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Chapter 6: Problem 61

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

Short Answer

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

Step by step solution

01

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 \( c \).
02

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 \((3, 8)\): \( 8 = a(3)^2 + b(3) + c \) => \( 8 = 9a + 3b + c \)
03

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):\( 1 + c = 3 \) => \( c = 2 \)
04

Write the Final Equation

Now that we have \( a = 1 \), \( b = -1 \), and \( c = 2 \), we can write the equation of the parabola as:\[ y = x^2 - x + 2 \]
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

To analyze population dynamics of the northern spotted owl, mathematical ecologists divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$\left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. $$\begin{aligned} &j_{n+1}=0.33 a_{n}\\\ &s_{n+1}=0.18 j_{n}\\\ &a_{n+1}=0.71 s_{n}+0.94 a_{n} \end{aligned}$$ (Source: Lamberson, R. H., R. McKelvey, B. R. Noon, and C. Voss, "A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape," Conservation Biology, Vol. \(6, \text { No. } 4 .)\) (a) Suppose there are currently 3000 female northern spotted owls: 690 juveniles, 210 subadults, and 2100 adults. Use the preceding matrix equation to determine the total number of female owls for each of the next 5 years. (b) Using advanced techniques from linear algebra, we can show that, in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In this model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the \(3 \times 3\) matrix. This number is low for two reasons: The first year of life is precarious for most animals living in the wild, and juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, due to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \leq\left(\frac{1}{2}\right)^{x}\\\ &y \geq 4 \end{aligned}$$

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x^{2}-y=0\\\&x+y^{2}=0\end{aligned}$$

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A+(B+C)=(A+B)+C\) (associative property)

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A(B+C)=A B+A C\) (distributive property)
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