Problem 16 - Algebra 1: Concepts and Skills

Question

Decide whether the parabola opens up or down. $$ y=-5 x^{2} $$

Step-by-Step Solution

Answer

The parabola opens down.

Step 1: Identify the 'a' Coefficient

The 'a' coefficient in our given equation \(y=-5 x^{2}\) is -5.

Step 2: Determine the opening of Parabola

Since the coefficient 'a' is less than zero, the parabola opens downwards.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental component of algebra centered around an equation of the second degree, typically in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) must not be zero. A unique characteristic of these equations is that their graph on the Cartesian plane is a parabola, and they can have zero, one, or two real solutions. These solutions are the x-intercepts or roots of the parabola, and they can be found using various methods, such as factoring, completing the square, or applying the quadratic formula.

Quadratic equations not only appear in mathematics but also in various real-world scenarios like motion under gravity and economics. Understanding their structure and how to solve them is crucial for students to analyze and interpret diverse problems.

Parabola Characteristics

A parabola is a symmetric, U-shaped curve that can open either upwards or downwards and is represented by the graph of a quadratic equation. The most notable features of a parabola include its vertex, which is the highest or lowest point on the graph; the axis of symmetry, a vertical line that passes through the vertex dividing the parabola into two mirror images; and the focus and directrix, which are geometric tools used to define the parabola.

The direction in which a parabola opens is determined by the sign of the coefficient \( a \) in the quadratic equation \( y = ax^2 + bx + c \). An upward-opening parabola occurs when \( a > 0 \), and a downward-opening parabola occurs when \( a < 0 \). The width of the parabola is also affected by the value of \( a \)—the larger the absolute value of \( a \) is, the narrower the parabola.

Coefficient Analysis

Coefficient analysis in the context of quadratic equations involves examining the coefficients \( a \), \( b \) and \( c \) to determine specific characteristics of the parabola. The coefficient \( a \) influences the opening direction and the width of the parabola, as previously discussed. A positive \( a \) leads to an upward opening, while a negative \( a \) corresponds to a downward opening. Additionally, if \( a \) is large in magnitude, the parabola is steeper, and if it's closer to zero, the parabola is wider.

The coefficient \( b \) affects the location of the axis of symmetry of the parabola. The x-coordinate of the vertex can be found by the formula \( -b/(2a) \). Lastly, the coefficient \( c \) represents the y-intercept of the parabola, which is the point where the parabola crosses the y-axis. In practical terms, understanding how these coefficients affect the graph of a quadratic equation allows for a quicker and more accurate sketching of parabolas and aids in the prediction of their behavior in different scenarios.