Problem 26 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the exponential function when \(x=2.5 .\) Round your answer to the nearest hundredth. $$y=3(4)^{x}$$

Step-by-Step Solution

Answer

The value of the function \(y = 3(4)^{2.5}\) rounded to the nearest hundredth is 94.87.

Step 1: Identify the variables given

Our function is \(y = 3(4)^x\). In our exercise, \(x = 2.5\).

Step 2: Substitute variables into the function

Now substitute \(2.5\) into the function for \(x\). So our function becomes \(y = 3(4)^{2.5}\)

Step 3: Calculate the function

Using a calculator, \(y = 3(4)^{2.5}\) equals 94.86833.

Step 4: Round to the nearest hundredth

We will now round to the nearest hundredth, giving us a final answer of 94.87.

Key Concepts

Understanding Calculator Usage

When working with exponential functions, a calculator can be your trusty companion. Calculators help simplify complex calculations, especially when dealing with high powers or non-integer exponents. To use a calculator for an expression like \( y = 3(4)^{2.5} \), you should:

• First, ensure your calculator is set to the correct mode (often "normal" or "scientific" mode).
• Enter the base number inside the parentheses, which is 4 in this instance.
• Use the exponent button (usually labeled as `^`, `EXP`, or `ELE`) to enter your exponent, which is 2.5 here.
• Calculate the result of the exponentiation, and then multiply this result by the leading coefficient, 3.
• Ensure accuracy by double-checking your input before hitting the calculation button.

These steps will help you achieve an accurate result for even complex exponential functions. It's always a good practice to familiarize yourself with your specific calculator model, as buttons and interface may vary.

Mastering Function Evaluation

Function evaluation involves replacing the variable in a function with a given number and performing the necessary calculations. In the case of the function \( y = 3(4)^x \) where \( x = 2.5 \), the process is straightforward once you know the steps:

• Identify the function, which in this exercise is \( y = 3(4)^x \), and note the value to substitute, \( x = 2.5 \).
• Replace \( x \) with 2.5 in the expression: \( y = 3(4)^{2.5} \).
• Use a calculator to evaluate \( (4)^{2.5} \).
• Take the computed result and multiply it with the coefficient 3.

This methodical approach ensures you substitute and calculate accurately, leading to the correct function value. Replace, compute, and multiply – these are your steps for effective function evaluation.

Rounding Numbers Effectively

Rounding numbers is an essential skill, especially in mathematics, to present data in a clearer and more comprehensible way. When rounding to the nearest hundredth, you are essentially limiting the decimal to two places:

• Find the third digit after the decimal point. This determines whether the second digit stays the same or rounds up.
• If the third digit is 5 or greater, add one to the second digit. For example, in 94.86833, the third digit is 8, so we round the second digit 6 up to 7.
• If the third digit is less than 5, simply drop all remaining digits beyond the hundredth place.
• In the current problem, after rounding, 94.86833 becomes 94.87.

Understanding these rules will assist you in precise rounding, ensuring your calculations remain accurate to the desired level of precision.