Problem 23 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

\text { Find the equation of the parabola through the points }(0,3),(1,0), \text { and }(2,-1) \text { . }

Step-by-Step Solution

Answer

The equation of the parabola through the points (0,3), (1,0), and (2,-1) is \(y = x^2 - 4x + 3\).

Step 1: Formulate the set of equations

Use the general equation of a parabola \(y = ax^2 + bx + c\) and substitute each of the given points into this equation to form three equations:1. From (0,3), we get: \(3 = 0a + 0b + c\), simplifying to \(c = 3\),2. From (1,0), we get: \(0 = a + b + c\),3. From (2,-1), we get: \(-1 = 4a + 2b + c\).

Step 2: Solve the set of equations

Now we have the set of equations:1. \(c = 3\),2. \(a + b + 3 = 0\),3. \(4a+2b+3 = -1\).Equation 2 can be simplified to: \(a + b = -3\). Equation 3 can be simplified to: \(4a + 2b = -4\).Divide equation 3 by 2 to get: \(2a + b = -2\). Now we have two systems of equations: 1. \(a + b = -3\) and 2. \(2a + b = -2\). Substract equation 1 from equation 2 to get: \(a = 1\) and substitute \(a = 1\) into equation 1 to find \(b = -4\).

Step 3: Get the parabola's equation

Now that we have the values: \(a = 1\), \(b = -4\), and \(c = 3\), we can substitute these into the general equation of a parabola \(y = ax^2 + bx + c\) which gives us the equation: \(y = x^2 - 4x + 3\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the second degree, which means they include a variable raised to the power of two. They follow the standard form of an equation: \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \) is not equal to zero. The graph of a quadratic equation is a parabola, which can either open upwards or downwards depending on the sign of the coefficient \( a \).

To solve quadratic equations, one can utilize various methods such as factoring, using the quadratic formula, completing the square, or graphically solving for the roots, which are the values of \( x \) where the parabola intersects the x-axis. In the context of finding the equation of a parabola through specific points, we use the provided points to generate a system of equations that helps determine the values of the coefficients \( a \), \( b \), and \( c \) that satisfy the particular curve.

Systems of Equations

Systems of equations consist of multiple equations that share variables and have common solutions. When working with a system, the goal is to find the set of values for the variables that will satisfy all equations in the system simultaneously.

There are several methods for solving systems of equations, such as graphing, substitution, elimination, and matrix operations. In our solved exercise example, we used substitution and elimination to solve a system of linear equations derived from the known points of the parabola. By substituting the coordinates of the points into the general parabola equation (quadratic equation), new linear equations were formulated and then solved step-by-step, leading to the values of \( a \), \( b \) and \( c \) that define the specific parabola equation.

Understanding Systems of Equations in Solving Parabolas

By applying the values from the points given into the parabola equation, systems of equations become a powerful tool in defining the precise quadratic equation of a parabola. When the points lie on the curve, they reveal the interdependent relationship among the coefficients that shape the parabola. Solving the system thus uncovers the exact parameters that describe the quadratic curve passing through the given points.

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients, structured as the sum of terms of non-negative integer exponents of variables. Essentially, it is a combination of monomials. Polynomial functions can take various forms depending on the highest degree of the variable, which also dictates the general shape of their graphs.

Parabolas are the graphs of polynomial functions of degree two, also known as quadratic functions. The standard form of a quadratic function is \( f(x) = ax^2 + bx + c \), where \( a \) determines the 'width' or 'narrowness' of the parabola and whether the parabola opens up (\( a > 0 \) or down (\( a < 0 \) ). Coefficient \( b \) affects the horizontal position of the vertex, and \( c \) gives the vertical position of the parabola's vertex.

Role of Polynomial Functions in Identifying Parabolas

A clear understanding of polynomial functions allows students to analyze the nature of the parabola and its directional movement. Knowing how to work with polynomial functions is crucial in various fields of science and engineering, as they help model and solve real-world problems. In our context, recognizing the parabola as a polynomial function with specific points enables us to deduce the particular quadratic equation that it represents.