Problem 2 - Algebra Form and Function

Question

In Exercises \(1-4\), write a formula for \(y\) in terms of \(x\) if \(y\) satisfies the given conditions. Proportional to the cube of \(x\), with constant of proportionality -0.35 .

Step-by-Step Solution

Answer

Question: Given that y is proportional to the cube of x with a constant of proportionality of -0.35, find a formula for y in terms of x. Answer: The formula for y in terms of x is y = -0.35x^3, where y is proportional to the cube of x with a constant of proportionality of -0.35.

Step 1: Determine the proportionality relationship

Since \(y\) is proportional to the cube of \(x\), we can represent the proportionality relationship as \(y \propto x^3\). Step 2: Introduce the constant of proportionality

Step 2: Include the constant of proportionality in the relationship

To introduce the constant of proportionality, we will replace the proportionality symbol "\(\displaystyle\propto\)" with an equal sign, and multiply the \(x^3\) term by the constant of proportionality. The constant of proportionality is given as -0.35. So, the equation becomes: \[y = -0.35x^3\] Step 3: Write the final formula

Step 3: Present the final formula for y in terms of x

The final formula for \(y\) in terms of \(x\) is \[y = -0.35x^3\], where \(y\) is proportional to the cube of \(x\) with a constant of proportionality of -0.35.

Key Concepts

Cubic Functions

A cubic function is a type of polynomial function where the highest degree is 3. In simpler terms, the variable \(x\) is raised to the power of three, so the basic form of a cubic function is \(y = ax^3 + bx^2 + cx + d\). Understanding cubic functions is essential for many areas of mathematics and physics because they can model various real-world situations, like the volume of a cube or the behavior of certain types of waves.
When dealing with cubic functions, you might notice their characteristic graph shape, a curve that typically has either two turning points or none, which makes it unique compared to quadratic functions that those with parabolas. The steep parts of the curve are due to the cube power, which increases faster than linear or quadratic relationships.

• Key Element: The coefficient of \(x^3\) determines how steep or wide the cubic curve is.
• Behavior: Depending on the sign of the leading coefficient, the graph could slope upward or downward.
• Roots: A cubic function can have up to three real roots, which are points where the curve crosses the \(x\)-axis.

Exploring cubic functions helps you grasp more complex algebraic concepts, especially when adding elements like the constant of proportionality.

Constant of Proportionality

The constant of proportionality is a number that relates two variables that are proportional to each other. When one variable \(y\) changes in a consistent way with another variable \(x\), they are said to be directly proportional if \(y = kx\) for some constant \(k\), which is the constant of proportionality.
In our cubic function case, the constant of proportionality is \(-0.35\). This tells us that every time \(x\) changes, \(y\) will change by \(-0.35\) times the cube of \(x\).

• Determining Factor: The constant's sign indicates direction—negative implies inverse relationship.
• Scale: It reflects the scale factor, determining how large or small \(y\) will be compared to \(x^3\).
• Units: If needed, it adjusts units of measurement between \(y\) and \(x^3\).

Understanding the constant of proportionality is crucial as it allows us to mathematically describe real-life applications, such as calculating forces or understanding economic models.

Algebraic Formulas

Algebraic formulas are equations that express relationships between different variables using algebraic expressions. These formulas serve as shortcuts to solving problems by giving us a structured way to manipulate numbers and capture relationships.
In the exercise we looked at, we derived the algebraic formula \(y = -0.35x^3\). This formula helps us quickly compute \(y\), the dependent variable, based on changes in \(x\), the independent variable.

• Simplification: Algebraic formulas can simplify complex real-world problems into solvable mathematical models.
• Prediction: They allow us to predict outcomes by plugging in variable values.
• Flexibility: Formulas can be adapted to different situations by changing constants and other parameters.

Algebraic formulas like the one in our example are foundational to higher-level mathematics, as they reveal how different quantities are interconnected, paving the way for problem solving across various scientific fields.