Problem 168 - Elementary Algebra

Question

In the following exercises, determine if the parabola opens up or down. $$ y=-9 x^{2}-24 x-16 $$

Step-by-Step Solution

Answer

The parabola opens downwards.

Step 1: Identify the quadratic equation's general form

The general form of a quadratic equation is \( y = ax^2 + bx + c \). In this specific equation, identify the coefficients corresponding to each term.

Step 2: Extract the coefficients

From the given equation \( y = -9x^2 - 24x - 16 \), identify the coefficient \( a = -9 \), \( b = -24 \), and \( c = -16 \).

Step 3: Determine the direction of the parabola

The direction in which the parabola opens is determined by the sign of the coefficient \( a \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), the parabola opens downwards. Since \( a = -9 \), which is less than 0, the parabola opens downwards.

Key Concepts

Parabola Direction

Understanding whether a parabola opens upwards or downwards is crucial in graphing quadratic equations. A quadratic equation is given in the form of \( y = ax^2 + bx + c \). The coefficient \( a \) dictates the parabola's direction.

• If \( a > 0 \), the parabola opens upwards, resembling a 'U' shape.
• If \( a < 0 \), the parabola opens downwards, resembling an inverted 'U'.

For the given equation, \( y = -9x^2 - 24x - 16 \), the coefficient \( a \) is -9. Since -9 is less than 0, the parabola opens downwards. This rule helps in understanding the symmetry and visual orientation of the graph.

General Form of Quadratic Equation

The general form of a quadratic equation is essential for identifying the coefficients and understanding the behavior of the graph. This form is represented as:

\ ( y = ax^2 + bx + c \ )

Here, \( a, b, \) and \( c \) are constants where:

• \( a \) is the quadratic coefficient (associated with \( x^2 \)).
• \( b \) is the linear coefficient (associated with \( x \)).
• \( c \) is the constant term.

This form provides a standardized way to describe any quadratic equation, making it easier to analyze graph properties like the vertex, axis of symmetry, and direction of opening. For example, in the equation \( y = -9x^2 - 24x - 16 \), you can easily pick out that \( a = -9 \), \( b = -24 \), and \( c = -16 \).

Coefficient Identification

Identifying the coefficients in a quadratic equation is the first step toward graphing and solving it. Given the general form \( y = ax^2 + bx + c \), let's extract the coefficients:

• \( a \): This is the coefficient of \( x^2 \). It defines the width and direction of the parabola. In the equation \( y = -9x^2 - 24x - 16 \), \( a = -9 \).
• \( b \): This coefficient is paired with the \( x-t \)ern, impacting the slope of the parabola's arms. Here, \( b = -24 \).
• \( c \): This is the constant term, which vertically shifts the parabola on the graph. For this equation, \( c = -16 \).

By carefully extracting and understanding these coefficients, you are better equipped to determine the shape and position of the parabola.