Problem 4 - College Algebra with Modeling and Visualization

Question

Evaluate the expression for the given \(f(x, y)\). $$ f(-1,3) \text { if } f(x, y)=2 x^{2}-y^{2} $$

Step-by-Step Solution

Answer

The expression \(f(-1,3)\) evaluates to -7.

Step 1: Understand the Expression

We start by understanding the given function \(f(x, y) = 2x^2 - y^2\). This is a two-variable function that involves both \(x\) and \(y\) with \(x\) being squared, multiplied by 2, and \(y\) being squared and subtracted.

Step 2: Substitute the Values

Substitute \(x = -1\) and \(y = 3\) into the function. This means we calculate the expression: \(f(-1,3) = 2(-1)^2 - 3^2\).

Step 3: Calculate \((-1)^2\)

First, calculate \((-1)^2\). The square of \(-1\) is 1, so we have: \(2(1) - 3^2\).

Step 4: Multiply the Result by 2

Multiply the result by 2: \(2 \times 1 = 2\). Our expression now is \(2 - 3^2\).

Step 5: Calculate \(3^2\)

Next, calculate \(3^2\). The square of 3 is 9, so we have: \(2 - 9\).

Step 6: Final Calculation

Perform the final calculation: \(2 - 9 = -7\). Therefore, \(f(-1,3) = -7\).

Key Concepts

Two-variable Function

A two-variable function is a mathematical expression that takes two inputs, usually represented as \(x\) and \(y\). It utilizes specific relationships between these variables to determine an output. Such functions are crucial in various fields, including physics and engineering, for modeling scenarios involving more than one changing variable.
For example, in the given exercise, the function is \(f(x, y) = 2x^2 - y^2\). Here, both \(x\) and \(y\) play a role in determining the value of the function.

• The term \(2x^2\) shows that \(x\) is squared and then multiplied by 2.
• The term \(-y^2\) indicates that \(y\) is squared, and this result is subtracted from the first term.

Two-variable functions like this can be visually represented with a contour plot or 3D surface, showing how output values change with different \(x\) and \(y\) combinations.

Substitution Method

The substitution method is a straightforward technique to evaluate expressions, especially when dealing with functions. It involves replacing each variable with actual numbers and then simplifying the expression. This method not only applies to mathematical functions but is also useful in solving equations and systems of equations.
In the context of your exercise, the substitution method means:

• Taking the original function \(f(x, y) = 2x^2 - y^2\).
• Replacing \(x\) with \(-1\) and \(y\) with \(3\), so it becomes \(f(-1, 3) = 2(-1)^2 - 3^2\).

Once the values are substituted, simply proceed with arithmetic operations to find the result. Substitution is a powerful method because it simplifies complex algebraic expressions into basic calculations.

Squaring Numbers

Squaring a number means multiplying the number by itself. It's a fundamental arithmetic operation essential in many areas of math, especially when working with quadratic equations and functions. Understanding squaring helps in evaluating expressions, like those given in your exercise.
Here’s how squaring works in your problem:

• Calculate \((-1)^2\): This means \(-1\) times \(-1\), which gives 1. So, \(2(-1)^2\) becomes \(2 \times 1 = 2\).
• Calculate \(3^2\): This means \(3\) times \(3\), which gives 9.

Once squared, use these results to perform further calculations as shown in the step-by-step solution. Mastering the concept of squaring numbers aids in solving a range of mathematical problems with ease.