Problem 30 - Precalculus Mathematics for Calculus

Question

Graph the equation \(y^{2}-9 x^{2}=1\) by solving for \(y\) and graphing the two equations corresponding to the positive and negative square roots.

Step-by-Step Solution

Answer

The graph of \(y^2 - 9x^2 = 1\) is a vertically oriented hyperbola.

Step 1: Rewrite the Equation

Start with the given equation, \(y^2 - 9x^2 = 1\). To isolate \(y^2\), add \(9x^2\) to both sides, resulting in \(y^2 = 9x^2 + 1\).

Step 2: Solve for y using Positive Square Root

Take the positive square root of both sides to solve for \(y\), resulting in \(y = \sqrt{9x^2 + 1}\). This represents the upper half of the hyperbola.

Step 3: Solve for y using Negative Square Root

Similarly, take the negative square root to solve for \(y\), yielding \(y = -\sqrt{9x^2 + 1}\). This represents the lower half of the hyperbola.

Step 4: Graph the Positive Root Equation

Graph the function \(y = \sqrt{9x^2 + 1}\). This will be the upper portion of your hyperbola, opening vertically.

Step 5: Graph the Negative Root Equation

Plot the function \(y = -\sqrt{9x^2 + 1}\). This will form the lower portion of your hyperbola, completing the shape.

Key Concepts

Solving Quadratic Equations

Quadratic equations often appear in the form of \[ax^2 + bx + c = 0\] where \(a\), \(b\), and \(c\) are constants. However, when graphing or isolating variables, as in the exercise above, we manipulate the equation differently. You might need to rearrange terms to isolate a particular square, like \(y^2\) in the equation \(y^2 - 9x^2 = 1\).

• Add or subtract terms on both sides of the equation to isolate the quadratic term.
• Recognize when you need to take square roots, as doing so will help solve for the desired variable.

Once the quadratic term is isolated, further solutions often involve square roots to resolve equations further, which is crucial for understanding hyperbolas and other conic sections.

Square Roots

Square roots are integral in solving and graphing quadratic-related problems, especially when tackling equations involving squares or powers of two. Taking the square root of a number reverses the squaring process. When isolating \(y^2\) in the previous hyperbola equation, \(y^2 - 9x^2 = 1\), it becomes crucial to understand both the positive and negative roots.

• Calculate the square root of each side separately when solving an equation.
• Remember, taking the square root includes both the positive and negative solutions, as indicated by \(y = \pm \sqrt{9x^2 + 1}\).
• In graphing, this process helps define both sections of a hyperbola or similar graph structures.

Using square roots effectively ensures accurate solutions and complete graphs.

Graphing Techniques

Graphing involves turning equations into visual representations on the coordinate plane. It's particularly insightful for understanding hyperbolas formed by functions like \(y = \sqrt{9x^2 + 1}\) and \(y = -\sqrt{9x^2 + 1}\). Each part of the equation contributes to constructing different parts of the graph.

• Graph positive root functions separately from negative root functions to differentiate graph sections.
• Note that in the case of hyperbolas, expect a symmetric pattern about the axis due to the nature of the functions being graphed.
• Utilize symmetrical properties to better anticipate the shape and position of the hyperbola on the coordinate plane.

Proficiency in graphing techniques is key to visualizing complex solutions and confirming the validity of algebraic manipulations.