Problem 40 - Algebra and Trigonometry

Question

\(37-40\) An equation and its graph are given. Find the \(x\) - and \(y\) -intercepts. $$ x^{2}+y^{3}-x^{2} y^{2}=64 $$

Step-by-Step Solution

Answer

X-intercepts: \((8, 0), (-8, 0)\); Y-intercept: \((0, 4)\).

Step 1: Identifying Intercepts: Understanding

The intercepts are the points where the graph crosses the axes. For the x-intercept, set \(y = 0\) and solve for \(x\). For the y-intercept, set \(x = 0\) and solve for \(y\).

Step 2: Finding the X-Intercept

Set \(y = 0\) in the equation \(x^2 + y^3 - x^2 y^2 = 64\). The equation simplifies to \(x^2 = 64\). Solve for \(x\):\[x^2 = 64\]\[x = \pm 8\]. Thus, the x-intercepts are the points \((8, 0)\) and \((-8, 0)\).

Step 3: Finding the Y-Intercept

Set \(x = 0\) in the equation \(x^2 + y^3 - x^2 y^2 = 64\). The equation simplifies to \(y^3 = 64\). Solve for \(y\):\[y^3 = 64\]\[y = 4\]. Thus, the y-intercept is the point \((0, 4)\).

Key Concepts

x-intercept

The x-intercept of a graph is the point where the graph meets the x-axis. At this point, the value of y is zero because it lies directly on the x-axis. To find the x-intercept, you substitute zero in for y in the given equation and solve for x.
For example, in the equation \(x^2 + y^3 - x^2 y^2 = 64\), setting \(y = 0\) results in \(x^2 = 64\). Solving this equation gives \(x = \pm 8\).
Therefore, the x-intercepts of this particular graph are at the coordinates \((8, 0)\) and \((-8, 0)\), where the graph crosses the x-axis.

y-intercept

The y-intercept is the point where the graph crosses the y-axis. Here, the value of x is zero because it is located directly on the y-axis. To find it, you replace x with zero in the original equation and solve for y.
From our provided equation \(x^2 + y^3 - x^2 y^2 = 64\), setting \(x = 0\) simplifies it to \(y^3 = 64\).
Solving for y gives \(y = 4\), meaning that the y-intercept of this graph is \((0, 4)\). This is the exact point where the graph touches the y-axis.

polynomial equation

A polynomial equation is an expression consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials can appear in various forms and degrees depending on the highest power of the variable.
In the provided equation, \(x^2 + y^3 - x^2 y^2 = 64\), there are terms with both x and y raised to different powers.
Such a combination shows that this is a complex polynomial equation due to the mixed degree of terms like \(x^2 y^2\). It's crucial to understand the uniqueness of each term, as solving the polynomial will require isolating terms involving one variable, as done in identifying intercepts.

graphing techniques

Understanding graphing techniques involves knowing how to visually represent equations on a graph. This helps in gaining insights into the behavior of the equation based on its graphical representation.
For the given polynomial equation, plotting involves:

• Finding intercepts as initial key points.
• Using additional points to ensure the curve's shape is accurate.
• Being attentive to symmetry and direction as defined by the equation.

These techniques assist in sketching the overall shape and intersections with the axes, producing a clearer image of the graph's layout, encapsulating the broad behavior predicted by the equation.