91Ó°ÊÓ

A parabola is an important type of curve described by a quadratic equation in the form of \( y = ax^2 + bx + c \). This equation is central to many applications in mathematics and physics, giving us a way to describe and analyze parabolic shapes. The function depends on three coefficients: \( a \), \( b \), and \( c \). These coefficients determine the shape and position of the parabola:

• The coefficient \( a \) affects the direction and the "width" of the parabola. If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
• The coefficient \( b \) influences the symmetry axis of the parabola, shifting it horizontally.
• The coefficient \( c \) represents the y-intercept, where the parabola crosses the y-axis.

These coefficients are vital for mathematically constructing the parabola's path, especially when it needs to pass through certain points.

Determining the coefficients of a parabola requires using points that lie on the curve. Given several points of the parabola, you can substitute their coordinates into the quadratic equation \( y = ax^2 + bx + c \) to form a system of linear equations. For instance, substituting the point \((-2, 11)\) into the equation yields \( 11 = a(-2)^2 + b(-2) + c \), which simplifies to \( 4a - 2b + c = 11 \). Repeat this substitution process with other given points to form a set of equations:

• Each point provides a unique equation reflecting its position on the parabola.
• This process is repeated until a system of equations is created sufficient to solve for the unknown coefficients \(a\), \(b\), and \(c\).

Through these derived equations, you have the foundation needed to determine the coefficients, ultimately unveiling the specific parabolic equation.

Solving a system of linear equations is crucial in determining the coefficients of the parabola. Each equation corresponds to a point on the parabola, and the goal is to find \(a\), \(b\), and \(c\). The process begins by eliminating one variable at a time. For example, subtracting one equation from another can eliminate a variable, simplifying the equations to a form that is easier to solve. Let's look at the steps:

• Identify which variable can be eliminated easily and perform necessary operations between equations.
• After finding values for one or two variables, substitute back into other equations to determine the remaining values.

In our case, once \(b\) is found to be \(-1\), it simplifies solving for \(a\) and \(c\). By subsequently substituting \(b\) into remaining equations, you can solve step-by-step until all coefficients are identified. The result is the complete parabolic equation which matches all original conditions or points provided.