Question

Find a quadratic \(a+b r+c x^{2}\) such that the graph of \(y=a+b x+c x^{2}\) contains each of the points \((-1,6),(2,0),\) and (3,2)

Short answer

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

Step-by-step solution

Write the General Form of the Quadratic Equation

Consider the standard form of the quadratic equation: \(y = ax^2 + bx + c\).

Set Up Equations Using Given Points

Substitute each given point into the quadratic equation to set up a system of equations based on the conditions.1. For the point \((-1, 6)\): \[ 6 = a(-1)^2 + b(-1) + c \] which simplifies to \( a - b + c = 6 \).2. For the point \((2, 0)\): \[ 0 = a(2)^2 + b(2) + c \] which simplifies to \( 4a + 2b + c = 0 \).3. For the point \((3, 2)\): \[ 2 = a(3)^2 + b(3) + c \] which simplifies to \( 9a + 3b + c = 2 \).

Solve the System of Equations

You now have a system of three linear equations:1. \( a - b + c = 6 \)2. \( 4a + 2b + c = 0 \)3. \( 9a + 3b + c = 2 \)Solve this system using substitution or elimination methods.

Use the Elimination Method to Reduce the System

Subtract equation 1 from equation 2 to eliminate \(c\):\( 4a + 2b + c - (a - b + c) = 0 - 6 \)which simplifies to \( 3a + 3b = -6 \)or \( a + b = -2 \).Subtract equation 2 from equation 3:\( 9a + 3b + c - (4a + 2b + c) = 2 - 0 \)which simplifies to \( 5a + b = 2 \).

Solve for One Variable

Use the equations \( a + b = -2 \) and \( 5a + b = 2 \). Subtract the first equation from the second:\( (5a + b) - (a + b) = 2 - (-2) \)which simplifies to \(4a = 4 \).Solve for \(a\):\( a = 1 \).

Substitute Back to Find Other Variables

Substitute \( a = 1 \) into the equation \( a + b = -2 \):\( 1 + b = -2 \)which gives \( b = -3 \).Now substitute \( a = 1 \) and \( b = -3 \) into the equation \( a - b + c = 6 \):\( 1 - (-3) + c = 6 \)which simplifies to \( 4 + c = 6 \),giving \( c = 2 \).

State the Final Quadratic Equation

The quadratic equation that passes through the points \((-1, 6)\), \((2, 0)\), and \((3, 2)\) is:\[ y = x^2 - 3x + 2 \]

Key concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics where algebra is used to study geometric properties. In this area, we use a system of coordinates to represent and analyze geometric figures.
This is often referred to as the Cartesian coordinate system, developed by René Descartes.
In the system, points are indicated by pairs of numbers (x, y). Each point denotes a specific location on a plane.

• The x-coordinate shows the horizontal position.
• The y-coordinate shows the vertical position.

A point like (-1, 6) indicates a position where the x-value is -1 and the y-value is 6.
When working with equations, you can plot points to form curves or shapes. For quadratic equations like the one in the exercise, such plots form a parabola. Parabolas have specific properties that you can study, such as their vertex, axis of symmetry, and direction of opening.

Systems of Equations

Systems of equations are sets of equations with multiple variables that you solve simultaneously. In the exercise, we encounter three equations that represent conditions offered by each given point on a quadratic curve.
This involves substituting the x and y values from each point into the general form of the quadratic equation. Each substitution yields an equation:

• Equation 1: \( a - b + c = 6 \)
• Equation 2: \( 4a + 2b + c = 0 \)
• Equation 3: \( 9a + 3b + c = 2 \)

The task is to find values of a, b, and c that satisfy all equations, which geometrically means finding a parabola passing through all given points.
Solving these involves methods such as substitution (where you solve one equation for one variable and replace it in others) or elimination (where you add or subtract equations to eliminate a variable). This effectively reduces the number of variables and simplifies the system.

Graphing Quadratics

Graphing quadratic equations reveals the shape, called a parabola. The general form of a quadratic equation is \( y = ax^2 + bx + c \) as seen in the step-by-step solution.
The coefficients a, b, and c affect the shape and position of the parabola:

• Coefficient \( a \) determines the direction of the parabola. If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
• Coefficient \( b \) affects the tilt or the symmetrical axis of the parabola, affecting its horizontal position.
• Coefficient \( c \) represents the y-intercept, showing where the parabola crosses the y-axis.

By substituting coordinates into the quadratic equation, we ensure our graph intersects specific points, as shown in the solution. The computed parabola, \( y = x^2 - 3x + 2 \), passes through points (-1, 6), (2, 0), and (3, 2), guaranteeing the accuracy of the solution.