Problem 2 - Calculus with Concepts in Calculus

Question

Determine all intercepts of the graph of the equation. Then decide whether the graph is symmetric with respect to the \(x\) axis, the \(y\) axis, or the origin. $$ 4 x^{2}+y^{2}=12 $$

Step-by-Step Solution

Answer

Intercepts: \((\sqrt{3}, 0)\), \((-\sqrt{3}, 0)\), \((0, 2\sqrt{3})\), \((0, -2\sqrt{3})\). Symmetry: x-axis, y-axis, origin.

Step 1: Determine x-intercepts

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

Step 2: Determine y-intercepts

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

Step 3: Test Symmetry with Respect to the x-axis

To test for symmetry with respect to the x-axis, replace \( y \) with \( -y \) in the equation and see if the equation remains unchanged:\[4x^2 + (-y)^2 = 12 \4x^2 + y^2 = 12\]Since the equation remains the same, the graph is symmetric with respect to the x-axis.

Step 4: Test Symmetry with Respect to the y-axis

To test for symmetry with respect to the y-axis, replace \( x \) with \( -x \) in the equation and check if the equation is equal to the original:\[4(-x)^2 + y^2 = 12 \4x^2 + y^2 = 12\]The equation is unchanged, indicating symmetry with respect to the y-axis.

Step 5: Test Symmetry with Respect to the Origin

To test for symmetry with respect to the origin, replace \( x \) with \( -x \) and \( y \) with \( -y \) and check the result:\[4(-x)^2 + (-y)^2 = 12 \4x^2 + y^2 = 12\]The equation remains unchanged, showing symmetry with respect to the origin as well.

Key Concepts

Equation of a Conic Section

Conic sections are curves that can be formed by intersecting a plane with a cone. They include ellipses, parabolas, hyperbolas, and circles. The given equation \(4x^2 + y^2 = 12\) is an example of a conic section. It can be rearranged to match the standard form of an ellipse.

To identify the type of conic, we need to rewrite the equation in a recognizable form. For ellipses, the standard form is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.

By dividing the entire equation by 12, we get: \[\frac{4x^2}{12} + \frac{y^2}{12} = 1\] simplifying gives:\[\frac{x^2}{3} + \frac{y^2}{12} = 1\]This indicates the graph is an ellipse because both terms are quadratic, and they are added together with positive coefficients.

Graph Symmetry

Symmetry in graphs helps us understand their shape and orientation. For the given equation \(4x^2 + y^2 = 12\), analyzing symmetry can simplify our understanding of the graph's properties.

There are three types of symmetry relevant here:

• Symmetry with respect to the x-axis: Occurs if replacing \(y\) with \(-y\) results in the same equation.
• Symmetry with respect to the y-axis: Occurs if replacing \(x\) with \(-x\) gives an unchanged equation.
• Symmetry with respect to the origin: Happens if replacing \(x\) with \(-x\) and \(y\) with \(-y\) retains the original form.

In our equation, each transformation results in the original equation, revealing symmetry with respect to the x-axis, y-axis, and the origin.

Intercepts of a Graph

Intercepts are points where a graph crosses the axes. They're critical for sketching the graph's base layout. To find intercepts, set the other variable to zero and solve.

**Finding x-intercepts:**

• Set \(y = 0\): \(4x^2 = 12\) yielding \(x = \pm \sqrt{3}\).
• The x-intercepts are \((\sqrt{3}, 0)\) and \((-\sqrt{3}, 0)\).

**Finding y-intercepts:**

• Set \(x = 0\): \(y^2 = 12\) results in \(y = \pm 2\sqrt{3}\).
• The y-intercepts are \((0, 2\sqrt{3})\) and \((0, -2\sqrt{3})\).

These points help us visualize where the curve touches or crosses the axes.

Coordinate Axes Symmetry

Understanding symmetry concerning the coordinate axes helps predict and simplify graph behavior. It's particularly useful in analytic geometry for reducing the algebraic work needed to assess a function or graph.

When a graph is symmetric:

• Respect to the x-axis: The upper and lower parts of the graph mirror each other.
• Respect to the y-axis: The left and right sides are identical mirrors.
• Respect to the origin: Rotating the graph 180 degrees around the origin leaves it unchanged.

For \(4x^2 + y^2 = 12\), symmetry with respect to all these reflects a well-balanced and evenly distributed geometric shape. Such insight aids in predicting values and additional points on the graph, enhancing comprehensibility.