Problem 5 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

Evaluating the function \(y=3 \sqrt{x}+4\) for x at 0, 1, 2, 3, and 4 provides the respective answers 4, 7, 7.2, 7.7, and 10.

Step 1: Evaluate function for x=0

Just replace x with 0 in the given function, \(y=3 \sqrt{x}+4\), which gives \(y=3 \sqrt{0}+4\) = 4

Step 2: Evaluate function for x=1

Replace x with 1 in the given function, \(y=3 \sqrt{x}+4\), which gives \(y=3 \sqrt{1}+4\) = 7

Step 3: Evaluate function for x=2

Replace x with 2 in the given function, \(y=3 \sqrt{x}+4\), which gives \(y=3 \sqrt{2}+4\) = 7.2 when rounded to the nearest tenth.

Step 4: Evaluate function for x=3

Replace x with 3 in the given function, \(y=3 \sqrt{x}+4\), which gives \(y=3 \sqrt{3}+4\) = 7.7 when rounded to the nearest tenth.

Step 5: Evaluate function for x=4

Replace x with 4 in the given function, \(y=3 \sqrt{x}+4\), which gives \(y=3 \sqrt{4}+4\) = 10

Key Concepts

Understanding the Square Root Function

The square root function is a fundamental mathematical operation often represented as \( \sqrt{x} \). In essence, it seeks a number which, when multiplied by itself, yields the product \( x \). For instance, since \( 2 \times 2 = 4 \), we say \( \sqrt{4} = 2 \).
In our exercise, the square root function is part of the overall function \( y = 3 \sqrt{x} + 4 \). Here, the square root function transforms the values of \( x \) by finding their square roots, which is then multiplied by 3 and increased by 4 to determine the final value of \( y \).
Understanding how the square root function works can help piece together how changes in \( x \) affect \( y \). For values of \( x = 0, 1, 2, 3, \) and \( 4 \), their square roots are \( 0, 1, \sqrt{2}, \sqrt{3}, \) and \( 2 \) respectively, affecting the result of the subsequent calculations.

Grasping the Substitution Method

The substitution method is a handy tool used to simplify expressions by replacing a variable with a particular value. In this exercise, substitution involves substituting specific numerical values for \( x \) within the function \( y = 3 \sqrt{x} + 4 \).
For example, when solving for \( x = 0 \), substitute 0 in for \( x \), modifying the expression to \( y = 3 \sqrt{0} + 4 \). Calculating further, we find \( y = 3 \times 0 + 4 = 4 \). By repeating this process for \( x = 1, 2, 3, \) and \( 4 \), you can calculate \( y \) by working through the same substitution and evaluation method for each value.
This method not only helps in evaluating functional values but also enhances your intuitive understanding of how functions behave when variable inputs change.

The Importance of Rounding Numbers

Rounding numbers is an essential mathematical technique used to simplify figures while retaining their approximate values. In many cases, especially within practical and real-world applications, precision to an exact decimal isn’t necessary, and rather, rounded numbers suffice.
In our exercise, you are tasked with rounding to the nearest tenth. This means adjusting the number to one decimal place. For instance, when the calculated result is \( 7.14 \), rounding it to the nearest tenth gives \( 7.1 \) since the number in the hundredths place is lower than 5.

• If the digit right after the desired decimal place (tenths in this case) is 5 or higher, you round up. Hence, \( 7.25 \) becomes \( 7.3 \).
• If the digit is lower than 5, you keep the number unchanged, such as \( 7.24 \) to \( 7.2 \).

Mastering the technique of rounding can simplify your mathematical procedures and help in making comparisons or approximations effectively.