Problem 45 - Algebra 2

Question

Mental Math Each given point is on the graph of the given equation. Use symmetry to find at least one more point on the graph. $$ (0, \sqrt{7}), x^{2}+2 y^{2}=14 $$

Step-by-Step Solution

Answer

By exploiting symmetry of the given ellipse, another point on the graph will be (0, -sqrt(7)).

Step 1: Identifying the Equation and Given Point

The given equation is \(x^{2}+2 y^{2}=14\), and the given point is (0, \( \sqrt{7}\))

Step 2: Understanding Symmetry in the Context of the Given Equation

The equation represents an ellipse, with symmetry about both x and y axes.

Step 3: Finding a New Point on the Graph by Using Symmetry

This symmetry implies that if (0, \( \sqrt{7}\)) is on the graph, then so is (0, -\( \sqrt{7}\)), the reflection of the given point over the x-axis.

Key Concepts

Graphing Ellipses

Graphing ellipses can be a fascinating journey into the realm of geometry. An ellipse is a curved shape that looks like an elongated circle and can be graphed just like other mathematical figures. To graph an ellipse, you need to understand its major (longest) and minor (shortest) axes. These axes are determined by the lengths in the ellipse's standard equation form. The center of the ellipse is typically the origin or where the axes intersect, and from there, the drawing of the shape can begin by marking points around this center.

Ellipses have properties such as:

• They are symmetrical about both their horizontal and vertical axes.
• The farther points (vertices) and closer points (co-vertices) to the center decide the stretch and size of the ellipse.

For example, let's consider the equation: \(x^2 + 2y^2 = 14\). Here, due to the symmetry, you would know you can find points around the center by understanding its axes and the circle's shape, making it possible to directly deduce other points from a given point.

Equation of Ellipse

Understanding the equation of an ellipse is crucial for both graphing the ellipse and exploring its symmetry. The equation \(x^2 + 2y^2 = 14\) is a form of an ellipse equation, albeit not the most standardized format. In general, an ellipse equation is represented as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), wherein \(a\) and \(b\) are the semi-axes lengths surrounding the center.

In our example, we have \(x^2\) and \(2y^2\) in the equation. It shows the ellipse is stretched differently along the x and y axes:

• The coefficient of \(y^2\) ensures that stretching is less on this axis than \(x\).
• The number 14 on the right-side scales the ellipse up to that value.

By dividing the complete equation by 14, the standard form can be identified and help better visualize the ellipse.

Reflection Symmetry

Reflection symmetry in ellipses is a wonderful feature that students use to identify other points on the graph without solving equations for each. If a point is known on the ellipse, its symmetrical counterpart can be easily determined by reflecting across axes. Ellipses have both x-axis and y-axis symmetry.

Take, for example, the point \((0, \sqrt{7})\) on the equation \(x^2 + 2y^2 = 14\). Due to reflection symmetry:

• Reflecting across the x-axis, the coordinates become\((0, -\sqrt{7})\).
• Similarly, the same principle applies horizontally, confirming the presence of other symmetric points.

Every point on the ellipse has its reflected pair, simplifying graph sketching and points identification on elliptical graphs.