Problem 29 - 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. $$y=6\left(\frac{1}{2}\right)^{x}$$

Step-by-Step Solution

Answer

The value of the function when \( x = 2.5 \) is \( y \approx 1.70 \).

Step 1: Substitute the given value.

Input the given value, \( x=2.5 \), into the exponential function, which gives: \( y=6(\frac{1}{2})^{2.5} \).

Step 2: Evaluation of the function.

Use a calculator to evaluate the right-hand side of the equation. This gives the value of \( y \) as \( y \approx 1.696 \).

Step 3: Round the result.

The problem instructs to round the answer to the nearest hundredth. So, the final value for \( y \) is \( y \approx 1.70 \).

Key Concepts

calculator use

Calculators can be incredibly handy tools when it comes to evaluating exponential functions. When you are faced with a function like \( y = 6\left(\frac{1}{2}\right)^{x} \), it involves both a fraction and a power. Here is a basic way to input an exponential function into a calculator:

• Locate and use the 'power' key, often marked as `^` or 'EXP', to input the exponent.
• Input the base of the exponent with your fraction. For our example, \(\frac{1}{2}\), make sure to use parentheses if needed to keep the correct order of operations.
• Next, input the exponent value, in this case, \( x = 2.5 \).
• Finally, multiply the result by 6 as given in the problem.

It is essential to check your calculator's display to ensure everything is entered correctly before hitting the equal sign or execute button. Calculators can vary, so it might be useful to familiarize yourself with your specific model’s functions.

rounding numbers

Rounding numbers correctly is crucial when the problem specifies rounding to a particular decimal place. In this example, the exponential function result is approximately 1.696 after calculation. The issue here is to round to the nearest hundredth. Here's a simple guide to rounding:

• Look at the third decimal place after the decimal point.
• If it is 5 or more, round the second decimal place up by one.
• If it is less than 5, keep the second decimal place as it is.

In our situation, the third decimal place is 6. Therefore, you will round 1.696 up to 1.70. Remember, rounding accurately not only follows mathematical convention but also ensures your answers meet problem requirements.

function evaluation

Evaluating a function involves replacing variables with numbers, followed by simplifying to find the outcome. In the given exercise, the exponential function \( y = 6\left(\frac{1}{2}\right)^{x} \) is evaluated with \( x = 2.5 \). To accomplish this:

• Substitute \( x \) with 2.5, resulting in \( y = 6\left(\frac{1}{2}\right)^{2.5} \).
• Next, calculate the power, \( \left(\frac{1}{2}\right)^{2.5} \). This requires using your calculator to find the correct value.
• Once you get this result, multiply it by 6 to obtain \( y \).

This results in approximately \( y = 1.696 \), which then needs rounding as instructed. Function evaluation often involves these three basic steps: substituting, calculating the function’s value, and interpreting the result as needed.