Problem 51 - Algebra and Trigonometry

Question

graph each relation. Use the relation’s graph to determine its domain and range. $$ \frac{x^{2}}{9}-\frac{y^{2}}{16}=1 $$

Step-by-Step Solution

Answer

The domain of the relation \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\) is \(-\infty

Step 1: Graph the equation

To graph the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\), plot the points at the intercepts. The x-intercepts are \(x=3\) and \(x=-3\), while there are no y-intercepts because when \(x=0\), there is no real value for y in the equation. The sketches of the hyperbola on both the positive and the negative side of the x-axis is symmetrical about the x-axis.

Step 2: Determine the domain and range

From the graph, it can be seen that the hyperbola extends indefinitely along both the positive and negative sides of the x-axis. Therefore, the domain of the function is all real numbers, represented as \(-\infty

Key Concepts

Domain and Range

Understanding the domain and range of a hyperbola involves looking at the spread of x and y values that are possible for the given equation. In the equation \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\), determining the domain means identifying all the x-values that satisfy the equation. In a hyperbola, the nature of the equation leads to **all real numbers** being possible for x, as the graph stretches infinitely along the x-axis. Thus, the domain is \(-\infty < x < \infty\).
For the range, we focus on the y-values. In this hyperbola, not all y-values correspond to a real x, which limits the range. From the solution, we learn that the y-values are bounded between -4 and 4, giving us a range of \(-4 < y < 4\). This limitation is due to the y-term in the equation, where certain y-values cause the equation to result in a negative, invalid for real numbers under a square root.

Graphing Equations

Graphing a hyperbola like \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\) involves sketching based on intercepts and symmetry properties. Begin by identifying the intercepts. For x-intercepts, set \(y=0\), yielding \(x=3\) and \(x=-3\). However, for y-intercepts, setting \(x=0\) leads to no real solutions, indicating there are none. This hyperbola is centered on the origin, as visible from the equation's form.
When graphing, symmetry plays a crucial role because hyperbolas are symmetric about their axes. The x-values extend from -3 to 3 at the center, but since there are no y-intercepts, the graph doesn't cross the y-axis, remaining separate between two branches. Each branch dips towards, but never reaches, the x-axis, creating a distinctive open curve. Utilizing these properties helps draw the hyperbola accurately.

Conic Sections

A hyperbola belongs to the group of shapes known as conic sections, formed by the intersection of a plane and a cone. Conic sections include circles, ellipses, parabolas, and hyperbolas. Specifically, a hyperbola represents a situation where the intersecting plane cuts through both halves of the cone, creating two mirrored, open curves.
Each conic section has its equation type, reflecting its unique geometry. For hyperbolas, the standard form is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\). This form allows us to grasp both the opening direction and the distance between the vertices. By analyzing the terms \(a\) and \(b\), we determine the orientation—if x has the positive term, the hyperbola opens horizontally; if y has it, the opening is vertical. Understanding conics unveils the intriguing interconnection between algebra and geometry in graphing tasks.

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Key Concepts

Domain and Range

Understanding the domain and range of a hyperbola involves looking at the spread of x and y values that are possible for the given equation. In the equation \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\), determining the domain means identifying all the x-values that satisfy the equation. In a hyperbola, the nature of the equation leads to **all real numbers** being possible for x, as the graph stretches infinitely along the x-axis. Thus, the domain is \(-\infty < x < \infty\).
For the range, we focus on the y-values. In this hyperbola, not all y-values correspond to a real x, which limits the range. From the solution, we learn that the y-values are bounded between -4 and 4, giving us a range of \(-4 < y < 4\). This limitation is due to the y-term in the equation, where certain y-values cause the equation to result in a negative, invalid for real numbers under a square root.

Graphing Equations

Graphing a hyperbola like \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\) involves sketching based on intercepts and symmetry properties. Begin by identifying the intercepts. For x-intercepts, set \(y=0\), yielding \(x=3\) and \(x=-3\). However, for y-intercepts, setting \(x=0\) leads to no real solutions, indicating there are none. This hyperbola is centered on the origin, as visible from the equation's form.
When graphing, symmetry plays a crucial role because hyperbolas are symmetric about their axes. The x-values extend from -3 to 3 at the center, but since there are no y-intercepts, the graph doesn't cross the y-axis, remaining separate between two branches. Each branch dips towards, but never reaches, the x-axis, creating a distinctive open curve. Utilizing these properties helps draw the hyperbola accurately.

Conic Sections

A hyperbola belongs to the group of shapes known as conic sections, formed by the intersection of a plane and a cone. Conic sections include circles, ellipses, parabolas, and hyperbolas. Specifically, a hyperbola represents a situation where the intersecting plane cuts through both halves of the cone, creating two mirrored, open curves.
Each conic section has its equation type, reflecting its unique geometry. For hyperbolas, the standard form is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\). This form allows us to grasp both the opening direction and the distance between the vertices. By analyzing the terms \(a\) and \(b\), we determine the orientation—if x has the positive term, the hyperbola opens horizontally; if y has it, the opening is vertical. Understanding conics unveils the intriguing interconnection between algebra and geometry in graphing tasks.