Problem 39 - Algebra and Trigonometry

Question

\(37-40\) An equation and its graph are given. Find the \(x\) - and \(y\) -intercepts. $$ x^{4}+y^{2}-x y=16 $$

Step-by-Step Solution

Answer

The \(x\)-intercepts are \((2, 0)\) and \((-2, 0)\). The \(y\)-intercepts are \((0, 4)\) and \((0, -4)\).

Step 1: Understanding the Equation

We are given the equation \(x^4 + y^2 - xy = 16\). We need to find the \(x\)- and \(y\)-intercepts of this equation.

Step 2: Find the x-intercept by setting y = 0

To find the \(x\)-intercept, set \(y = 0\) in the equation. This gives us \(x^4 + 0^2 - x \cdot 0 = 16\), simplifying to \(x^4 = 16\).

Step 3: Solve for x

Solve the equation \(x^4 = 16\) by taking the fourth root of both sides. This results in \(x = \pm 2\). Thus, the \(x\)-intercepts are the points \((2, 0)\) and \((-2, 0)\).

Step 4: Find the y-intercept by setting x = 0

To find the \(y\)-intercept, set \(x = 0\) in the equation. This gives us \(0^4 + y^2 - 0 \cdot y = 16\), simplifying to \(y^2 = 16\).

Step 5: Solve for y

Solve \(y^2 = 16\) by taking the square root of both sides. This results in \(y = \pm 4\). Thus, the \(y\)-intercepts are the points \((0, 4)\) and \((0, -4)\).

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that involve one or more variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. They often look like a "balance" between two expressions, shown by an equals sign.
In our exercise, the equation given is \( x^4 + y^2 - xy = 16 \). This represents a relationship between the two variables, \(x\) and \(y\), and a constant, 16. Solving algebraic equations generally involves finding the value of the variable that makes the equation true.
Here, we aim to find intercepts, which gives us the points where the curve cuts the x-axis and y-axis.

• The x-intercepts are found by setting \(y = 0\) and solving for \(x\). This tells us where the curve crosses the x-axis.
• The y-intercepts are found by setting \(x = 0\) and solving for \(y\). This indicates where the curve crosses the y-axis.

Understanding these intercepts is crucial as they give vital information about where the graph starts on the axes in a coordinate plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a way of representing geometric figures and analyzing their properties using a coordinate system. This system uses numbers, called coordinates, to determine the positions of points on a plane.
The intercepts found in our exercise are coordinates that define specific points where the graph of the equation crosses the axes. In our example, the results were the points \((2, 0)\) and \((-2, 0)\) for the x-intercepts and \((0, 4)\) and \((0, -4)\) for the y-intercepts.

• Graphing these points: The coordinate plane consists of a horizontal axis (x-axis) and a vertical axis (y-axis). Points are plotted using an ordered pair \((x, y)\). The x-coordinate tells you how far to move horizontally from the origin, and the y-coordinate tells you how far to move vertically.
• Axes intercepts: Finding intercepts helps us quickly sketch the rough shape of the graph, indicating where it touches the axes.

This visual representation helps better understand how the equation behaves.

Polynomial Functions

Polynomial functions are a type of algebraic expression that involves variables raised to whole number powers. Such functions can be classified based on their degree, which is the highest power of the variable in the expression. A polynomial of degree four would look like \(x^4\), as seen in our exercise.
In these functions, the coefficients and exponents define the function's shape and position.

• Characteristics: Polynomial functions are smooth, continuous, and have specific patterns like turning points, intercepts, and sometimes symmetry.
• Application in intercepts: In our problem, finding the intercepts involved manipulating the polynomial terms by setting specific variables to zero and solving simple polynomial equations like \(x^4 = 16\) or \(y^2 = 16\).

By solving these, we used concepts of both square and fourth roots, showcasing how polynomial equations of different degrees interact with the coordinate plane. Understanding these interactions is key in both calculus and physics as they relate to real-world systems and scenarios.