To understand how the modification to the function affects it, let's analyze each option.
The original function is:
[tex]\[ F(x) = 650,000 (1.04)^x \][/tex]
This represents annual sales increasing at a rate of 4% per year.
Let's break down the modified function:
[tex]\[ F(x) = 650,000 (1.04)^x \cdot 3 \][/tex]
Now let's consider the effects one by one:
Option A: It changes the interest rate from 4% per year to 12% per year.
- The interest rate in the original function is 4%, represented by the factor [tex]\((1.04)\)[/tex]. This means there is a compound growth of 4% per year.
- Multiplying by 3 does not affect the growth rate itself; it only scales the function. The growth rate remains 4%, not 12%.
Option B: It changes the x-intercept of the graph of the function from 1 to 3.
- The x-intercept of a function is the value of [tex]\( x \)[/tex] when [tex]\( F(x) = 0 \)[/tex].
- For exponential functions of this type, there is no finite x-intercept because the exponential function never actually reaches zero.
Option C: It changes the amount of time that the money is invested for from [tex]\( x \)[/tex] to [tex]\( 3x \)[/tex].
- The term [tex]\( x \)[/tex] in the exponential function represents time.
- Multiplying the entire function by 3 affects the output value of the function but it does not affect the exponent [tex]\( x \)[/tex], which still represents the same amount of time.
Option D: It changes the y-intercept of the graph of the function from 650,000 to 1,950,000.
- The y-intercept of a function is the value of [tex]\( F(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- For the original function, substituting [tex]\( x = 0 \)[/tex] gives:
[tex]\[ F(0) = 650,000 (1.04)^0 = 650,000 \][/tex]
- For the modified function, substituting [tex]\( x = 0 \)[/tex] gives:
[tex]\[ F(0) = 650,000 (1.04)^0 \cdot 3 = 650,000 \times 3 = 1,950,000 \][/tex]
So, the correct answer is:
D. It changes the y-intercept of the graph of the function from 650,000 to 1,950,000.
