Question

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

Short answer

The equation represents an ellipse centered at the origin with intercepts at \((\pm \sqrt{3}, 0)\) and \((0, \pm 2)\).

Step-by-step solution

Analyze the Equation for Symmetry

The given equation is \(4x^2 + 3y^2 = 12\). To determine symmetry, note that the equation is in the form \(Ax^2 + By^2 = C\). Such equations represent ellipses and are symmetric with respect to both the x-axis and y-axis. Therefore, the graph has symmetry about both axes.

Determine the Intercepts

To find the x-intercepts, set \(y = 0\) in the equation: \(4x^2 = 12\). Solving for \(x\) gives \(x = \pm \sqrt{3}\). So, the x-intercepts are \((\sqrt{3}, 0)\) and \((-\sqrt{3}, 0)\). To find the y-intercepts, set \(x = 0\) in the equation: \(3y^2 = 12\). Solving for \(y\) gives \(y = \pm 2\). Hence, the y-intercepts are \((0, 2)\) and \((0, -2)\).

Rewrite the Equation in Standard Form

Divide the entire equation by 12 to express it in standard form: \(\frac{x^2}{3} + \frac{y^2}{4} = 1\). This resembles the standard form of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a = \sqrt{3}\) and \(b = 2\).

Determine Key Features of the Ellipse

From the standard form, the major axis is along the y-axis with length \(2b = 4\), and the minor axis is along the x-axis with length \(2a = 2\sqrt{3}\). The center of the ellipse is at the origin \((0, 0)\).

Plot the Graph

Using the intercepts and key features, plot the ellipse centered at the origin. Ensure the distances along the major and minor axes are represented: Extend 2 units up and down from the origin for the major axis, and \(\sqrt{3}\) units left and right for the minor axis. Draw a smooth curve to form the ellipse.

Key concepts

Symmetry in graphs

In mathematics, the concept of symmetry is widely used to simplify calculations and understand geometric shapes better. An ellipse, as seen in the equation \(4x^2 + 3y^2 = 12\), often displays symmetry. Symmetry in graphs means that one portion of the graph mirrors another across a line or around a point.

For ellipses, symmetry is often about both the x-axis and y-axis. This means that if you were to fold the graph along these axes, the portions would match perfectly. Symmetrical graphs help in predicting the shape and behavior of the ellipse easily. It implies that calculations for one side can be mirrored or repeated for another, reducing repetition in tasks like plotting.

Remember:

• Graphs symmetrical about the x-axis mirror above and below this axis.
• Graphs symmetrical about the y-axis mirror left and right of this axis.
• Checking symmetry is a great first step in graph analysis to foresee how the graph will look.

Intercepts

Intercepts are key to understanding points where a graph intersects the axes. In our ellipse equation \(4x^2 + 3y^2 = 12\), intercepts indicate precise locations where the curve crosses the x-axis and y-axis.

X-intercepts are found by setting \(y=0\) in the equation, while Y-intercepts are located by setting \(x=0\). Solving these conditions reveals specific points where the graph touches the horizontal or vertical axis.

For the given ellipse:

• X-intercepts: by setting \(y=0\), we find \((\pm \sqrt{3}, 0)\).
• Y-intercepts: by setting \(x=0\), we find \((0, \pm 2)\).

Intercepts are fundamental as they anchor the graph in space and provide reference points when plotting. This is why they are often among the first features to calculate when graphing.

Standard form of ellipse

The standard form of an ellipse offers a formulaic representation making graphing and identifying features much simpler. The given equation \(4x^2 + 3y^2 = 12\) can be transformed into the standard form \(\frac{x^2}{3} + \frac{y^2}{4} = 1\) by dividing both sides by 12.

This standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) articulates an ellipse centered at the origin. Here, \(a\) and \(b\) represent the semi-major and semi-minor axis lengths, respectively.

The transformations lead to:

• \(a = \sqrt{3}\)
• \(b = 2\)

These values provide quick access to key dimensions of the ellipse, facilitating analysis and ensuring that all graphed dimensions are accurate within the simple coordinate grid. It is this power of the standard form that enables consistent and clear plotting.

Graph plotting

Plotting the graph of an ellipse from its equation requires a series of structured steps to accurately represent its features and shape. Knowing the intercepts and the transformed standard form, the task becomes a straightforward geometric exercise.

To plot the given ellipse \(\frac{x^2}{3} + \frac{y^2}{4} = 1\):

• Placing the center at the origin \((0, 0)\).
• From the center, extend \(b=2\) units upward and downward along the y-axis for the major axis.
• Similarly, extend \(a=\sqrt{3}\) units outward along the x-axis for the minor axis.

This gives a skeleton outline of the ellipse's shape.

Next, drawing a smooth curve that connects these boundary points results in a substantial representation of the ellipse. Always remember, the intercepts ensure that the graph is positioned accurately, while the standard form provides consistency in dimension.