Problem 2 - Intermediate Algebra

Question

The graph of each equation is a parabola. Determine whether the parabola opens upward, downward, to the left, or to the right. Do not graph. $$y=-x^{2}+16$$

Step-by-Step Solution

Answer

The parabola opens downward.

Step 1: Identify the standard form of the parabola equation

The given equation is \( y = -x^2 + 16 \). This is the standard form of a parabola equation with respect to the variable \( y \), which is typically written as \( y = ax^2 + bx + c \). In this form, \( a = -1 \), \( b = 0 \), and \( c = 16 \).

Step 2: Determine the direction of opening based on coefficient 'a'

The sign of the coefficient \( a \) in the equation \( y = ax^2 + bx + c \) determines the direction the parabola opens. If \( a > 0 \), the parabola opens upward. If \( a < 0 \), the parabola opens downward. Here, \( a = -1 \), which is less than zero.

Step 3: Conclude the direction of the parabola's opening

Since \( a = -1 \) is negative, the parabola described by the equation \( y = -x^2 + 16 \) opens downward.

Key Concepts

Standard Form

When working with parabolas, the standard form of their equation is crucial. This standard form usually follows the equation \( y = ax^2 + bx + c \), where:

• \( a \), \( b \), and \( c \) are coefficients.
• \( y \) is typically dependent on the variable \( x \).

For example, the equation \( y = -x^2 + 16 \) is a standard form. Here, note that \( a = -1 \), \( b = 0 \), and \( c = 16 \). These values help us understand how the parabola behaves on a graph. Having it in this standard form simplifies identifying key features such as direction and vertex without having to graph the equation.
Although different formulations exist, such as the vertex form \( y = a(x-h)^2 + k \), starting with the standard form gives a solid grounding in the basic properties of the parabola.

Direction of Opening

Understanding the direction in which a parabola opens is fundamental to learning about its graphical representation. This direction is primarily dictated by the coefficient \( a \) in the equation's standard form. Here's what you should remember:

• If \( a > 0 \), the parabola opens upwards. Think of it as a smile.
• If \( a < 0 \), the parabola opens downwards. It's this scenario that resembles a frown.

In the case of our example \( y = -x^2 + 16 \), with \( a = -1 \), which is less than zero, the parabola opens downward. This means that as you move along the x-axis, it arches from high to low, creating a downward curve, like an upside down "U" shape. Recognizing the sign and implication of \( a \) is how you can determine this without graphing.

Coefficient Analysis

The analysis of coefficients in the standard form of a parabola lets us uncover essential features of the graph. Let's break down these roles:

• Coefficient \( a \): Determines the direction of the parabola's opening and affects its width. A larger absolute value of \( a \) means a steeper graph, whereas a smaller absolute value indicates a broader parabola.
• Coefficient \( b \): Affects the symmetry and position of the parabola along the x-axis. While it is zero in our example, a nonzero \( b \) means the parabola doesn't necessarily peak or dip directly at the y-axis.
• Coefficient \( c \): Represents the y-intercept, the point where the graph crosses the y-axis. In \( y = -x^2 + 16 \), the parabola crosses the y-axis at \( y = 16 \).

By examining these coefficients, you can predict the overall shape and position of the parabola in a graphical context. Understanding the interplay and effects of these coefficients is crucial in mastering the graphing and analysis of parabolic equations.