Problem 77 - College Algebra

Question

Exercises \(77-79\) will help you prepare for the material covered in the first section of the next chapter. Consider the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}-1\) a. Set \(y-0\) and find the \(x\) -intercepts. b. Set \(x-0\) and find the \(y\) -intercepts.

Step-by-Step Solution

Answer

The x-intercepts are at (3,0) and (-3,0). The y-intercepts are at (0,2) and (0,-2).

Step 1: Find the x-intercepts

Set \(y = 0\) in the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}-1=0\). This gives \(\frac{x^{2}}{9}-1 = 0\). Solving for \(x\), we get: \(x^{2} = 9\) or \(x = \pm 3\). Therefore, the x-intercepts are at (3,0) and (-3,0).

Step 2: Find the y-intercepts

Set \(x = 0\) in the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}-1=0\). This gives \(\frac{y^{2}}{4}-1 = 0\). Solving for \(y\), we get: \(y^{2} = 4\) or \(y= \pm 2\). Therefore, the y-intercepts are at (0,2) and (0,-2).

Key Concepts

Ellipse

An ellipse is a type of conic section that you encounter when slicing through a cone at an angle. It appears as an elongated circle and is defined by its unique geometric properties. Mathematically, an ellipse can be represented by the equation of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are constants that determine the shape and size of the ellipse.

Ellipses have two focal points and exhibit symmetry about both their major and minor axes. The longest diameter of an ellipse is its major axis, and the shortest is its minor axis. These axes intersect at the center of the ellipse, which is also the origin point for the coordinates if the ellipse is centered at the origin. Understanding the equation of an ellipse helps in determining its intercepts, size, and shape.

• The value of \(a\) represents half of the length of the major axis.
• The value of \(b\) represents half of the length of the minor axis.

Ellipses are quite common in physics and astronomy, often used to describe the orbits of planets and comets around the sun.

X-intercepts

X-intercepts are the points where a curve or line crosses the x-axis on the coordinate plane. At these points, the value of \(y\) is always zero. Identifying x-intercepts is crucial because they reveal where a function hits the horizontal axis.

In the original exercise, to find the x-intercepts of the ellipse, we set \(y = 0\) in the ellipse’s equation \(\frac{x^2}{9} + \frac{y^2}{4} - 1 = 0\). This simplification allows for solving the x-value directly:

• Substitute \(y = 0\) into the equation, yielding \(\frac{x^2}{9} - 1 = 0\).
• Solve for \(x\), resulting in \(x^2 = 9\).
• The solutions for \(x\) are \(x = 3\) and \(x = -3\).

This means the ellipse intercepts the x-axis at points (3,0) and (-3,0). Finding intercepts this way helps to visualize the graph of the equation and understand the behavior of the function.

Y-intercepts

Y-intercepts are points at which the curve or line crosses the y-axis on the coordinate plane. For y-intercepts, the value of \(x\) is always zero. Finding y-intercepts is essential to understand the points where a function intersects the vertical axis.

In the given exercise, to determine the y-intercepts, we set \(x = 0\) in the equation for the ellipse: \(\frac{x^2}{9} + \frac{y^2}{4} - 1 = 0\). Doing so simplifies the y-computation:

• By setting \(x = 0\), the equation becomes \(\frac{y^2}{4} - 1 = 0\).
• Solve for \(y\), leading to \(y^2 = 4\).
• The solutions are \(y = 2\) and \(y = -2\).

Thus, the y-intercepts are located at (0,2) and (0,-2). Understanding the method to calculate intercepts helps to graph equations and examine the behavior and characteristics of curves.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane defined by a horizontal line (x-axis) and a vertical line (y-axis) that intersect at the origin point \((0,0)\). This plane is fundamental in mathematics because it allows the graphical representation of equations and functions.

On the coordinate plane, each point can be described by an ordered pair \((x,y)\). The first value represents the position along the x-axis, and the second represents the position along the y-axis. This setup enables us to map out shapes, functions, and data points visually.

• The x-axis is horizontal, and values increase from left to right.
• The y-axis is vertical, and values increase from bottom to top.

The coordinate plane is essential for graphing solutions to equations, finding intercepts, and visualizing geometrical shapes like ellipses. It serves as a foundation for much of algebra, calculus, and geometry, making it an indispensable tool for understanding and working with mathematical concepts.