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Magazine Chapter 11: Problem 40
Exer. 39-40: Find an equation for the parabola that has a vertical axis and passes through the given points. $$ P(3,-1), \quad Q(1,-7), \quad R(-2,14) $$
Short Answer
Expert verified
The equation of the parabola is \( y = 2x^2 - 5x - 4 \).
Step by step solution
01
Understand the Parabola Equation
A parabola with a vertical axis can be represented by the equation \( y = ax^2 + bx + c \). We need to determine the values of \( a \), \( b \), and \( c \) using the given points \( P(3, -1) \), \( Q(1, -7) \), and \( R(-2, 14) \).
02
Set Up Equations for Each Point
Substitute each of the given points into the parabola equation:1. For point \( P(3, -1) \): \[ -1 = a(3)^2 + b(3) + c \rightarrow -1 = 9a + 3b + c \]2. For point \( Q(1, -7) \): \[ -7 = a(1)^2 + b(1) + c \rightarrow -7 = a + b + c \]3. For point \( R(-2, 14) \): \[ 14 = a(-2)^2 + b(-2) + c \rightarrow 14 = 4a - 2b + c \].
03
Write the System of Equations
We now have a system of three equations:1. \( 9a + 3b + c = -1 \)2. \( a + b + c = -7 \)3. \( 4a - 2b + c = 14 \).
04
Solve the System of Equations
We'll solve the system using the elimination method:1. Subtract equation 2 from equation 1: \[ (9a + 3b + c) - (a + b + c) = -1 - (-7) \rightarrow 8a + 2b = 6 \] Simplify to \( 4a + b = 3 \).2. Subtract equation 2 from equation 3: \[ (4a - 2b + c) - (a + b + c) = 14 - (-7) \rightarrow 3a - 3b = 21 \] Simplify to \( a - b = 7 \).3. Solve \( a - b = 7 \) for \( a \): \[ a = b + 7 \].
05
Substitute and Solve for \( a \), \( b \), \( c \)
Substitute \( a = b + 7 \) in \( 4a + b = 3 \):\[ 4(b + 7) + b = 3 \rightarrow 4b + 28 + b = 3 \rightarrow 5b = -25 \rightarrow b = -5 \].Using \( b = -5 \) to find \( a \):\[ a = -5 + 7 = 2 \].Substitute \( a = 2 \) and \( b = -5 \) in \( a + b + c = -7 \):\[ 2 - 5 + c = -7 \rightarrow -3 + c = -7 \rightarrow c = -4 \].
06
Write the Final Equation of the Parabola
Combine the values of \( a \), \( b \), and \( c \) to form the equation:\[ y = 2x^2 - 5x - 4 \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertical Axis Parabola
A vertical axis parabola is a curve represented by the general equation \( y = ax^2 + bx + c \). This equation describes a parabola that opens either upwards or downwards, depending on the sign of \( a \). When \( a \) is positive, the parabola opens upwards; if \( a \) is negative, it opens downwards.
Understanding its structure is key in analyzing its behavior:
Understanding its structure is key in analyzing its behavior:
- \( a \) affects the direction and width of the parabola.
- \( b \) and \( c \) influence its position relative to the x-axis and y-axis.
- The vertex form of this equation, \( y = a(x-h)^2 + k \), shows the vertex at \( (h, k) \).
System of Equations
In mathematics, a system of equations is a collection of two or more equations with a set of unknowns. In solving parabola-related problems, you often deal with a system of equations to find the parabola's parameters \( a \), \( b \), and \( c \).
For each known point that the parabola passes through, you substitute the \( x \) and \( y \) values into the parabola equation, creating separate equations. From our exercise problem:
For each known point that the parabola passes through, you substitute the \( x \) and \( y \) values into the parabola equation, creating separate equations. From our exercise problem:
- The point \( (3, -1) \) gives one equation.
- The point \( (1, -7) \) another.
- And the point \( (-2, 14) \) the last one.
Elimination Method
The elimination method is a popular technique for solving a system of linear equations. It simplifies this process by eliminating one variable at a time, making it easier to find the values of the unknowns.
In our given problem, the elimination method was used successfully:
In our given problem, the elimination method was used successfully:
- We subtracted one equation from another to eliminate a variable. This reduces the number of variables and simplifies the equations.
- This sequential elimination helped to express one variable in terms of another, allowing straightforward substitution back into a simplified equation.
- Ultimately, it led us to separate values for each unknown in the parabola equation, \( a, b, \) and \( c \).
Quadratic Equation
A quadratic equation, generally expressed as \( ax^2 + bx + c = 0 \), is pivotal in describing parabolas. Key characteristics include:
- Solutions or "roots" of this equation indicate the x-intercepts of a parabola.
- The vertex of the parabola provides either the maximum or minimum value of the quadratic expression, depending on whether it opens upwards or downwards.
- Quadratic equations are usually solved using methods such as factoring, completing the square, or applying the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
