Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 17 through 21, nd the equation of the parabola with the speci cations given. \(x\) -intercepts of \(\pi\) and \(3 \pi ; y\) -intercept of 6

Step-by-Step Solution

Answer

The equation of the parabola is \(y = \frac{2}{\pi^2}(x - \pi)(x - 3\pi)\)

Step 1: Understanding the problem

Given that the x-intercepts are \(\pi\) and \(3\pi\) and the y-intercept is 6. We know that the quadratic equation must pass through these points. So, the quadratic equation should be in the form of \(y=a(x-b)(x-c)\) where \(b\) and \(c\) are the x-intercepts. In this case \(b = \pi\) and \(c = 3\pi\). We're asked to find the value of \(a\).

Step 2: Finding a using y-intercept

To find \(a\), plug in the y-intercept (0,6) into the quadratic equation: \(6 = a * 0-b * 0-c\). So the y-intercept gives us \(6 = a * -\pi * -3\pi\) which simplifies to \(6 = 3\pi^2*a\). Solving for \(a\) gives \(a = \frac{6}{3\pi^2} = \frac{2}{\pi^2}\).

Step 3: Writing the Equation of the parabola

Substitute \(a\), \(b\) and \(c\) back into the quadratic equation \(y = a(x - b)(x - c)\), giving us the equation of the parabola: \(y = \frac{2}{\pi^2}(x - \pi)(x - 3\pi)\)

Key Concepts

X-Intercepts and Y-Intercepts

In the context of quadratic equations represented on the Cartesian plane, x-intercepts are the points where the graph crosses the x-axis, whereas the y-intercept is the singular point where the graph crosses the y-axis. These intercepts provide critical information about the graph's shape and position.

To find the x-intercepts, we set the value of y to zero and solve the equation for x. In our problem, the given x-intercepts are unique values, \(\pi\) and \(3\pi\), which indicates that the parabola touches the x-axis at these points. When graphed, these intercepts manifest as the 'roots' or 'zeros' of the quadratic function.

Conversely, finding the y-intercept involves setting x to zero and calculating the resultant y value. In this particular scenario, the y-intercept is specified as 6, making it the point (0,6). This tells us that when the graph is extended vertically, it will cross the y-axis six units above the origin.

Quadratic Equations

Quadratic equations are fundamental for defining the behavior of parabolas. They can be represented by the standard form \(ax^2 + bx + c = 0\), with a, b, and c being constants, and 'a' must not be zero. The solution to these equations, through factoring, completing the square, or using the quadratic formula, gives us the x-intercepts of the parabola.

In our working example, the equation of the parabola was initially formed as \(y = a(x - b)(x - c)\), showing a relationship where 'b' and 'c' are the x-intercepts. After identifying 'a' through the given y-intercept, we have a complete quadratic equation describing the parabola's trajectory. The ability to manipulate and understand these equations is crucial for algebra students, as they encapsulate a wide range of mathematical phenomena.

Graphing Parabolas

Graphing parabolas is a visual interpretation of quadratic equations, providing a clear picture of how the variables in the equation shape the curve. For every quadratic equation of the form \(y = ax^2 + bx + c\), the graph will always exhibit a U-shaped curve known as a parabola.

To graph a parabola, one typically starts by identifying the axis of symmetry, x-intercepts, y-intercept, and the vertex. These components help layout the most prominent features of the parabola. As an illustration, with the given x-intercepts \(\pi\) and \(3\pi\), and a y-intercept of 6, we would plot these three points and draw a symmetrical curve passing through them, ensuring the curve opens upwards or downwards depending on the sign of 'a' in the equation.

The equation \(y = \frac{2}{\pi^2}(x - \pi)(x - 3\pi)\) from the problem results in a parabola that opens upwards because the coefficient \(\frac{2}{\pi^2}\) is positive. This method of using intercepts and a calculated 'a' value simplifies the graphing process, allowing for a precise representation of the quadratic function in question.