Problem 79 - College Algebra

Question

In Exercises 79-80, write each equation as a quadratic equation in \(y\) and then use the quadratic formula to express \(y\) in terms of \(x\). Graph the resulting two equations using a graphing utility. What effect does the \(xy\)-term have on the graph of the resulting parabola? $$16 x^{2}-24 x y+9 y^{2}-60 x-80 y+100=0$$

Step-by-Step Solution

Answer

Following the above steps, y is expressed in terms of x as follows: \(y = \frac{24x - 80 \pm \sqrt{(-96x + 320)^2 - 36*(16x^{2} - 60x + 100)}}{18}\). The xy-term in the equation sets a tilt or rotation in the graph of the parabola, altering the axis of symmetry.

Step 1: Arranging the equation in quadratic form

Rewrite the given equation in terms of y, i.e., set the equation to zero by moving all terms not involving y to the other side: \(9y^{2} - 24xy + 16x^{2} - 60x + 100 = 80y\).

Step 2: Applying the quadratic formula

Proceed to express y in terms of x using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this case the equation is of the form \(ay^2 + by + c = 0\), where \(a=9\), \(b=-24x + 80\) and \(c=16x^{2} - 60x + 100\). Substituting the values into the quadratic formula, we obtain \(y = \frac{24x - 80 \pm \sqrt{(-24x + 80)^2 - 4*9*(16x^{2} - 60x + 100)}}{2*9}\)

Step 3: Simplifying the quadratic formula

Simplify the equation to finalize the expression of y in terms of x. The simplified term gives us \(y = \frac{24x - 80 \pm \sqrt{(-96x + 320)^2 - 36*(16x^{2} - 60x + 100)}}{18}\)

Step 4: Discussion on the effect of the xy-term on the graph

With regards to the xy-term effect on the graph, it triggers a tilt or rotation of the graph. This is because when the quadratic expression has an xy-term, the axis of symmetry of the parabola will no longer be vertical (or horizontal), but rather, at an angle. This creates a more complex graph with a tilt.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. It is given by \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula allows you to find the solutions, also called roots, of any quadratic equation of the form \( ay^2 + by + c = 0 \). Let's break it down further:

• \( a, b, \) and \( c \) are coefficients from the quadratic equation.
• \( b^2 - 4ac \) is called the discriminant. It determines the nature of the roots.

The formula is a simple plug-and-play tool, which means once you identify \( a, b, \) and \( c \), you can solve the equation by substituting these values into the formula. This process not only provides numerical solutions but also helps in understanding the behavior of the parabola when it is graphed.

In our example, the quadratic equation in terms of \( y \) presents \( a = 9 \), \( b = -24x + 80 \), and \( c = 16x^2 - 60x + 100 \). After substitution into the quadratic formula, we derive an expression for \( y \) as it depends on \( x \). This fundamental approach showcases how each term affects the overall solution.

Effects of xy-term

The presence of an \( xy \)-term in a quadratic equation can alter the graph's orientation and shape.

When you have a term like \( -24xy \), it complicates the graph since it creates a nonlinear relationship between \( x \) and \( y \). The fascinating aspect is how this added term affects the parabola:

• Without the \( xy \)-term, a parabola would typically open upwards or downwards along a vertical line.
• The inclusion of the \( xy \)-term disrupts this pattern, resulting in the parabola being tilted, meaning the axis of symmetry no longer aligns vertically.

This transformation is not just a shift but a rotation, leading to a different geometric presentation of the parabola. It makes understanding the graph more challenging but also more interesting, as it introduces aspects of symmetry and orientation beyond the basic quadratic term's effects.

Graphing Parabolas

Graphing parabolas involves plotting the solutions of quadratic equations visually. When you graph these kinds of equations, you often consider several properties:

• The vertex of the parabola, which is the turning point.
• The axis of symmetry, which divides the parabola into mirror images.
• Direction of Opening, which tells if the parabola opens upwards, downwards, or tilts due to the \( xy \)-term.

Using a graphing utility simplifies this process, as it can compute and display the curve quickly. In the exercise, after determining \( y \) in terms of \( x \) using the quadratic formula, plotting using a tool gives a tangible image of how the \( xy \)-term affects the parabola's appearance on the graph. This graphical representation complements the algebraic solution, giving a more comprehensive understanding of the equation's implications.

Parabola Transformation

Parabola transformation involves changing the shape and position of the graph of a quadratic equation with certain modifications like stretches, compressions, and rotations.

In the given example, the transformation through the \( xy \)-term leads to:

• A rotated axis of symmetry. Instead of the typical straight vertical or horizontal orientation, the parabola's axis tilts.
• This rotation means the graph will no longer appear as the standard upward or downward curve, but instead might resemble an ellipsoid path, contingent upon the extent of the \( xy \) interaction.

Understanding these transformations is crucial for interpreting more complex quadratic graphs. It helps predict how adjustments to the equation's coefficients and terms will potentially affect the graph's shape and orientation. Practically, these transformations illustrate the dynamic nature of quadratic functions and the diverse representations they can take in various contexts.