Problem 17 - Algebra 1: Concepts and Skills

Question

Decide whether the parabola opens up or down. $$ y=-7 x^{2}+5 $$

Step-by-Step Solution

Answer

The parabola opens downward.

Step 1: Identifying the Standard Form

The given equation \(y = -7x^2 + 5\) is already in the standard form of a parabola, \(y = ax^2 + b\), where \(a\) and \(b\) are constants.

Step 2: Identifying the Value of \(a\)

The coefficient of \(x^2\) in the given equation is \(a = -7\).

Step 3: Determining the Direction of Opening

Since the value of \(a\) is negative, \(a < 0\), it means the parabola opens downward.

Key Concepts

Standard Form of a Parabola

Understanding the standard form of a parabola is crucial for identifying its properties. The standard form of a parabola's equation is expressed as \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients and \( x \) and \( y \) are variables. In this standard equation, the coefficient \( a \) is responsible for the parabola's vertical stretch and direction of opening, while \( b \) affects the horizontal placement, and \( c \) determines the vertical position of the vertex.

When dealing with the simplified version without the \( bx \) term such as \( y = ax^2 + c \), as in the given exercise \( y = -7x^2 + 5 \), it's even more straightforward to analyze the parabola's behavior. This form clearly showcases the parabola's symmetry about the y-axis since the absence of the \( bx \) term indicates there is no horizontal shift away from the origin.

Coefficient of Quadratic Term

The coefficient of the quadratic term in a parabola's equation has a profound impact on the graph's shape and orientation. This coefficient is represented by the variable \( a \) in the standard form equation \( y = ax^2 + bx + c \). When \( a \) is positive, the parabola opens upward, forming a 'smile', and when \( a \) is negative, the parabola opens downward like a 'frown'. A greater absolute value of \( a \) indicates a narrower parabola, while a smaller absolute value suggests a wider opening.

In our exercise, the coefficient of the quadratic term is \( a = -7 \). This negative value immediately tells us that the parabola opens downward. Additionally, because the value of \( |a| \) is relatively large, we can infer that the parabola will be quite narrow, steeply dropping off from the vertex. This single coefficient shapes much of the parabola's overall appearance on the graph.

Identifying Parabola Orientation

Identifying the orientation of a parabola is a straightforward task once you know what to look for. As mentioned earlier, the key lies in the sign of the coefficient \( a \) in the parabola's standard form. If \( a \) is positive, expect the parabola to open upward, and if negative, anticipate a downward opening.

However, there's more to orientation than just up or down. The orientation also gives insight into a parabola's potential minimum or maximum value. For upward-opening parabolas (\( a > 0 \)), the vertex represents the minimum point, providing the smallest y-value. Conversely, for downward-opening parabolas (\( a < 0 \)), the vertex is the maximum point, offering the largest y-value on the graph.

By applying this knowledge to the exercise with the equation \( y = -7x^2 + 5 \), we can confidently conclude that the parabola will open downward, and the vertex will signify the highest point on the graph. This understanding is essential for graphing the equation and analyzing its characteristics in algebraic and geometric contexts.