Graph-3x2 + 12y2 = 84. What Are The Domain And Range?
Understanding the domain and range of a graph is fundamental in algebra and calculus, as it helps us interpret the behavior and limitations of the equations. In this article, we will explore the specific equation 3x² + 12y² = 84, analyze its geometric characteristics, and determine its domain and range. This equation represents a conic section—specifically, an ellipse—and understanding its properties provides valuable insights into its graphing and applications.
---
Understanding the Equation 3x² + 12y² = 84
Before diving into the domain and range, it’s essential to understand the form and nature of the given equation.
Identifying the Type of Conic Section
The equation:
\[ 3x^2 + 12y^2 = 84 \]
is quadratic in both variables and resembles the standard form of an ellipse:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
To confirm this, we can manipulate the given equation into the standard form.
Rearranging into Standard Form
Divide both sides of the equation by 84:
\[ \frac{3x^2}{84} + \frac{12y^2}{84} = 1 \]
Simplify each term:
\[ \frac{x^2}{28} + \frac{y^2}{7} = 1 \]
Expressed as:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
we identify:
\[ a^2 = 28 \quad \Rightarrow \quad a = \sqrt{28} \approx 5.29 \]
\[ b^2 = 7 \quad \Rightarrow \quad b = \sqrt{7} \approx 2.65 \]
Therefore, the equation describes an ellipse centered at the origin with semi-major axis \(a \approx 5.29\) along the x-axis and semi-minor axis \(b \approx 2.65\) along the y-axis.
---
Plotting the Ellipse: Visual Interpretation
Understanding the shape and extent of the ellipse is crucial for determining its domain and range.
Ellipse Characteristics
- The ellipse is centered at the origin (0,0).
- It extends horizontally from \(-a\) to \(a\), i.e., approximately \(-5.29\) to \(5.29\).
- It extends vertically from \(-b\) to \(b\), i.e., approximately \(-2.65\) to \(2.65\).
---
Determining the Domain of the Equation
The domain of a function or relation describes all possible x-values for which the equation yields real y-values.
Methodology for Finding the Domain
Given the equation:
\[ \frac{x^2}{28} + \frac{y^2}{7} = 1 \]
or equivalently,
\[ 3x^2 + 12y^2 = 84 \]
We fix an x-value and analyze whether the corresponding y-values are real.
- Isolate y²:
\[ \frac{y^2}{7} = 1 - \frac{x^2}{28} \]
- Rewrite:
\[ y^2 = 7 \left( 1 - \frac{x^2}{28} \right) \]
- For y² to be non-negative (since y² ≥ 0), the right side must be ≥ 0:
\[ 7 \left( 1 - \frac{x^2}{28} \right) \geq 0 \]
Dividing both sides by 7:
\[ 1 - \frac{x^2}{28} \geq 0 \]
- Rearranged:
\[ \frac{x^2}{28} \leq 1 \]
Multiply both sides by 28:
\[ x^2 \leq 28 \]
- Take the square root:
\[ -\sqrt{28} \leq x \leq \sqrt{28} \]
Since \(\sqrt{28} \approx 5.29\), the domain is:
\[ \boxed{ -\sqrt{28} \leq x \leq \sqrt{28} } \]
or approximately:
\[ -5.29 \leq x \leq 5.29 \]
Therefore, the domain of the ellipse is:
- x-values from \(-\sqrt{28}\) to \(\sqrt{28}\), approximately \(-5.29\) to \(5.29\).
---
Determining the Range of the Equation
The range describes all possible y-values for which the equation is valid, given the x-values within the domain.
Methodology for Finding the Range
Starting again from:
\[ y^2 = 7 \left( 1 - \frac{x^2}{28} \right) \]
Since y² ≥ 0:
\[ y^2 \leq 7 \left( 1 - \frac{x^2}{28} \right) \]
The maximum value of y² occurs at the minimal x², which is zero (x=0):
\[ y^2_{\max} = 7 \left( 1 - 0 \right) = 7 \]
Similarly, the minimum y-value occurs at the maximum x-value (x=±√28), where:
\[ y^2 = 7 \left( 1 - \frac{28}{28} \right) = 0 \]
Thus, y ranges from:
\[ -\sqrt{7} \leq y \leq \sqrt{7} \]
since y² ≥ 0 and y can be positive or negative.
Therefore, the range of the ellipse is:
- y-values from \(-\sqrt{7}\) to \(\sqrt{7}\), approximately \(-2.65\) to \(2.65\).
---
Summary of Domain and Range
| Property | Values | Approximate Values |
| --- | --- | --- |
| Domain (x-values) | \(-\sqrt{28}\) to \(\sqrt{28}\) | \(-5.29\) to \(5.29\) |
| Range (y-values) | \(-\sqrt{7}\) to \(\sqrt{7}\) | \(-2.65\) to \(2.65\) |
---
Implications for Graphing the Equation
Knowing the domain and range allows for precise plotting of the ellipse, ensuring the graph accurately reflects the limits of the equation.
Graphing Tips
- Plot the center at (0,0).
- Mark key points at the extremities:
- Horizontal endpoints: \((-5.29, 0)\) and \((5.29, 0)\).
- Vertical endpoints: \((0, -2.65)\) and \((0, 2.65)\).
- Sketch a smooth, symmetrical ellipse passing through these points.
---
Applications of the Equation and Its Domain and Range
Understanding the domain and range of an ellipse like this has practical applications in various fields:
Physics and Engineering
- Modeling orbital paths where elliptical orbits are common.
- Designing lenses or reflective surfaces with elliptical shapes.
Mathematics and Geometry
- Analyzing conic sections and their properties.
- Solving optimization problems constrained by elliptical boundaries.
Computer Graphics
- Rendering elliptical shapes accurately.
- Implementing collision detection within elliptical boundaries.
Conclusion
The equation 3x² + 12y² = 84 describes an ellipse centered at the origin with semi-major axis approximately 5.29 along the x-axis and semi-minor axis approximately 2.65 along the y-axis. Its domain and range are directly linked to these axes: the domain spans from \(-\sqrt{28}\) to \(\sqrt{28}\), and the range from \(-\sqrt{7}\) to \(\sqrt{7}\). Understanding these limits is crucial for graphing, analyzing, and applying the ellipse in various scientific and mathematical contexts. Whether for academic study or practical application, knowing the domain and range of this conic section enhances comprehension of its geometric properties and real-world relevance.
