Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the parabola (with vertical axis that passes through the data points shown or specified. Check your answer. $$(2,14),(0,0),(-1,-1)$$

Step-by-Step Solution

Answer

The equation is \( y = 2x^2 + 3x \).

Step 1: Identify the General Equation

To find the equation of a parabola with a vertical axis, we start with the general form: \[ y = ax^2 + bx + c \]

Step 2: Write System of Equations

Use the given points \((2, 14)\), \((0, 0)\), and \((-1, -1)\) to create a system of equations based on the general equation:- Substituting \((2, 14)\) gives: \[ 4a + 2b + c = 14 \] - Substituting \((0, 0)\) gives: \[ c = 0 \] - Substituting \((-1, -1)\) gives: \[ a - b + c = -1 \]

Step 3: Simplify the System Using Given Equations

Since \(c = 0\), substitute \(c\) in the other equations:- Equation 1 becomes: \[ 4a + 2b = 14 \]- Equation 3 becomes: \[ a - b = -1 \]

Step 4: Solve the Simplified System

Solve for \(a\) and \(b\) using substitution or elimination:1. From \( a - b = -1 \), we can express \(a\) as: \[ a = b - 1 \]2. Substitute \(a = b - 1\) into the first equation: \[ 4(b - 1) + 2b = 14 \] \[ 4b - 4 + 2b = 14 \] \[ 6b = 18 \] \[ b = 3 \]3. Substitute \(b = 3\) back to find \(a\): \[ a = 3 - 1 = 2 \]

Step 5: Write the Parabola Equation

Using the values \(a = 2\), \(b = 3\), and \(c = 0\), substitute back into the general equation:\[ y = 2x^2 + 3x \]

Key Concepts

System of Equations

A system of equations is a set of two or more equations that share common variables. The goal when dealing with a system is to find the value of the variables that satisfy all equations in the set simultaneously. In our exercise, the given data points

• (2, 14)
• (0, 0)
• (-1, -1)

are used to form a system based on the general equation of a parabola.
Each point gives us an equation by substituting the x and y values into the general equation. Here, the system of equations formed was:

• 4a + 2b + c = 14
• c = 0
• a - b + c = -1

The task then becomes solving for the variables a, b, and c that make all three equations true. Once a value is found for c, it simplifies the system, making it easier to solve for a and b, as demonstrated in our solution.

General Equation of a Parabola

The general equation of a parabola with a vertical axis is expressed as \[ y = ax^2 + bx + c \].This form is quite powerful because it lets us describe any parabola by simply adjusting the values of a, b, and c.

• The coefficient \(a\) controls the width and the direction of the parabola. If \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards.
• The coefficient \(b\) influences the direction in which the parabola shifts along the x-axis, causing it to tilt one way or another.
• The term \(c\) represents the y-intercept of the parabola, which is where it crosses the y-axis.

By having the specific values of these coefficients determined by the given data points, we can uniquely define the parabola passing through the points.

Substitution Method

The substitution method is a common algebraic technique used to find solutions for a system of equations. It involves solving one equation for one variable and then substituting this expression into the other equations. This has the effect of reducing the number of equations and variables, making the system easier to solve.
In our case, after substituting the given point (0,0) and finding \(c = 0\), we simplified the system:

• 4a + 2b = 14
• a - b = -1

We then solved the second equation for \(a\) in terms of \(b\): \[ a = b - 1 \].
When we substituted this into the first equation, it allowed us to find the value of \(b\) directly; thereafter, we could substitute \(b = 3\) back to determine \(a\). This step-by-step substitution process is what ultimately leads us to find the exact coefficients for our specific parabola.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is a mathematical discipline that utilizes algebraic approaches to study geometric objects. It uses a coordinate system as the basis to discuss geometric shapes and their relationships. In this exercise, we used the principles of coordinate geometry to explore how specific points define a parabolic shape.
The given points

• (2, 14)
• (0, 0)
• (-1, -1)

serve as anchors to determine the unique path of the parabola. By substituting the x and y values of these points into the general equation format of a parabola, we establish a clear, algebraic path that the geometric shape (parabola) will follow. Understanding coordinate geometry empowers us to transition smoothly between geometric intuition and algebraic expression, enhancing problem-solving capabilities in both dimensions.