Problem 55 - Precalculus Mathematics for Calculus

Question

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

x-intercepts: (2,0) and (-2,0); y-intercepts: (0,4) and (0,-4).

Step 1: Understand the Problem

The equation given is \( x^4 + y^2 - xy = 16 \). We need to find the points where this equation intersects the x-axis and the y-axis.

Step 2: Find the x-intercepts

The x-intercept is found when \( y = 0 \). Substitute \( y = 0 \) into the equation: \( x^4 + (0)^2 - x(0) = 16 \), which simplifies to \( x^4 = 16 \). To find \( x \), solve \( x^4 = 16 \) which gives \( x = \pm 2 \).

Step 3: Find the y-intercepts

The y-intercept is found when \( x = 0 \). Substitute \( x = 0 \) into the equation: \( (0)^4 + y^2 - (0)y = 16 \), which simplifies to \( y^2 = 16 \). To find \( y \), solve \( y^2 = 16 \) which gives \( y = \pm 4 \).

Key Concepts

Equation Solving

Solving equations is a fundamental part of algebra. It involves finding the unknown values that satisfy the given mathematical statement. When we have an equation like \( x^4 + y^2 - xy = 16 \), we need to determine the values of \( x \) and \( y \) that make this equation true. This involves substituting conditions and simplifying.
For x-intercepts, we set \( y = 0 \) since x-intercepts happen where the graph crosses the x-axis. Substituting \( y = 0 \), the equation simplifies to \( x^4 = 16 \). Here, we just need to solve \( x^4 = 16 \) which results in \( x = \pm 2 \).
Similarly, for y-intercepts, set \( x = 0 \), simplifying the equation to \( y^2 = 16 \), giving us \( y = \pm 4 \). Using substitution and simplification, we find the intercepts which are crucial as they show where the function interacts with the coordinate axes.

Graphing Functions

Graphing functions allows us to visualize equations and their solutions on the coordinate plane. With a graph, we can see where a function crosses the x-axis and y-axis, which correspond to the intercepts.
When graphing the function given by \( x^4 + y^2 - xy = 16 \), we plot the points where it meets each axis. For x-intercepts, plot (2, 0) and (-2, 0). For y-intercepts, plot (0, 4) and (0, -4).
By marking these intercepts, we start to understand the shape and trajectory of the graph. Intercepts are key points that guide us towards sketching more complex components of the function's graph. They serve as anchors and reference points when drawing the graph.

Polynomial Equations

Polynomial equations are mathematical expressions involving variables raised to whole number powers. These types of equations are prevalent in algebra and calculus.
In our example, the equation \( x^4 + y^2 - xy = 16 \) can be broken down into polynomial components like the terms \( x^4 \) and \( y^2 \).
Polynomial equations are defined by their degree, which is the highest power of the variable within the equation. In our case, \( x^4 \) suggests a fourth-degree polynomial, indicating that there might be up to four real roots when looking for x-intercepts.
Understanding the degree of a polynomial helps in predicting the behavior of its graph. More specifically, the higher the degree, the more complex the graph, as it can have multiple turning points or inflection points. Recognizing these characteristics aids in solving and graphing polynomial equations effectively.