Problem 3 - Algebra 1: Concepts and Skills

Question

Evaluate the function for x 0, 1, 2, 3, and 4. Round your answers to the nearest tenth. $$y=4 \sqrt{x}$$

Step-by-Step Solution

Answer

The function \(y=4 \sqrt{x}\) evaluates to 0, 4, 5.6, 6.9, and 8 for \(x = 0, 1, 2, 3, 4\) respectively.

Step 1: Substitute x = 0 into the function

This is performed by replacing \(x\) in the function \(y=4 \sqrt{x}\) with 0, yielding \(y = 4 \sqrt{0}\), so \(y = 0\).

Step 2: Substitute x = 1 into the function

This is performed by replacing \(x\) in the function \(y=4 \sqrt{x}\) with 1, yielding \(y = 4 \sqrt{1}\), so \(y = 4\).

Step 3: Substitute x = 2 into the function

This is performed by replacing \(x\) in the function \(y=4 \sqrt{x}\) with 2, yielding \(y = 4 \sqrt{2}\). As \(\sqrt{2}\) is approximately 1.4, \(y = 4 \times 1.4 = 5.6\). Rounding to the nearest tenth gives \(y = 5.6\).

Step 4: Substitute x = 3 into the function

This is performed by replacing \(x\) in the function \(y=4 \sqrt{x}\) with 3, yielding \(y = 4 \sqrt{3}\). As \(\sqrt{3}\) is approximately 1.732, \(y = 4 \times 1.732 = 6.928\). Rounding to the nearest tenth gives \(y = 6.9\).

Step 5: Substitute x = 4 into the function

This is performed by replacing \(x\) in the function \(y=4 \sqrt{x}\) with 4, yielding \(y = 4 \sqrt{4}\), so \(y = 8\).

Key Concepts

Understanding the Square Root Function

The square root operation is key to evaluating functions like the one in the exercise, where the equation involves a square root: \( y = 4 \sqrt{x} \).
It helps us find a number which, when multiplied by itself, equals the given value.
For example, \( \sqrt{4} \) equals 2, because 2 multiplied by itself equals 4.When you're solving exercises involving square roots, it's vital to understand that these operations are nonlinear.
That means changes in the input \( x \) can result in nonuniform changes in the output \( y \).
For instance, while \( 4 \sqrt{x} \) at \( x = 1 \) equals 4, at \( x = 4 \), the result is double, which is 8.
This shows how the function scales differently as \( x \) increases.Recognizing these properties helps when predicting the behavior of functions in mathematical modeling.

Using the Substitution Method

The substitution method is a reliable and straightforward approach to evaluating functions.
It involves replacing a variable with a given value and solving the resulting equation.
Let's break it down with a simple example from the function \( y = 4 \sqrt{x} \).

• Take the value of \( x \), for instance, \( x = 2 \).
• Replace \( x \) in the function with 2: \( y = 4 \sqrt{2} \).
• Calculate \( \sqrt{2} \), which approximates to 1.4, and multiply it by 4 to get 5.6.
• Rounding gives us the final answer.

This method offers an equation specific to each \( x \) value, allowing us to directly compute corresponding \( y \) values.
Understanding substitution helps in breaking down complex expressions into manageable parts, making problem-solving more straightforward.

The Importance of Rounding Numbers

Rounding is an essential skill in mathematics used to simplify numbers while preserving their value as much as possible.
In our exercise, rounding is necessary to express answers to the nearest tenth.
This keeps the numbers easier to handle without significant loss of precision. To round a number like 6.928 to the nearest tenth:

• Identify the digit in the tenths place, which is 9 here.
• Look at the digit in the hundredths place, which is 2.
• Since 2 is less than 5, the tenths place remains 9, so 6.928 rounds to 6.9.

Rounding is particularly useful when dealing with decimals derived from operations like square roots.
This technique ensures the result is both understandable and practically useful, especially in fields like engineering or data science where precision is important yet simplicity is key.