Problem 12 - 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. $$x^{2}-6 x+y+5=0$$

Step-by-Step Solution

Answer

Answer: The given equation represents a parabola, and a suitable viewing window for its graph is [2.5, 3.5] x [3, 5].

Step 1: Identify the coefficients of the given equation

The given equation is in the general form: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ For the given equation, $$x^{2}-6 x+y+5=0$$: - A = 1 - B = 0 - C = 0 - D = -6 - E = 1 - F = 5

Step 2: Calculate the discriminant

The discriminant of a conic equation is given by the formula: $$\Delta = B^2 - 4AC$$ For our equation, we have: $$\Delta = (0)^2 - 4(1)(0) = 0$$

Step 3: Identify the conic section using the discriminant

Using the discriminant, we can determine the type of conic section as follows: - If \(\Delta > 0\), the equation represents a hyperbola - If \(\Delta = 0\), the equation represents a parabola - If \(\Delta < 0\), the equation represents an ellipse or a circle Since the discriminant \(\Delta = 0\) for our equation, it represents a parabola.

Step 4: Find the viewing window

A suitable viewing window for a parabola should display its vertex, as well as the part where it opens up or down, left or right. To do this, we can re-write the equation in vertex form: $$(x - h)^2 = -4p(y - k)$$ Given the equation $$x^{2}-6 x+y+5=0$$, let's add \(9\) on both sides and move \(y\) to the other side to get the vertex form: $$x^{2}-6 x + 9 = -y + 4$$ $$(x-3)^2 = -y + 4$$ Now, the equation is in vertex form with \(h = 3\), \(k = 4\), and \(-4p = -1\), so \(p = 0.25\). The vertex of the parabola is \((h, k) = (3, 4)\), and since the coefficient of \(y\) is negative, it opens downwards. To get a complete view of the graph, we will set the window as follows: - Xmin = h - 2(p) = 3 - 2(0.25) = 2.5 - Xmax = h + 2(p) = 3 + 2(0.25) = 3.5 - Ymin = k - 4(p) = 4 - 4(0.25) = 3 - Ymax = k + 4(p) = 4 + 4(0.25) = 5 Viewing window: [2.5, 3.5] x [3, 5]

Key Concepts

Discriminant of a Conic Section

Understanding the discriminant of a conic section is essential for identifying the type of conic represented by a second-degree equation. The discriminant, symbolized by \(\Delta\), is calculated from the coefficients of the quadratic terms in the general form of a conic equation, \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).

The formula for the discriminant is \(\Delta = B^2 - 4AC\). The value of the discriminant indicates the nature of the conic section:

• If \(\Delta > 0\), the equation represents a hyperbola.
• If \(\Delta = 0\), the equation represents a parabola.
• If \(\Delta < 0\), the equation represents an ellipse, and if additionally \(A = C\) and \(B = 0\), it is a circle.

In the exercise, we determined that the discriminant is zero, which means the conic section is a parabola. This step is crucial as it informs the approach to graphing and analyzing the conic section.

Identifying Conic Sections

Identifying conic sections in their general form can initially seem daunting due to the presence of several terms and coefficients. However, the discriminant simplifies this process. Once the discriminant has been calculated, the conic can be classified as a parabola, ellipse, or hyperbola.

The classification process is only the beginning. The next step is to manipulate the equation into a form that reveals more details about the specific conic section. For a parabola, this often means completing the square to convert the equation to its vertex form, which clearly identifies the location of the vertex and the direction in which the parabola opens.

By systematically applying these strategies, one can determine key features of the conic section from its equation—information that's vital for graphing the conic or solving related problems. For students, it's helpful to practice this process with various equations to become proficient in quickly identifying and understanding different conic sections.

Graphing Parabolas

Graphing parabolas is a visualization of the equation's properties and behavior. Once a parabolic equation has been identified, the goal is to graph it accurately. To achieve a precise graph, rewriting the equation in vertex form, \((x - h)^2 = 4p(y - k)\), is advantageous, as it immediately provides the vertex \((h, k)\) and the direction in which the parabola opens based on the sign of \(p\).

The process often involves completing the square, which was demonstrated in the exercise by converting the given equation to vertex form and finding the vertex. With the vertex known, a judicious choice of a viewing window around the vertex allows the most important features of the parabola to be seen: the vertex, axis of symmetry, and the portion that contains the parabola's concavity.

Selecting an appropriate viewing window is essential. For downward-opening parabolas like the one analyzed, ensure the window is set below the vertex; for upward-opening ones, above the vertex. Always include points on either side of the vertex to display the symmetry of the parabola. Breakdowns such as these help students incrementally build their graphing skills—enabling them to tackle not only homework problems but also to apply their understanding in real-world situations where parabolic paths are relevant, such as in projectile motion.