Problem 49 - Precalculus with Limits

Question

In Exercises 43-50, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for \(y\), and (c) use a graphing utility to graph the equation. \(x^2+4xy+4y^2-5x-y-3=0\)

Step-by-Step Solution

Answer

The graph of the equation is a parabola, which is classified using a discriminant value of 0. The quadratic formula yields two solutions for y, given by \(y_1 = \frac{-(x - 1.25)+ \sqrt{(x - 1.25)^2 - 1(x - 1)}}{2}\) and \(y2 = \frac{-(x - 1.25) - \sqrt{(x - 1.25)^2 - 1(x - 1)}}{2}\).

Step 1: Classify the Graph using the Discriminant

The given equation can be expressed in general form as \(Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0\). Here, \(A=1\), \(H=2\), \(B=4\), \(G=-2.5\), \(F=-0.5\) and \(C=-3\). We can classify the graph of the equation using the discriminant, which is \(D = H^2 - AB\). Substituting the known values, we get \(D = 2^2 - 1*4 = 4 - 4 = 0\). Since \(D = 0\), the graph is a parabola.

Step 2: Solve for y using the Quadratic Formula

Now, rearrange the given equation into a form suitable for the quadratic formula, \(y = ax^2 + bx + c\). In this case, y is the subject of the formula. So, we rearrange the equation to \(4y^2 + 4xy +x^2 - 5x - y - 3 = 0\) then divide by 4, \(y^2 + yx + 0.25x^2 - 1.25x -0.25 - 0.75 = 0\). Applying the quadratic formula, \(y = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = x - 1.25\) and \(c = 0.25x^2 -1\). Solving for y gives two solutions \(y_1 = \frac{-(x - 1.25)+ \sqrt{(x - 1.25)^2 - 1(x - 1)}}{2}\) and \(y2 = \frac{-(x - 1.25) - \sqrt{(x - 1.25)^2 - 1(x - 1)}}{2}\).

Step 3: Graphing the Equation

The graphing step is omitted in here but could be done using graphing software, substituting the values of x to get the corresponding values of y, then plot the points to draw the graph of the equation.

Key Concepts

Discriminant in Conic Sections

Understanding the discriminator in conic sections is vital when classifying the graphs of quadratic equations. In the context of conic sections, the discriminant is given by the formula \( D = H^2 - AB \), where \(A\) and \(B\) are the coefficients of \(x^2\) and \(y^2\), respectively, and \(H\) is the coefficient of the \(xy\) term.

When working with the equation \( Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 \), classifying the graph as a circle, ellipse, hyperbola, or parabola depends on the value of the discriminant. A discriminant of zero (\(D = 0\)) indicates that the graph is a parabola. If \(D > 0\), we're looking at a hyperbola, while \(D < 0\) suggests an ellipse or circle. In our exercise, since the discriminant is zero, we instantly know that the graph represents a parabola.

Quadratic Formula

The Quadratic Formula is a mathematical lifeline for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It is expressed as \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

The beauty of this formula lies in its ability to provide exact solutions for \(y\), known as the roots of the equation, regardless of whether they are real or complex numbers. The part under the square root sign, \( b^2 - 4ac \), is known as the discriminant and informs us about the nature of the roots. If the discriminant is positive, we have two distinct real roots. If it's zero, we get one real root (since the roots coincide), and a negative value gives us complex roots. In our exercise, the application of the Quadratic Formula allows us to find the \(y\)-values for the parabolic graph based on the given \(x\)-values.

Parabola

A parabola is one of the fundamental shapes in conic sections and is formed by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. It has a unique U-shaped curve with an apex point called the vertex.

Parabolas are represented by a quadratic equation in its standard form \( y = ax^2 + bx + c \). The orientation and width of the parabola are determined by the coefficient, \(a\). When \(a > 0\), the parabola opens upwards, and when \(a < 0\), it opens downwards. The vertex can be found by using the formula \( x = -\frac{b}{2a} \), and then substituting \(x\) back into the equation to find the \(y\)-coordinate. Parabolas have significant applications in physics, engineering, and other sciences due to their reflective properties and the concept of a focal point.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double napped cone. They represent essential shapes in geometry and are divided into four categories: circles, ellipses (which include circles as a special case), parabolas, and hyperbolas.

• Circles arise when the intersecting plane is at a right angle to the cone's axis.
• Ellipses form when the angle is oblique, but not steep enough to produce a hyperbola.
• Parabolas result from a plane parallel to a generating line of the cone.
• Hyperbolas appear when the plane is steeply inclined to the axis.

Each conic can be described by a second-degree polynomial equation. Their diverse properties and behaviors are not only mathematically intriguing but also have practical applications in various fields such as astronomy, physics, and satellite dish design. Understanding the underlying principles of conic sections is key to mastering many concepts in mathematics and related disciplines.