Problem 19 - Algebra 1: Concepts and Skills

Question

Decide whether the parabola opens up or down. $$ y=-8 x^{2}-9 $$

Step-by-Step Solution

Answer

The parabola opens downwards.

Step 1: Identify the Equation Type

Recognizing that the given equation is of the form \(y=ax^2+bx+c\). The coefficient of \(x^2\) is the value of \(a\).

Step 2: Review the Coefficient of the Parabola

The coefficient of the \(x^2\) term in the equation is -8, which is less than zero.

Step 3: Determine the Direction of the Parabola

Since the coefficient of the \(x^2\) term is less than 0, the parabola will open downward.

Key Concepts

Parabolas

Parabolas are curves on a graph described by quadratic equations. In simpler terms, these are the U or inverted U shaped graphs that we frequently see in algebra. The general form of a quadratic equation is given by:\[ y = ax^2 + bx + c \]This equation tells us how the graph will look on paper. The term "parabola" refers to this U-shaped graph created by the quadratic function. When you plot a quadratic equation on a Cartesian plane, you'll notice its distinctive shape. Great examples can be found in satellite dishes or even the paths of thrown balls.

• Vertices: The peak or the lowest point of a parabola.
• Axis of symmetry: A line that vertically cuts the parabola into two mirror-image halves.
• Roots: Points where the parabola intersects the x-axis, helping determine the solutions to the quadratic equation.

Parabolas are essential for understanding topics like projectile motion, and even in lenses and optics.

Direction of Opening

The direction in which a parabola opens—upward or downward—is determined by the sign of the coefficient \(a\) in its equation. If you look at the quadratic equation in the form \(y = ax^2 + bx + c\), the coefficient \(a\) dictates this behavior.

• a > 0: The parabola opens upward, resembling a smile.
• a < 0: The parabola opens downward, forming a frown.

For the given equation \(y = -8x^2 - 9\), the coefficient \(-8\) is less than zero. Consequently, the parabola will open downward. This is a useful property because it directly impacts how we predict the shape and graph of the parabola without needing to plot points manually.

Coefficient Analysis

Analyzing the coefficients in a quadratic equation gives us meaningful insights into the graph's shape and position. The term \(a\) from the quadratic equation \(y = ax^2 + bx + c\) is the coefficient that influences the parabola's opening direction and width.

• Width: A larger absolute value of \(a\) makes the parabola more narrow, whereas a smaller absolute value makes it wider.
• Direction: As discussed earlier, if \(a > 0\), it opens up, whereas \(a < 0\) means it opens down.

In cases with only the \(x^2\) and constant term, like \(y = -8x^2 - 9\), we focus primarily on \(a\). Here, with \(a = -8\), the parabola opens downwards, and the shape will be relatively narrow compared to smaller values of \(a\). Understanding these coefficients allows one to draw quick generalizations about the parabola's overall form before seeing the actual graph.