Problem 15 - Contemporary Precalculus

Question

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$17 x^{2}-48 x y+31 y^{2}+49=0$$

Step-by-Step Solution

Answer

Answer: The given equation represents a hyperbola, and an appropriate viewing window for its graph would be -10 ≤ x ≤ 10 and -5 ≤ y ≤ 5.

Step 1: Identify the conic section using the discriminant

The discriminant of a conic section with equation \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\) is given by \(B^2 - 4AC\). Let's find the discriminant of the given equation: $$17 x^{2}-48 x y+31 y^{2}+49=0$$ We have \(A = 17\), \(B = -48\), \(C = 31\). Calculate the discriminant: $$\Delta = B^2 - 4AC = (-48)^2-4(17)(31) = 2304 - 2128 = 176$$ Since \(\Delta >0\), the conic section is a hyperbola.

Step 2: Find the domain and range for the graph

Now we will try to find the domain and range for the hyperbola. Since it is not in the standard form, it's not easy to find them directly. However, we can find an approximate domain and range by plotting the graph or using a graphing calculator. We observe the plotted graph and then decide an appropriate viewing window for the hyperbola.

Step 3: Choose a viewing window

After plotting the graph of the hyperbola using a graphing calculator, we observe that the hyperbola is much wider than it is tall. Based on this observation, we can decide an appropriate viewing window for the graph. We choose the following viewing window that shows a complete graph of the hyperbola: $$-10 \le x \le 10$$ $$-5 \le y \le 5$$ This viewing window should capture the complete graph and allow us to effectively analyze the conic section.

Key Concepts

Discriminant

The discriminant is a key element in determining the type of conic section represented by a quadratic equation in two variables. For an equation in the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is given by the formula \(\Delta = B^2 - 4AC\). This value helps you identify whether the equation represents a parabola, circle, ellipse, or hyperbola.

**Understanding the Discriminant Values:**

• If \(\Delta = 0\), the conic section could be a parabola, or it could be degenerate (meaning it may not form a real curve).
• If \(\Delta > 0\), like in our example where \(\Delta = 176\), the conic section is a hyperbola.
• If \(\Delta < 0\), the conic section is an ellipse or a circle.

Knowing the discriminant simplifies the process of sketching or analyzing the conic section without necessarily graphing it initially.

Hyperbola

A hyperbola is a type of conic section characterized by two mirrored opens arcs. These arcs extend indefinitely in opposite directions. A hyperbola forms when you have a quadratic equation where the discriminant \(\Delta > 0\).

**Recognizing a Hyperbola:**

• Its general form can be \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), with specific conditions.
• In the standard form, a hyperbola is expressed as \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) or \(\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\).
• The center of the hyperbola is at \((h, k)\), with "a" and "b" determining the width of the arcs.

Understanding these properties helps in identifying hyperbolas simply by looking at an equation or a graph. They provide the necessary properties, making it easier to predict how the graph will look.

Graphing Conic Sections

Graphing conic sections involves translating the mathematical expression of the conic into a visual curve on a graph. To accurately plot a conic section, especially when dealing with hyperbolas, knowledge of the appropriate viewing window is important.

**Steps for Graphing a Conic Section:**

• Identify the type of conic using the discriminant as discussed earlier. For hyperbolas and ellipses, reorganize the equation into the standard form.
• Determine the center or the vertex of the conic depending on the type. For hyperbolas, calculate the vertices based on the values in its standard equation form.
• Select a suitable viewing window. The viewing window is an approachable range for x and y that ensures the entire conic section can be seen.

In our example, determining a viewing window of \(-10 \le x \le 10\) and \(-5 \le y \le 5\) effectively shows the complete graph of the hyperbola. By doing this, you gain a full perspective of the conic section, ensuring no important features like vertices or asymptotes are missed.