Problem 4 - Algebra 1

Question

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

Step-by-Step Solution

Answer

For the given function, the values of \(y\) when \(x = 0, 1, 2, 3,\) and \(4\) are \(0.0, 0.5, 0.7, 0.9,\) and \(1.0\) respectively.

Step 1: Evaluate for \(x = 0\)

Substitute \(x = 0\) into the equation to get \(y = \frac{1}{2} \sqrt{0} = 0.0\)

Step 2: Evaluate for \(x = 1\)

Substitute \(x = 1\) into the equation to get \(y = \frac{1}{2} \sqrt{1} = 0.5\)

Step 3: Evaluate for \(x = 2\)

Substitute \(x = 2\) into the equation to get \(y = \frac{1}{2} \sqrt{2} = 0.7\) when rounded to the nearest tenth

Step 4: Evaluate for \(x = 3\)

Substitute \(x = 3\) into the equation to get \(y = \frac{1}{2} \sqrt{3} = 0.9\) when rounded to the nearest tenth

Step 5: Evaluate for \(x = 4\)

Substitute \(x = 4\) into the equation to get \(y = \frac{1}{2} \sqrt{4} = 1.0\)

Key Concepts

Understanding the Square Root Function

The square root function is represented by the symbol \( \sqrt{} \). It is used to find a number that, when multiplied by itself, gives the original number under the root. So, when you see \( \sqrt{x} \), it means finding a number which squared gives \( x \).
For example, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \). Each number has both a positive and a negative square root, but in this context, we only consider the positive value. It is important in many mathematical areas, especially when dealing with quadratic equations and geometry.
This function is continuous and defined for all non-negative real numbers. It's a fundamental part of evaluating expressions, particularly when you encounter it in algebraic equations like the one in the exercise where \( y = \frac{1}{2} \sqrt{x} \). Understanding how to apply it is crucial for solving these expressions accurately.

Mastering the Substitution Method

The substitution method is a simple yet powerful tool in algebra. It involves replacing a variable with a number to simplify or solve an equation. This method helps in evaluating functions, especially when specific values are given for a variable.
In the exercise, we are substituting different values of \( x \) into the expression \( y = \frac{1}{2} \sqrt{x} \). Steps include:

• Identify the value of \( x \) to be substituted.
• Replace \( x \) in the expression with this value.
• Perform the necessary arithmetic operations.

For example, if \( x = 2 \), substitute 2 into the function to get \( y = \frac{1}{2} \sqrt{2} \). This makes the expression much simpler and allows for easy computation of \( y \). Substitution makes tackling complex functions straightforward and manageable.

The Art of Rounding Numbers

Rounding is the process of reducing the digits of a number while keeping its value close to what it was. It's useful when you want to simplify numbers, making them easier to read or when a precise number isn't necessary.
In the problem, after finding \( y = \frac{1}{2} \sqrt{x} \), results needed to be rounded to the nearest tenth. Here’s how to round numbers:

• Look at the number in the place immediately to the right of the place value you're rounding to.
• If this number is 5 or greater, increase the rounding place by 1.
• If it's less than 5, leave the rounding place value as it is.

For instance, when calculating \( y = \frac{1}{2} \sqrt{2} \), you get approximately 0.707. Rounding this to the nearest tenth becomes 0.7. Rounding improves the neatness of results, making them more presentable or practical for further application.

Exploring Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They do not have an equal sign, which differentiates them from equations.
The given function is \( y = \frac{1}{2} \sqrt{x} \), which is an algebraic expression. Here, \( y \) represents the output of the function, and it's defined in terms of \( x \). The expression involves both a constant multiplier (\( \frac{1}{2} \)) and a square root function.
Understanding how to evaluate such expressions is critical because they appear frequently in mathematical problems and real-world situations. You need to be comfortable substituting numbers into these and performing calculations to find desired values. Practicing with different values of \( x \), as in this exercise, helps solidify your grasp of algebraic manipulation and functional evaluation. It encourages flexibility and confidence in problem-solving scenarios.