Problem 3 Does the parabola open upward or... [FREE SOLUTION]

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Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 9: Problem 3

Does the parabola open upward or downward? Explain. $$ y=-2 x_{2}+5 x+12 $$

Short Answer

Expert verified

The parabola opens downward because the coefficient of $ x^2 $ is negative.

Step by step solution

Identify the Quadratic Coefficient

The given quadratic equation is in the form $ y = ax^2 + bx + c $. Here, $ a = -2 $, $ b = 5 $, and $ c = 12 $. The coefficient $ a $ is the key in determining the direction in which the parabola opens.

Analyze the Sign of 'a'

For a parabola in the form $ y = ax^2 + bx + c $, the parabola opens:- Upward if $ a > 0 $- Downward if $ a < 0 $In this problem, $ a = -2 $, which is less than 0. Therefore, the parabola opens downward.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Parabola

A parabola is a symmetrical, curved shape often seen in graphs of quadratic equations. It represents the graph of an equation of the form $ y = ax^2 + bx + c $. This curve has an important property called the "vertex," which is its highest or lowest point. Parabolas are prevalent in mathematics and even in real life, appearing in archways, satellite dishes, and the paths of thrown objects.

Symmetrical about a vertical line called the axis of symmetry.
Contains a vertex, where it either turns upward or downward.
Roots (or zeros) where the curve crosses the x-axis.

Understanding how changes in the quadratic equation affect the parabola's shape and position can be incredibly valuable, especially when analyzing data or solving real-world problems.

The Role of Coefficient 'a'

The coefficient 'a' plays a crucial role in shaping a parabola in a quadratic equation $ y = ax^2 + bx + c $. Specifically, it determines the parabola's "width" and direction. When assessing a quadratic equation:

The size or magnitude of 'a' affects how "narrow" or "wide" the parabola appears. Smaller values of $ |a| $ make the parabola wider, while larger values result in a more narrow shape.
The sign of 'a' (positive or negative) is pivotal in determining whether the parabola opens upward or downward, which we will explore further in the next section.

Understanding $ a $ helps in predicting and sketching the graph accurately, as well as interpreting the behavior of the quadratic function at various input values.

Direction of Parabola Opening

To determine which direction the parabola opens, we look at the sign of coefficient 'a' in the quadratic equation $ y = ax^2 + bx + c $. The general rule is:

The parabola opens upward if $ a > 0 $. This results in a shape akin to a "smile."
If $ a < 0 $, the parabola opens downward, resembling a "frown."

In the given problem, the quadratic equation is $ y = -2x^2 + 5x + 12 $. Here, the coefficient $ a $ is $-2$, which is negative. As per the rule, a negative $ a $ means the parabola opens downward. This understanding can help predict how a function behaves, describe its graphical representation, and solve quadratic problems more effectively.

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91影视

Does the parabola open upward or downward? Explain. $$ y=-2 x_{2}+5 x+12 $$

Short Answer

Step by step solution

Identify the Quadratic Coefficient

Analyze the Sign of 'a'

Key Concepts

Understanding the Parabola

The Role of Coefficient 'a'

Direction of Parabola Opening

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