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Chapter 8: Problem 51

Use a system of equations to find the parabola of the form \(y=a x^{2}+b x+c\) that goes through the three given points. $$(0,0),(1,3),(2,2)$$

Short Answer

Expert verified
The equation of the parabola is \(y = -2x^2 + 5x\).

Step by step solution

01

Set Up the Equations

Use the general form of a parabola, \(y = ax^2 + bx + c\). Substitute each given point into the equation to create a system of three equations. For point \((0,0)\): \(0 = a(0)^2 + b(0) + c\), which simplifies to \(c = 0\). For point \((1,3)\): \(3 = a(1)^2 + b(1) + c\), which simplifies to \(3 = a + b + c\). For point \((2,2)\): \(2 = a(2)^2 + b(2) + c\), which simplifies to \(2 = 4a + 2b + c\).
02

Substitute Known Values

From Step 1, we know \(c = 0\). Substitute \(c = 0\) into the remaining equations: \(3 = a + b\) and \(2 = 4a + 2b\).
03

Solve the System of Equations

We now have two equations with two variables: \(3 = a + b\) and \(2 = 4a + 2b\). Solve the first equation for one variable: \(b = 3 - a\). Substitute this into the second equation: \(2 = 4a + 2(3 - a)\). Simplify and solve for \(a\): \(2 = 4a + 6 - 2a\) or \(2 = 2a + 6\) or \(-4 = 2a\) or \(a = -2\).
04

Find Remaining Variable

Substitute \(a = -2\) back into \(b = 3 - a\): \(b = 3 - (-2)\) or \(b = 5\).
05

Write the Parabola Equation

Combine the values of \(a\), \(b\), and \(c\) into the general form \(y = ax^2 + bx + c\). Therefore, the equation of the parabola is \(y = -2x^2 + 5x\).

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Key Concepts

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

parabola
A parabola is a U-shaped curve that can open upwards or downwards. It is the graph of a quadratic function. The most basic form of a parabola opens upwards and is symmetrical around a vertical axis of symmetry. When graphed on a coordinate plane, it generally looks like a bowl or a hill, depending on its concavity.

Parabolas have several key features:
  • **Vertex**: The highest or lowest point of the parabola, also the point where the axis of symmetry intersects the parabola.
  • **Axis of Symmetry**: The vertical line that splits the parabola into two mirror-image halves.
  • **Focus and Directrix**: Geometric traits used to define the parabola. The distance from any point on the parabola to the focus equals the distance from that point to the directrix.
A parabola can be represented by the equation: \(y = ax^2 + bx + c\) where 'a', 'b', and 'c' are coefficients. Key changes occur when manipulating these coefficients. For instance:
  • If 'a' is positive, the parabola opens upwards.
  • If 'a' is negative, it opens downwards.
  • The values of 'b' and 'c' affect the position of the vertex and the parabola's intersection with the y-axis.
quadratic functions
Quadratic functions are fundamental in algebra and feature prominently in many real-life scenarios. They can be used to model phenomena such as projectile motion, profit maximization in business, and many more. The standard form of a quadratic function is: \(y = ax^2 + bx + c\)

Let's delve into the details:
  • **Coefficient 'a'**: Determines the
systems of linear equations
To find the equation of a parabola passing through three points, we must use systems of linear equations. This might sound complicated, but it's manageable with a few steps.

Any quadratic function can be expressed as: \(y = ax^2 + bx + c\).
Suppose you have three points given, say \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\). To find the quadratic equation, substitute these points into the quadratic function:
  • For \((x_1, y_1)\): \(y_1 = a x_1^2 + b x_1 + c\).
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    Most popular questions from this chapter

    Solve each system. $$\begin{aligned} x-y+z &=7 \\ 2 y-3 z &=-13 \\ 3 x-2 z &=-3 \end{aligned}$$.

    Solve each problem by using a system of three linear equations in three variables. Cooperative Learning Write a word problem for which a system of three linear equations in three unknowns can be used to find the solution. Ask a classmate to solve the problem.

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    Solve each problem using two variables and a system of two equations. Solve the system by the method of your choice. Note that some of these problems lead to dependent or inconsistent systems. Mr. Thomas and his three children paid a total of \(\$ 65.75\) for admission to Water World. Mr. and Mrs. Li and their six children paid a total of \(\$ 131.50 .\) What is the price of an adult's ticket and what is the price of a child's ticket?

    Write a dependent system of two linear equations for which \(\\{(t, t+5) | t \text { is any real number }\\}\) is the solution set. Ask a classmate to solve your system.
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