Problem 61 - A Graphical Approach to College Algebra

Question

Find the equation of the parabola with vertical axis that passes through the data points shown or specified. Check your answer. $$(-1,4),(1,2),(3,8)$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Standard Form of a Parabola

The standard form of a parabola with a vertical axis is given by \( y = ax^2 + bx + c \). To find the equation, we need to determine the values of \( a \), \( b \), and \( c \).

Step 2: Set Up the System of Equations

Using the given points \((-1,4),(1,2),(3,8)\), we substitute \( x \) and \( y \) values into the parabola equation:1. For \((-1, 4)\): \( 4 = a(-1)^2 + b(-1) + c \) => \( 4 = a - b + c \)2. For \((1, 2)\): \( 2 = a(1)^2 + b(1) + c \) => \( 2 = a + b + c \)3. For \((3, 8)\): \( 8 = a(3)^2 + b(3) + c \) => \( 8 = 9a + 3b + c \)

Step 3: Solve the System of Equations

We have three equations:1. \( a - b + c = 4 \)2. \( a + b + c = 2 \)3. \( 9a + 3b + c = 8 \)Subtract equation 1 from equation 2 to eliminate \( c \):\( (a + b + c) - (a - b + c) = 2 - 4 \)\( 2b = -2 \) => \( b = -1 \)Use \( b = -1 \) in equation 2:\( a - 1 + c = 2 \) => \( a + c = 3 \) (i)Use \( b = -1 \) in equation 3:\( 9a - 3 + c = 8 \) => \( 9a + c = 11 \) (ii)Subtract equation (i) from equation (ii) to solve for \( a \):\( (9a + c) - (a + c) = 11 - 3 \)\( 8a = 8 \) => \( a = 1 \)Substitute \( a = 1 \) into equation (i):\( 1 + c = 3 \) => \( c = 2 \)

Step 4: Write the Final Equation

Now that we have \( a = 1 \), \( b = -1 \), and \( c = 2 \), we can write the equation of the parabola as:\[ y = x^2 - x + 2 \]

Step 5: Verify the Equation with Given Points

Substitute the given points back into the equation \( y = x^2 - x + 2 \) to verify:- For \((-1, 4)\): \( y = (-1)^2 - (-1) + 2 = 1 + 1 + 2 = 4 \)- For \((1, 2)\): \( y = 1^2 - 1 + 2 = 1 - 1 + 2 = 2 \)- For \((3, 8)\): \( y = 3^2 - 3 + 2 = 9 - 3 + 2 = 8 \)All points satisfy the equation, confirming that the solution is correct.

Key Concepts

System of Equations

When working with multiple equations, a system of equations is a collection of equations that share the same set of variables. In many problems, like the one involving a parabola, you'll encounter more than one equation that needs to be solved simultaneously.
Solving a system of equations means finding the values of the variables that satisfy all the given equations at the same time. There are several methods to solve these systems, but some commonly used ones include:

• Substitution Method: Solve one of the equations for one variable and substitute it into the other equations.
• Elimination Method: Add or subtract equations to eliminate one of the variables, making it easier to solve for the other variables.
• Graphical Method: Graph the equations and intersect their graphs to find the solution.

In our exercise, we specifically used the substitution method to find the values of the variables that gave us the correct parabolic equation.

Standard Form

The standard form of a parabola equation, particularly for a parabola with a vertical axis, is expressed as \( y = ax^2 + bx + c \). This form is very useful since it clearly displays the quadratic nature of the parabola and allows for straightforward substitution and calculation with given points.

• Coefficient \( a \): Determines the vertical stretch or compression and the direction of the parabolic opening. If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
• Coefficient \( b \): Affects the parabola's symmetry and its horizontal displacement.
• Constant \( c \): Represents the y-intercept, or the point where the parabola intersects the y-axis.

In our example, the values of \( a = 1 \), \( b = -1 \), and \( c = 2 \) yield the quadratic equation \( y = x^2 - x + 2 \). Understanding these components aids in customizing the parabola to fit specific points.

Substitution Method

The substitution method is a powerful technique used to find the solution to a system of equations efficiently. This method involves solving one equation for one variable in terms of the others, and then substituting this expression into the remaining equations in the system.
In the parabola equation problem, after setting up the system of equations with the form \( a - b + c = 4 \), \( a + b + c = 2 \), and \( 9a + 3b + c = 8 \), the substitution method played a crucial role as follows:

• First, solve one equation to express \( b \).
• Substitute the value of \( b \) into the other equations to further reduce the number of variables.
• Systematically simplify until each variable has a determined value.

This method is particularly useful when equations are relatively simple and can be easily manipulated algebraically. In this exercise, it allowed us to find \( a, b, \) and \( c \) values quickly and verify that the final parabola equation matches the data points exactly.