Problem 4 - Introductory Algebra for College Students

Question

Determine if the parabola whose equation is given opens upward or downward. $$y=-2 x^{2}-4 x+6$$

Step-by-Step Solution

Answer

The given parabola opens downward.

Step 1: Identify the quadratic form

The given equation \(y=-2 x^{2}-4 x+6\) is in standard quadratic form, which is \(ax^{2} + bx + c\).

Step 2: Identify the coefficient of \(x^{2}\)

In the equation, we see that the coefficient of \(x^{2}\), denoted as 'a', is -2.

Step 3: Determine the direction of the parabola

The coefficient of \(x^{2}\) in this case is negative (-2). When 'a' is negative in the standard quadratic form, it represents that the parabola opens downward.

Key Concepts

Quadratic Equations

A quadratic equation is an algebraic expression of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The highest power of the variable \(x\) in a quadratic equation is 2, which creates a characteristic curve called a parabola. This type of equation frequently shows up in various mathematical problems and has numerous practical applications, such as calculating projectile motion or finding the optimal area or cost in business problems.

• Coefficient \(a\) determines the parabola's width and direction.
• Coefficient \(b\) affects the parabola's axis of symmetry.
• Constant \(c\) represents the y-intercept, where the parabola crosses the y-axis.

Quadratic equations can be solved using methods such as factoring, completing the square, or applying the quadratic formula. Each approach provides ways to identify the roots or solutions of the equation, which are the x-values where the parabola crosses the x-axis.

Standard Form

The standard form of a quadratic equation is the expression \(ax^2 + bx + c = 0\). This standardization is essential for easily analyzing and solving quadratic equations by revealing important characteristics.
- **'a' value**: This coefficient affects the shape and direction of the parabola.- **'b' value**: By playing a role in the equation of the axis of symmetry, it influences where the parabola is vertically centered.- **'c' value**: As the constant term, it indicates the point at which the parabola intersects the y-axis.
Converting a quadratic equation into its standard form could involve rearranging terms or performing simple algebraic manipulations. Recognizing this form allows for straightforward application of solving techniques and better understanding of the parabola's properties.

Parabola Direction

The direction in which a parabola opens is determined by the sign of the leading coefficient, \(a\), in the standard quadratic form \(ax^2 + bx + c\). Simply put:

• If \(a > 0\), the parabola opens upward, forming a U-shape.
• If \(a < 0\), the parabola opens downward, forming an upside-down U-shape.

This information is critical, as it influences how we interpret the maximum or minimum points of the parabola. For example, when \(a\) is negative, the maximum point is located at the vertex because the parabola curves downwards. Conversely, when \(a\) is positive, the minimum point is the vertex since the parabola curves upwards.
In practical applications, understanding the direction of a parabola can help predict behavior, like determining the highest point a thrown ball reaches or the optimal price point for maximizing profits.