Problem 36 - Basic Technical Mathematics with Calculus

Question

Solve the given applied problem. Find the equation of the parabola that contains the points \((-1,14),(1,9),\) and (2,8).

Step-by-Step Solution

Answer

The equation of the parabola is \( y = 0.5x^2 - 2.5x + 11 \).

Step 1: Understand the General Form of a Parabola

The general form of a parabola equation is given by \( y = ax^2 + bx + c \). We need to find the values of \( a \), \( b \), and \( c \) that satisfy all three given points.

Step 2: Set up the System of Equations

Substitute each point into the equation \( y = ax^2 + bx + c \). For point \((-1,14)\), substitute to get \( 14 = a(-1)^2 + b(-1) + c \), simplifying to \( 14 = a - b + c \). For point \((1,9)\), substitute to get \( 9 = a(1)^2 + b(1) + c \), simplifying to \( 9 = a + b + c \). For point \((2,8)\), substitute to get \( 8 = a(2)^2 + b(2) + c \), simplifying to \( 8 = 4a + 2b + c \).

Step 3: Solve the System of Equations

We have three equations: \( 14 = a - b + c \), \( 9 = a + b + c \), and \( 8 = 4a + 2b + c \). Solve this system using the method of elimination or substitution to find \( a \), \( b \), and \( c \). First, subtract the second equation from the first: \( 5 = -2b \), giving \( b = -2.5 \). Next, subtract the third equation from the second: \( 1 = -3a - b \). Substitute \( b = -2.5 \) into this: \( 1 = -3a + 2.5 \), giving \( 3a = 1.5 \) so \( a = 0.5 \). Finally, substitute \( a = 0.5 \) and \( b = -2.5 \) into \( 9 = a + b + c \) to find \( c \): \( 9 = 0.5 - 2.5 + c \), so \( c = 11 \).

Step 4: Write the Equation of the Parabola

Now that we have \( a = 0.5 \), \( b = -2.5 \), and \( c = 11 \), plug these values into the parabola equation: \( y = 0.5x^2 - 2.5x + 11 \). This is the equation of the parabola that passes through the given points.

Key Concepts

Understanding Systems of Equations

A system of equations is a collection of two or more equations that have a common set of variables. In many real-world problems, including those involving parabolas, it's essential to find a unique solution that satisfies all available equations. For instance, when determining the parabolic equation through three points, we express each point in terms of the parabola's general form:

• Each point gives us one equation in terms of the constants we want to find.
• Multiple equations together form the system of equations.

In relation to our problem, this system is developed by substituting the points into the parabola's general form equation, meaning each point provides a linear equation that must be satisfied by the unknowns. Solving such systems often involves techniques like substitution or elimination. By systematically aligning equations and methodically solving them, students can find unknown constants that define the parabola uniquely.

Exploring Quadratic Functions

Quadratic functions are fundamental objects of study in the field of mathematics. Represented by the equation \( y = ax^2 + bx + c \), a quadratic function forms a parabola when graphed in the coordinate plane. Parabolas can open upwards or downwards, depending on the sign of the leading coefficient \( a \). When solving problems involving parabolas:

• The vertex of the parabola provides key insight into its maximum or minimum value.
• The axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \).
• Intercepts with the axes offer important solutions and typically involve simple substitution to find where the parabola crosses the axes.

In the context of determining the parabola through specific points, understanding how each coefficient \( a, b,\) and \( c \) manipulates the shape and position of the parabola in the plane is important. This will guide any adjustments needed when fitting the function through given points.

Problem Solving in Mathematics

Mathematics problem solving is a methodical process that transforms a real-world question or requirement into a mathematical solution. This involves understanding the question, modeling it using appropriate mathematical representations, and carrying out the necessary computations. Key steps in mathematical problem-solving include:

• Understanding the problem: Clearly define what is asked.
• Forming a model: Use appropriate mathematical structures to express the problem.
• Executing a solution: Perform calculations systematically, ensuring each step builds logically on the last.
• Interpreting the result: Verify if the solution makes sense in the original context.

In the given problem, determining the equation of a parabola passing through defined points is a practical example of this process. By identifying information such as the general form, creating a system of equations, and methodically solving them, we obtain a people-appropriate solution linked back to the problem's context.