Problem 76 - Precalculus with Limits

Question

In Exercises 73 - 78, find a quadratic model for the sequence with the indicated terms. \( a_0 = 3, a_2 = 0, a_6 = 36 \)

Step-by-Step Solution

Answer

The quadratic model for the given sequence is \(y = x^2 - 3x + 3\)

Step 1: Set up the system of equations

The three equations derived from the sequences and the model form \(ax^2+bx+c\) are: \\ For \(a_0 = 3\), substitute \(x=0\) in model equation, we have equation (1) as \(a(0)^2+b(0)+c=3\) or \(c=3\) \\ For \(a_2 = 0\), substitute \(x=2\) in model equation, we have equation (2) as \(a(2)^2+b(2)+c=0\) or \(4a+2b+c=0\) \\ For \(a_6 = 36\), substitute \(x=6\) in model equation, we have equation (3) as \(a(6)^2+b(6)+c=36\) or \(36a+6b+c=36\)

Step 2: Solve the system of equations

We have \(c=3\) from the first equation, we can substitute this into second equation (2) and equation (3) to solve for \(a\) and \(b\). After substitution, equation (2) now reads \(4a+2b+3=0\), simplified to \(2a+b=-1\). Substituting into equation (3) gives \(36a+6b+3=36\), simplified to \(6a+b=5\). Solving these two equations simultaneously gives\[a=1, b=-3\]

Step 3: Write the quadratic model equation

The coefficients \(a\), \(b\), and \(c\) in the quadratic model are now known. So the quadratic model is \(y=ax^2+bx+c\), by substituting the values of \(a\), \(b\), and \(c\), the equation becomes \(y=x^2-3x+3\)

Key Concepts

Quadratic Model

A quadratic model is an equation that offers a way to represent a particular set of data or sequence using a quadratic function. It's particularly useful when the data exhibits a parabolic trend. Each quadratic model is expressed in the standard form:\[ y = ax^2 + bx + c \]Here, \(a\), \(b\), and \(c\) are coefficients that determine the shape and position of the parabola. The quadratic model is essential in various fields like physics, economics, and engineering to predict and analyze data patterns.
To find a quadratic model for a given sequence, we typically need at least three points. For the exercise, the points given were \( a_0 = 3, a_2 = 0, a_6 = 36 \). By substituting these points into the quadratic equation model, you can generate a system of equations to solve for \(a\), \(b\), and \(c\). This model captures the trend or pattern in the sequence data.

System of Equations

A system of equations involves multiple equations that are solved together since they share common variables. In the context of finding a quadratic model, we derive these equations by inserting known values of \(x\) and their corresponding outputs (like \(a_0 = 3\)) into our quadratic model:- For \(a_0 = 3\), substituting in gives: \(c=3\)- For \(a_2 = 0\), substituting in gives: \(4a + 2b + c = 0\)- For \(a_6 = 36\), substituting in gives: \(36a + 6b + c = 36\)These equations are simultaneously solved to find the values of \(a\), \(b\), and \(c\). Solving a system of equations typically requires either substitution or elimination methods. The system helps find a unique set of values that satisfy all the given equations, allowing us to construct our quadratic model.

Solving Quadratic Equations

Solving quadratic equations is a fundamental part of mathematics that involves finding the values of the variable that make the equation true. Once we've substituted point values into the quadratic model, we're left with a system of linear equations. The steps involved in solving these equations include:

• Substitution of known values: e.g., since \(c = 3\), we replace \(c\) in other equations.
• Elimination or substitution: reduce the system to isolate one variable.
• Solve for the unknowns: Solve the simplified equations, yielding values for \(a\), \(b\), and \(c\).

In our example, substituting \(c = 3\) into the other equations and solving them simultaneously led us to find that \(a = 1\) and \(b = -3\). By successfully finding these values, we construct the specific quadratic equation: \[ y = x^2 - 3x + 3 \]This illustrates how quadratic equation solving allows us to model patterns and predictions based on existing data, offering a deeper understanding of quadratic functions and their applications.