Problem 24 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find the coordinates of the vertex of the parabola passing through the points \((2,0)\), \((-1,9)\), and \((1,-5)\) Decide upon a strategy for doing this problem.

Step-by-Step Solution

Answer

The coordinates of the vertex of the parabola that passes through the points (2,0), (-1,9), and (1,-5) are (0.75, -1.125).

Step 1: Formulate the System of Equations

Using the standard equation for a parabola \(y = ax^2 + bx + c\), substitute each given point into the equation to get three equations. With (2,0): \(0=4a+2b+c\), with (-1,9): \(9=a-b+c\), with (1,-5): \(-5=a+b+c\).

Step 2: Solve the System of Equations

Next, solve the system of equations to find the values of a, b, and c. Doing so, we get: \(a = -2, b = 3, c = -1\). Therefore, the equation of our parabola is \(y = -2x^2 + 3x -1\).

Step 3: Find the Vertex，s Coordinates

The x-coordinate of the vertex of a parabola given by the equation \(y = ax^2 + bx + c\) is given by \(-b/2a\). Substituting our obtained values of a and b, we get \(x = -3 /-4 = 0.75\). Now, find the y-coordinate by substituting this x-value back into the found quadratic equation: \(y = -2(0.75)^2 + 3*0.75 -1 = -1.125\). Thus, the vertex's coordinates are (0.75, -1.125).

Key Concepts

Understanding the System of Equations

In mathematics, a system of equations is a set of two or more equations with the same set of variables. When trying to find the coordinates of a vertex of a parabola, one can utilize the fact that the parabola's equation can be represented by the quadratic form y = ax^2 + bx + c. Each point that lies on the parabola provides a unique equation when substituted into this form.

To solve the system, one can use methods such as substitution, elimination, or matrix operations. In our exercise, the formation of a system based on three given points leads to a simultaneous solution, revealing the coefficients a, b, and c that define the parabola's curve. Remember, while equations look daunting, they're just sentences in the language of mathematics, waiting for you to uncover their secrets.

The Quadratic Equation and Its Solutions

Quadratic equations are polynomial equations of the second degree, typically taking the form of ax^2 + bx + c = 0. The solutions to these equations are the points where the parabola intersects the x-axis, also known as its roots.

Understanding how to manipulate this form to find the vertex is crucial. A parabola can open upwards or downwards, which is determined by the sign of the a coefficient. Interestingly, while the quadratic equation can seem complex, it can be solved using different techniques such as factoring, using the quadratic formula, completing the square or graphing. It turns out, the quadratic equation isn't just about finding zeros; it's also a treasure map to the parabola's peak or trough--the vertex.

Locating the Vertex with the Parabola Vertex Formula

The parabola vertex formula is a straightforward way to find the turning point of a parabola. This turning point, or vertex, is crucial as it represents the maximum or minimum point on the graph of the quadratic function.

The formula for the x-coordinate of the vertex is -b/2a. Once you have the x-coordinate, you can substitute it back into the original quadratic equation to get the y-coordinate. If you find yourself mixed up in the signs or the calculation, think of it this way: the vertex formula is like a mirror reflecting the apex of a parabolic arch. By understanding the formula's origin and application, you can conquer any parabola problem that comes your way.