Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ 4 x^{2}+3 y^{2}=12 $$

Step-by-Step Solution

Answer

The equation describes an ellipse with symmetry about both axes, x-intercepts at \((\pm\sqrt{3}, 0)\) and y-intercepts at \((0, \pm2)\).

Step 1: Check for Symmetries

To check for symmetry, we can analyze the equation with respect to the x-axis, y-axis, and origin. First, check with respect to the x-axis by replacing \(y\) with \(-y\) in the equation:\[4x^2 + 3(-y)^2 = 12\] This simplifies back to the original equation \(4x^2 + 3y^2 = 12\), showing symmetry about the x-axis. Next, check with respect to the y-axis by replacing \(x\) with \(-x\):\[4(-x)^2 + 3y^2 = 12\]This simplifies to the original equation, indicating symmetry about the y-axis.Finally, check for origin symmetry by both \(x\) and \(y\), replacing with \(-x\) and \(-y\):\[4(-x)^2 + 3(-y)^2 = 12\]Again, the equation remains the same after simplification, showing no symmetry about the origin since we previously confirmed symmetry about both axes not the origin.

Step 2: Identify the Type of Conic Section

The given equation resembles the general form of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), which means it could graph as an ellipse. To express it as such, start by dividing the entire equation by 12:\[\frac{4x^2}{12} + \frac{3y^2}{12} = 1\]This simplifies to: \[\frac{x^2}{3} + \frac{y^2}{4} = 1\]This confirms the graph is an ellipse centered at the origin.

Step 3: Find the Intercepts

To find the x-intercepts, set \(y = 0\) and solve for \(x\):\[4x^2 = 12\]\[x^2 = 3\]\[x = \pm \sqrt{3}\]Thus, the x-intercepts are \((\sqrt{3}, 0)\) and \((-\sqrt{3}, 0)\).To find the y-intercepts, set \(x = 0\) and solve for \(y\):\[3y^2 = 12\]\[y^2 = 4\]\[y = \pm 2\]Thus, the y-intercepts are \((0, 2)\) and \((0, -2)\).

Step 4: Sketch the Ellipse

Using the intercepts and identified symmetries, sketch the ellipse. Major axis is along the y-axis from \(y = -2\) to \(y = 2\), and the minor axis is along the x-axis from \(x = -\sqrt{3}\) to \(x = \sqrt{3}\). Since it is symmetric about both the x-axis and y-axis, replicate the first quadrant to the other three quadrants. Label the axes, intercepts, and ensure it's centered at the origin.

Key Concepts

Conic Sections

Conic sections are a fascinating and important part of geometry that describe the curves formed by the intersection of a plane with a cone. These curves include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has its own unique properties and equations. In this exercise, the equation \(4x^2 + 3y^2 = 12\) represents an ellipse.

Ellipses are the set of all points where the sum of the distances to two fixed points (called foci) is constant. They look like stretched circles and can be recognized in their standard form by the equation:

• \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

In our given problem, after dividing by 12, the equation becomes \(\frac{x^2}{3} + \frac{y^2}{4} = 1\), confirming it's in the form of an ellipse. The values \(a^2\) and \(b^2\) are constants that determine the length of the axes.

Understanding conic sections helps us analyze and graph curves on a coordinate plane, revealing insights into both theoretical and real-world phenomena.

X-Intercepts

To find the x-intercepts of a graph, we set \(y = 0\) in the equation and solve for \(x\). This tells us where the graph crosses the x-axis. For the equation \(4x^2 + 3y^2 = 12\), setting \(y = 0\) simplifies it to:

• \(4x^2 = 12\)
• \(x^2 = 3\)
• \(x = \pm \sqrt{3}\).

These solutions mean the graph intersects the x-axis at \((\sqrt{3}, 0)\) and \((-\sqrt{3}, 0)\).

X-intercepts are crucial for sketching graphs as they mark one of the points where the curve meets the horizontal axis. For an ellipse, knowing the intercepts assists in accurately depicting the width and ensuring the correct orientation of the graph on a coordinate plane.

Y-Intercepts

Y-intercepts are determined by setting \(x = 0\) in the equation and solving for \(y\). This identifies the points where the graph crosses the y-axis. With the given ellipse equation, if we let \(x = 0\):

• \(3y^2 = 12\)
• \(y^2 = 4\)
• \(y = \pm 2\).

Hence, the y-intercepts are \((0, 2)\) and \((0, -2)\).

Similarly to x-intercepts, y-intercepts are essential for plotting graphs as they denote where the curve intersects the y-axis. This information helps when scaling and locating the ellipse on the plane, contributing to an accurate presentation of its dimensions and orientation.

Symmetry in Graphs

Symmetry in graphs can simplify the process of graphing by showing how the graph reflects across certain axes. This ellipse is symmetric about both the x- and y-axes.

• Symmetry about the x-axis means that the graph looks the same if flipped over the x-axis. To check this, replace \(y\) with \(-y\) and see if the equation remains unchanged. Here, it does since \(3y^2 = 3(-y)^2\).


• Symmetry about the y-axis is verified by replacing \(x\) with \(-x\). The equation still holds, as \(4x^2 = 4(-x)^2\).

By checking for origin symmetry, replacing both \(x\) and \(y\) with their negatives, the equation remains the same as well.
Understanding symmetry helps with graphing, as any point on one side of an axis of symmetry will have a corresponding point on the other side. This ensures that once part of the curve is drawn, the rest can easily be mirrored, making the process quicker and more accurate.