Use the properties of exponents to rewrite the exponential function y=750(1.6)t4 in the form y=a(1+r)x .Select the correct answers from the lists.(1 point)The function is growing, decaying,regreten s at a constant percent rate of change of about 88 8.8 12 1.2%.

Answer 1

To rewrite the given exponential function \( y = 750(1.6)^{t4} \) in the form \( y = a(1+r)^x \), let's start by clarifying the expression.

Step 1: Rewrite the exponential function

The function can be interpreted as:

• \( a = 750 \)
• The expression \( 1.6^{t4} \) appears to mix variables; however, it can be rewritten to match the required form.

Using the properties of exponents, we can factor out the constant from \( t4 \): \[ y = 750 \cdot (1.6)^{(t \cdot 4)} = 750 \cdot (1.6^4)^{t} \]

Now we calculate \( 1.6^4 \): \[ 1.6^4 = 6.5536 \] So the function can now be written as: \[ y = 750 \cdot (6.5536)^{t} \]

Step 2: Identify \( a \) and \( r \)

In the \( y = a(1 + r)^x \) form:

• Here, \( a = 750 \)
• Let \( r = 6.5536 - 1 = 5.5536 \) (since we're looking at how \( 6.5536 \) relates to \( 1 + r \))

Step 3: Convert \( r \) to a percentage

\[ r = 5.5536 \implies \text{Percentage} = 5.5536 \cdot 100 \approx 555.36% \]

Step 4: Determine the function's behavior

Since the base of the exponent \( 1.6 > 1 \), this indicates that the function is growing.

Conclusion

Thus, we summarize:

• The function is growing.
• The constant percent rate of change is approximately 555.36%.

Note: It appears that there might have been an error with the initial data options available (88, 8.8, 12, 1.2%). If those options are intended to refer to a different context or base, please clarify, as your input suggests a discrepancy. This analysis concludes the growth rate in the context provided.