Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{2}+9(y+2)^{2}=36 $$

Step-by-Step Solution

Answer

Find intercepts: (0,0) and (0,-4). Plot centered at (0,-2) with axes 6 and 2.

Step 1: Rewrite Equation in Standard Form

The given equation is \( x^2 + 9(y+2)^2 = 36 \). We can rewrite it to make it clearer: \( \frac{x^2}{36} + \frac{(y+2)^2}{4} = 1 \). This is in the form of the equation of an ellipse where the center is (0,-2), with semi-major axis 6 (horizontal) and semi-minor axis 2 (vertical).

Step 2: Check for Symmetries

Since the ellipse's equation does not contain any mixed products or additional \(x\) or \(y\) terms other than shifting \(y\) by 2, it is symmetric with respect to the x-axis. When \(x\) is replaced by \(-x\), the equation remains the same, indicating symmetry with respect to the y-axis as well.

Step 3: Find x-intercepts

To find the x-intercepts, set \(y = 0\) in the equation \(\frac{x^2}{36} + \frac{(0+2)^2}{4} = 1 \), which simplifies to \(\frac{x^2}{36} + 1 = 1\). Solving for \(x^2\), we get \(x^2 = 0\), so \(x = 0\). The ellipse has one x-intercept at (0,0).

Step 4: Find y-intercepts

To find the y-intercepts, set \(x = 0\) in the equation \(\frac{0}{36} + \frac{(y+2)^2}{4} = 1\). This simplifies to \(\frac{(y+2)^2}{4} = 1\). Solving for \(y\), we get \((y+2)^2 = 4\), or \(y+2 = \pm 2\). Therefore, \(y = 0\) or \(y = -4\). The y-intercepts are at (0,0) and (0,-4).

Step 5: Plot the Graph

With the center known at (0,-2), semi-major axis along the x-direction of 6, and semi-minor axis of 2 in the y-direction, an ellipse is plotted around this center through the intercepts (0,0) and (0,-4). The ellipse extends from (-6,-2) to (6,-2) horizontally and from (0,0) to (0,-4) vertically.

Key Concepts

Symmetry

Symmetry in geometry refers to a balanced arrangement, where one half mirrors the other. For ellipses, detecting symmetry helps simplify graphing. The given ellipse equation, after transformation, is \[ \frac{x^2}{36} + \frac{(y+2)^2}{4} = 1 \].
The absence of mixed terms (\( xy \) component) indicates it is symmetric.

• X-axis Symmetry: When swapping \( y \) with \(-y \) and the equation remains unchanged, it means the ellipse is symmetric along the x-axis.
• Y-axis Symmetry: Similarly, swapping \( x \) with \(-x \) leaves the equation unaltered, confirming y-axis symmetry.

The symmetry of an ellipse about both axes allows easier plotting and ensures the ellipse looks the same in all quadrants, aiding in predicting intercept positions too.

Intercepts

Intercepts are points where the ellipse crosses the axes. For ellipses, knowing intercepts helps in understanding the shape and symmetry better. Intercepts are determined by setting one variable to zero and solving for the other.

• X-intercepts: By setting \( y = 0 \), the equation \[ \frac{x^2}{36} + 1 = 1 \] results in \[ x^2 = 0 \], producing an x-intercept at \( (0, 0) \).
• Y-intercepts: Conversely, setting \( x = 0 \) gives \[ \frac{(y+2)^2}{4} = 1 \], which leads to \( y = 0 \) or \( y = -4 \). Therefore, the y-intercepts are \( (0, 0) \) and \( (0, -4) \).

These intercepts not only confirm symmetry but also help in accurately sketching the ellipse on a graph.

Graphing

Graphing an ellipse involves plotting points that represent the shape precisely on a coordinate grid. Understanding the semi-major and semi-minor axes lengths is crucial.

• Center: For our equation, the center of the ellipse is at \( (0, -2) \), derived by the \( y+2 \) term in the equation.
• Semi-Major Axis: This is the largest diameter and, in this equation, lies along the x-axis with a length of 12 (since semi-major axis \( = 6 \).
• Semi-Minor Axis: The shorter axis is vertical, with a total length of 4 (semi-minor axis equal to 2).

Graphing involves placing the ellipse centered at \( (0, -2) \), extending 6 units left and right, and 2 units up and down, connecting these regions smoothly to form the elliptical shape.

Conic Sections

Ellipses belong to a group of shapes known as conic sections. These shapes result from slicing a cone at different angles, resulting in circles, ellipses, parabolas, or hyperbolas.

• Ellipses: Formed when a plane cuts through a cone at an angle, but not perpendicularly to the base.
• Diverse Features: These shapes have a distinct center, major and minor axes, and they display symmetry, making them important in areas like astronomy and physics.

Understanding the basic conic section concept helps in recognizing ellipses among other shapes and linking their geometric properties to real-world applications.