Problem 69 - Precalculus: Functions and Graphs

Question

Describe the part of a hyperbola given by the equation. $$y=-\frac{9}{4} \sqrt{x^{2}-16}$$

Step-by-Step Solution

Answer

The hyperbola part given is the downward branches for \( x \leq -4 \) or \( x \geq 4 \), below the x-axis.

Step 1: Understand the Hyperbola Equation

The given equation is \( y = -\frac{9}{4} \sqrt{x^2 - 16} \), where the term \( x^2 - 16 \) indicates a relation to the hyperbola structure which has transverses axis along the x-axis.

Step 2: Identify the Domain of the Function

To find the domain of the function, \( x^2 - 16 \) must be non-negative for the square root to be defined. This means \( x^2 \geq 16 \). Solving gives \( x \leq -4 \) or \( x \geq 4 \).

Step 3: Analyze the Square Root Term

Within the domain found, the expression \( \sqrt{x^2 - 16} \) gives real values, forming part of the hyperbola which extends from the points where \( |x| > 4 \).

Step 4: Consider the Range and Coefficient

The term \(-\frac{9}{4} \) indicates that the part of the hyperbola described is below the x-axis (since it is multiplied by the square root term), affecting how far down the y-values go from the x-axis.

Step 5: Assemble Characteristics of the Hyperbola

Given domains \( x \leq -4 \) or \( x \geq 4 \), and the downward orientation due to the \(-\frac{9}{4}\) factor, this hyperbola has branches opening downwards along the y-axis for \( |x| > 4 \).

Key Concepts

Domain of the Function

When dealing with functions, especially those involving square roots, understanding the domain is crucial. The domain refers to all the possible input values (in this case, x-values) that won't make the function undefined.
In the hyperbola equation given, we have:

• \( y = -\frac{9}{4} \sqrt{x^2 - 16} \)
• The term \( x^2 - 16 \) must be positive or zero to ensure that the square root is a real number.

To find the domain, solve the inequality:

• \( x^2 - 16 \geq 0 \)
• This breaks down to \( x^2 \geq 16 \), meaning \( x \leq -4 \) or \( x \geq 4 \).

This tells us that all x-values in the intervals \([-\infty, -4]\) and \([4, \infty]\) are part of the domain. Avoiding values between -4 and 4 is necessary as they would make the expression under the square root negative, which is undefined for real numbers.

Range of the Function

The range of a function pertains to all the possible output values (y-values) that the function can produce.
In the hyperbola equation \( y = -\frac{9}{4} \sqrt{x^2 - 16} \), the square root term \( \sqrt{x^2 - 16} \) always yields positive or zero values within the domain.

• This is due to the fact that square roots of non-negative numbers are non-negative themselves.

However, since the square root is multiplied by \(-\frac{9}{4}\), all resulting y-values are negative or zero. This flips the values below the x-axis.

• Thus, the range can be written as \( (-\infty, 0] \).
• This includes all possible negative y-values, going as low as necessary depending on how large \(|x|\) grows beyond 4.

Hence, the graph of the hyperbola will extend infinitely downwards while never crossing above the x-axis.

Quadratic Inequality

The term "quadratic inequality" refers to equations or inequalities involving squared terms. For this hyperbola, determining when \( x^2 - 16 \geq 0 \) is essential in finding which x-values are allowed.
This inequality reads:

• \( x^2 - 16 \geq 0 \)
• It requires that the parabola \( x^2 - 16 \) remains above or at y=0.

To solve it:

• Identify critical points by isolating \( x^2 \): \( x^2 = 16 \).
• This simplifies to \( x = \pm4 \), the boundary points where \( x^2 - 16 \) hits zero.
• For intervals, consider when a parabola opens upwards, as this one does.

The intervals that satisfy the inequality are beyond the critical points because the parabola is above the x-axis at \( x \leq -4 \) and \( x \geq 4 \). This analysis ensures a solid grasp of the domain derived from the quadratic inequality.