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Chapter 2: Problem 8

7鈥�10 ? An equation and its graph are given. Find the x- and y-intercepts. $$ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$

Short Answer

Expert verified
x-intercepts: 3, -3; y-intercepts: 2, -2.

Step by step solution

01

Identifying the x-intercept

The x-intercept is where the graph crosses the x-axis. At this point, the value of y is 0. Substitute y = 0 into the equation: \( \frac{x^2}{9} + \frac{0^2}{4} = 1 \), which simplifies to \( \frac{x^2}{9} = 1 \).
02

Solving for x-intercept

Solve the equation \( \frac{x^2}{9} = 1 \) to find x. Multiply both sides by 9 to get \( x^2 = 9 \). Take the square root of both sides to find \( x = \pm 3 \). Thus, the x-intercepts are x = 3 and x = -3.
03

Identifying the y-intercept

The y-intercept is where the graph crosses the y-axis. At this point, the value of x is 0. Substitute x = 0 into the equation: \( \frac{0^2}{9} + \frac{y^2}{4} = 1 \), which simplifies to \( \frac{y^2}{4} = 1 \).
04

Solving for y-intercept

Solve the equation \( \frac{y^2}{4} = 1 \) to find y. Multiply both sides by 4 to get \( y^2 = 4 \). Take the square root of both sides to find \( y = \pm 2 \). Thus, the y-intercepts are y = 2 and y = -2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Finding X-Intercepts
To find the x-intercepts of an ellipse, we need to determine where the ellipse crosses the x-axis. The x-intercepts occur where the value of y is zero, as any point on the x-axis has a y-coordinate of 0.
In the equation given, \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), substituting \( y = 0 \) results in the simplified equation \( \frac{x^2}{9} = 1 \).
Here's how to solve it:
  • Multiply both sides of the equation by 9 to eliminate the fraction: \( x^2 = 9 \).
  • To find \( x \), take the square root of both sides: \( x = \pm 3 \).
This means the ellipse crosses the x-axis at the points where \( x = 3 \) and \( x = -3 \). Remember, the plus-minus sign (\( \pm \)) indicates there are two intercepts.
Finding Y-Intercepts
Finding y-intercepts involves discovering where the ellipse touches the y-axis. When finding the y-intercepts, we assume that x is zero because every point on the y-axis has an x-coordinate of zero.
Starting with the same ellipse equation, \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), substitute \( x = 0 \) to get \( \frac{y^2}{4} = 1 \).
The steps to solve it are:
  • Multiply both sides by 4 to remove the fraction: \( y^2 = 4 \).
  • Take the square root of both sides to find \( y \): \( y = \pm 2 \).
Thus, the y-intercepts of the ellipse are at \( y = 2 \) and \( y = -2 \). As before, the plus-minus sign tells us we have two intercept points.
Ellipse Equation
The equation of an ellipse can tell us a great deal about its shape and position on a graph. The standard form of an ellipse equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively.
In our specific equation, \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \):
  • The number 9 is \( a^2 \), which means \( a = 3 \).
  • The number 4 is \( b^2 \), implying \( b = 2 \).
This tells us that the ellipse is elongated along the x-axis since \( a \) (3) is greater than \( b \) (2).
An understanding of the ellipse equation allows us to find intercepts and analyze the ellipse鈥檚 dimensions and orientation on a graph.

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