Question

two functions are described below.

• function a: a taxi driver charges customers a base amount of $3.00 and also an amount of $2.00 per mile, x, for a total charge, y, to a customer.
• function b: the equation y = 3x + 2 represents the relationship between the number of miles, x, a taxi travels and the total charge, y, to a customer.

which statement correctly compares the relationship between function a and function b ?
a function a has both a greater rate of change and a greater initial value.
b function b has both a greater rate of change and a greater initial value.
c function a has a greater rate of change than function b, but the initial value for function a is less than the initial value for function b.
d function b has a greater rate of change than function a, but the initial value for function b is less than the initial value for function a.

Explanation:

Step1: Find Function A's equation

Function A: base $3, $2 per mile. So equation is $y = 2x + 3$.

Step2: Identify rate of change (slope) and initial value (y - intercept) for both functions

• Function A: slope ($m$) = 2, y - intercept ($b$) = 3.
• Function B: equation $y = 3x + 2$, so slope ($m$) = 3, y - intercept ($b$) = 2.

Step3: Compare rate of change and initial value

• Rate of change: 3 (Function B) > 2 (Function A).
• Initial value: 3 (Function A) > 2 (Function B). So Function B has greater rate of change, but less initial value than Function A.

Answer:

D. Function B has a greater rate of change than Function A, but the initial value for Function B is less than the initial value for Function A.