Problem 28 - Algebra Form and Function

Question

Put the functions in the form \(Q=k t\) and state the value of \(k\). $$ Q=\frac{1}{2} t \sqrt{3} $$

Step-by-Step Solution

Answer

Answer: The value of \(k\) is \(\frac{1}{2}\sqrt{3}\).

Step 1: Identify the Constant

The given function is \(Q=\frac{1}{2} t \sqrt{3}\). We can see that the constant \(\frac{1}{2}\sqrt{3}\) is multiplying the variable \(t\).

Step 2: Write the Function in the Form \(Q=kt\)

To write the function in the form \(Q=kt\), we can simply let \(k=\frac{1}{2}\sqrt{3}\). Thus, the function becomes \(Q = kt = \left(\frac{1}{2}\sqrt{3}\right) t\).

Step 3: State the Value of \(k\)

The value of \(k\) in the given function is \(\frac{1}{2}\sqrt{3}\).

Key Concepts

Constant Multiplication

When we talk about constant multiplication in the context of functions, we're referring to multiplying a variable by a fixed number, which is known as the constant. In the given problem, the function is expressed as \( Q = \frac{1}{2}t \sqrt{3} \). Here, the constant is \( \frac{1}{2}\sqrt{3} \), which is multiplied by the variable \( t \).
In a linear function of the form \( Q = kt \), the constant multiplier \( k \) determines the rate, or slope, at which the dependent variable \( Q \) changes with respect to the independent variable \( t \). A larger absolute value of \( k \) indicates that \( Q \) changes more rapidly.
Understanding how constants impact functions is key:

• A positive constant \( k \) means the function increases with \( t \).
• A negative constant would imply that it decreases as \( t \) increases.
• The size of this constant will tell you how steeply or gently the function increases or decreases.

Function Transformation

Function transformation discusses how a function can be manipulated to show different forms, such as translation, stretching, or reflection. In our example, the main transformation at play is the function's scaling.
Given the function \( Q = \frac{1}{2}t \sqrt{3} \), it can be rewritten for clarity as \( Q = kt = (\frac{1}{2}\sqrt{3})t \). This change highlights how the function is transformed by a constant multiplier.
Function transformations can often include:

• **Vertical shifts**, moving the graph up or down. This is not applicable here as the constant is directly multiplying the variable, not added to it.
• **Horizontal shifts**, moving the graph right or left, also irrelevant to this situation.
• **Scaling**, which is the effect of our constant multiplier causing the graph to expand or contract vertically.
• **Reflections**, that flip the graph over an axis, would occur if our constant were negative.

Transformations are essential for revealing the behavior and characteristics of functions.

Algebraic Manipulation

Algebraic manipulation involves operations that alter the form of an equation or expression without changing its value or solution. For the function \( Q = \frac{1}{2}t \sqrt{3} \), the goal was to express it in a specific form, i.e., \( Q = kt \).
To achieve this, we identified the constant \( k \) as \( \frac{1}{2}\sqrt{3} \) and rewrote the function accordingly as \( Q = (\frac{1}{2}\sqrt{3}) t \). This step shows how algebraic manipulation lets you adjust expressions to meet required conditions or forms.
Through algebraic manipulation, various actions can be taken:

• **Rearranging terms** to clarify expressions.
• **Combining like terms** to simplify algebraic expressions.
• **Factoring** expressions to reveal underlying structures.

These skills are vital in solving equations and understanding the relationships within mathematical expressions.