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:

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 7 days? (2 points)

Explanation:

Part A:

Step1: Calculate the slope

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(0,600)\) and \((x_2,y_2)=(3,720)\). Then \(m=\frac{720 - 600}{3-0}=\frac{120}{3}=40\).

Step2: Interpret the slope

The slope \(m = 40\) means that the balance in the bank account increases by \(40\) dollars per day.

Part B:

Step1: Point - slope form

The point - slope form is \(y - y_1=m(x - x_1)\). Using the point \((0,600)\) and \(m = 40\), we get \(y-600=40(x - 0)\) or \(y - 600=40x\).

Step2: Slope - intercept form

The slope - intercept form is \(y=mx + b\). Since \(m = 40\) and \(b = 600\) (from the point \((0,600)\)), the equation is \(y=40x+600\).

Step3: Standard form

The standard form is \(Ax+By = C\). Starting from \(y=40x + 600\), we subtract \(40x\) from both sides to get \(-40x+y=600\) or \(40x-y=- 600\).

Part C:

Step1: Function notation

Since \(y = g(x)\) and \(y=40x + 600\), the function notation is \(g(x)=40x+600\).

Part D:

Step1: Substitute \(x = 7\)

Substitute \(x = 7\) into \(g(x)=40x+600\). Then \(g(7)=40\times7+600\).

Step2: Calculate the value

\(g(7)=280 + 600=880\).

Answer:

• Part A: The slope is \(40\), which means the balance increases by \(40\) dollars per day.
• Part B:
• Point - slope form: \(y - 600=40x\)
• Slope - intercept form: \(y=40x + 600\)
• Standard form: \(40x-y=-600\)
• Part C: \(g(x)=40x + 600\)
• Part D: The balance after \(7\) days is \(\$880\)