Question

(02.04 LC)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$3255$40010$475Part A: Find and interpret the slope of the function.Part B: Write the equation of the line in point-slope, slope-intercept, and standard forms. Part C: Write the equation of the line using function notation. Part D: What is the balance in the bank account after 12 days?

Answer 1

Part A

The slope m of a linear function can be found by dividing the difference between two outputs by the difference between the respective inputs:

`m=(g(x_2)-g(x_1))/(x_2-x_1)`

In this case, we can use:

`\begin{gathered} x_1=0 \\ x_2=5 \\ g(x_1)=325 \\ g(x_2)=400 \end{gathered}`

So, we obtain:

`m=(400-325)/(5-0)=(75)/(5)=15`

Notice that, since x is the number of days and g(x) is the balance in dollars, this slope means that the balance increases $15 per day.

Part B

Point-slope form

The equation of the line with slope m, passing through point (x1, y1), in point-slope form is:

`y-y_1=m(x-x_1)`

Using the previous result for m, and the point (5, 400), we obtain the equation:

`y-400=15(x-5)`

Slope-intercept form

The slope-intercept form of a linear equation is

`y=mx+b`

where b is the y-intercept, i.e., the value of y when x = 0. Since y = 325 for x = 0, we have the following equation:

`y=15x+325`

Standard form

The standard form of a linear equation is:

`Ax+By=C`

where A, B, and C are constants.

Rearranging the terms in the previous equation, we obtain:

`y-15x=325`

Part C

Using function notation, we can replace y by g(x) in the slope-intercept. We obtain:

`g(x)=15x+325`

Part D

After 12 days, the balance in the bank account can be found by using x = 12 in the above equation:

`g(12)=15\cdot12+325=180+325=505`

Therefore, the balance in the bank account after 12 days is $505.