Problem 54 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

X-intercepts: (3, 0) and (-3, 0); Y-intercepts: (0, 2) and (0, -2).

Step 1: Introduction

Given the equation \(\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1\), we need to find both the x-intercept and the y-intercept. This equation represents the graph of an ellipse centered at the origin.

Step 2: Finding the x-intercepts

To find the x-intercepts, set \(y = 0\) and solve for \(x\). Substitute into the equation: \(\frac{x^{2}}{9} + \frac{0^{2}}{4} = 1\). This simplifies to \(\frac{x^{2}}{9} = 1\). Multiply both sides by 9 to get \(x^{2} = 9\). Solving for \(x\) gives \(x = \pm 3\). Therefore, the x-intercepts are \((3, 0)\) and \((-3, 0)\).

Step 3: Finding the y-intercepts

To find the y-intercepts, set \(x = 0\) and solve for \(y\). Substitute into the equation: \(\frac{0^{2}}{9} + \frac{y^{2}}{4} = 1\). This simplifies to \(\frac{y^{2}}{4} = 1\). Multiply both sides by 4 to get \(y^{2} = 4\). Solving for \(y\) gives \(y = \pm 2\). Therefore, the y-intercepts are \((0, 2)\) and \((0, -2)\).

Step 4: Conclusion

The x-intercepts of the graph are at \((3, 0)\) and \((-3, 0)\), while the y-intercepts are at \((0, 2)\) and \((0, -2)\). These points define where the ellipse crosses the axes.

Key Concepts

Finding x-intercepts

Finding x-intercepts involves discovering points where the graph touches the x-axis. At these points, the y-coordinate is always zero. To find the x-intercepts for an ellipse, start by setting \( y = 0 \) in the equation. In our example, the equation is \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \). By substituting \( y = 0 \) into it, you simplify to \( \frac{x^{2}}{9} = 1 \). Next, solve for \( x \) by multiplying both sides by 9. This gives \( x^{2} = 9 \). Finally, take the square root of both sides to find \( x = \pm 3 \).

Hence, the x-intercepts are \( (3, 0) \) and \((-3, 0)\). These points are where the ellipse crosses the x-axis, emphasizing the symmetric nature of the ellipse about the origin. Remember: finding x-intercepts is always about setting \( y \) to zero and solving for \( x \). This pattern applies to any equation touching the x-axis.

Finding y-intercepts

To identify the y-intercepts of an ellipse, you will need to determine where the graph intersects the y-axis. At these points, the x-coordinate is zero. For the provided ellipse equation, the process of finding the y-intercepts begins by setting \( x = 0 \). Using the given equation \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \), substituting \( x = 0 \) simplifies the equation to \( \frac{y^{2}}{4} = 1 \).

Multiply both sides by 4, yielding \( y^{2} = 4 \). By solving this equation, you find \( y = \pm 2 \). Thus, the y-intercepts are \( (0, 2) \) and \( (0, -2) \). These placements illustrate the points where the ellipse crosses the y-axis.

Always keep in mind that when finding y-intercepts, set \( x \) to zero. Then, proceed to resolve the equation for \( y \). This crucial concept aids in isolating the values where a shape touches the y-axis.

Equation of an Ellipse

The equation of an ellipse is key to understanding its shape and properties. Ellipses are conic sections similar to stretched circles, and they are often centered at the origin. Their standard equation form is \[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, \] where \( a \) is the semi-major axis length, determining the horizontal stretch, and \( b \) is the semi-minor axis length, dictating vertical compression.

Ellipses that are centered at the origin display symmetry around both the x-axis and y-axis, making intercept calculations predictable by simple manipulation of variables set to zero. Understanding this equation helps in graphing the ellipse, assessing maximum and minimum distances from the center, and finding necessary intercepts.

• The coefficients \( a \) and \( b \) represent the distances from the center at \( (0,0) \) to the vertices along the axes.
• Each intercept point occurs at positions defined when either variable corresponding to the axis is set to zero.

Grasping these principles helps in effectively addressing and solving any exercise related to ellipses.