Problem 51 - Precalculus Mathematics for Calculus

Question

Determine whether the given points are on the graph of the equation. $$x^{2}+x y+y^{2}=4 ; \quad(0,-2),(1,-2),(2,-2)$$

Step-by-Step Solution

Answer

Points (0, -2) and (2, -2) are on the graph; point (1, -2) is not.

Step 1: Verify Point (0, -2)

To determine if the point \((0, -2)\) lies on the graph of the equation, substitute \(x = 0\) and \(y = -2\) into the equation:\[0^2 + 0(-2) + (-2)^2 = 0 + 0 + 4 = 4\]Since the left side equals the right side of the equation (4 = 4), the point \((0, -2)\) is on the graph of the equation.

Step 2: Verify Point (1, -2)

To check if the point \((1, -2)\) is on the graph, substitute \(x = 1\) and \(y = -2\) into the equation:\[1^2 + 1(-2) + (-2)^2 = 1 - 2 + 4 = 3\]The left side does not equal the right side (3 ≠ 4), so the point \((1, -2)\) is not on the graph of the equation.

Step 3: Verify Point (2, -2)

To determine if point \((2, -2)\) is on the graph, substitute \(x = 2\) and \(y = -2\) into the equation:\[2^2 + 2(-2) + (-2)^2 = 4 - 4 + 4 = 4\]Since the left side equals the right side of the equation (4 = 4), the point \((2, -2)\) is on the graph of the equation.

Key Concepts

Equation Verification

Equation verification is a critical step in determining whether a given point lies on the graph of an equation. This process requires substituting the coordinates of the point into the equation and checking if both sides of the equation are equal after substitution.
For example, take the equation \(x^2 + xy + y^2 = 4\). To verify whether the point \((0, -2)\) is on this graph, we will substitute \(x = 0\) and \(y = -2\), resulting in:

• Calculate \(0^2 + 0(-2) + (-2)^2\)
• The computation becomes \(0 + 0 + 4 = 4\)

Since the left side equals the right side (4 = 4), the point \((0, -2)\) indeed lies on the graph. This process, called verification, is useful because it directly checks the validity of a point against the equation.

Graph of the Equation

Understanding the graph of an equation involves knowing how different values of variables affect the structure and shape of the graph. The equation given is a quadratic in both \(x\) and \(y\): \(x^{2}+xy+y^{2}=4\).
When graphing such equations, it's helpful to pick several points and check if they meet the condition set by the equation, as shown in verification.

Graphically, identifying whether points like \((0, -2)\) or \((2, -2)\) lie on the graph helps us see the pattern the graph follows. Such visualization can:

• Provide insight into the symmetry or range of the graph.
• Help in understanding the nature of intersections of the curve with the axes.
• Aid in visualizing solutions of the equation in a geometric context.

The power resulting from graphing is seeing theoretical solutions in a visual manner and making predictions about other points.

Substitution Method

The substitution method is a straightforward approach to simplify problems, particularly when verifying points on a graph. By substituting one point at a time, we can focus on evaluating the equation with concrete values.
In this method:

• One substitutes the \(x\) and \(y\) values of the point into the equation.
• Performs the arithmetic and checks the equality.

Built on a simple formula: Plug the point into the expression and solve to check both sides.

For instance, to verify \((1, -2)\) against \(x^2 + xy + y^2 = 4\):

• Substitute \(x = 1\), \(y = -2\): \(1^2 + 1(-2) + (-2)^2 = 3\).

Since 3 does not equal 4, the point does not lie on the graph, demonstrating how substitution provides a decisive answer.