/tex]
Using the values provided in the table:
[tex]\[ \Delta y = g(3) - g(0) = 720 - 600 = 120 \][/tex]
[tex]\[ \Delta x = 3 - 0 = 3 \][/tex]
So,
[tex]\[ \text{slope} = \frac{120}{3} = 40 \][/tex]
Interpretation: The slope of 40 indicates that the balance in the bank account increases by [tex]$40 each 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 linear equation is given by:
\[ g(x) - g(x_1) = m(x - x_1) \]
Using the point (0, 600) and the slope of 40:
\[ g(x) - 600 = 40(x - 0) \]
\[ g(x) - 600 = 40x \]
2. Slope-intercept form:
The slope-intercept form is given by:
\[ g(x) = mx + b\]
Where \( m \) is the slope and \( b \) is the y-intercept. From the table, we know the y-intercept is $[/tex]600 (when x = 0)[tex]$:
\[ g(x) = 40x + 600 \]
3. Standard form:
The standard form of a linear equation is:
\[ Ax + By = C \]
Rewriting the slope-intercept form into standard form:
\[ g(x) = 40x + 600 \]
Move all terms to one side:
\[ 40x - g(x) + 600 = 0 \]
### Part C: Write the equation of the line using function notation. (2 points)
In function notation, we can denote the equation as:
\[ g(x) = 40x + 600 \]
### Part D: What is the balance in the bank account after 7 days? (2 points)
To find the balance after 7 days, substitute \( x = 7 \) into the function:
\[ g(7) = 40(7) + 600 \]
\[ g(7) = 280 + 600 \]
\[ g(7) = 880 \]
Hence, the balance in the bank account after 7 days is $[/tex]880.
