Problem 3 - College Algebra with Corequisite Support

Question

If the equation of a parabola is written in standard form and \(p\) is negative and the directrix is a horizontal line, then what can we conclude about its graph?

Step-by-Step Solution

Answer

The parabola opens downwards.

Step 1: Understanding the Standard Form

The standard form of a parabola with a horizontal directrix is \(x^2 = 4py\). This means the parabola opens either upwards or downwards.

Step 2: Analyze the Sign of p

In the equation \(x^2 = 4py\), the parameter \(p\) influences the direction in which the parabola opens. If \(p > 0\), the parabola opens upwards, while if \(p < 0\), it opens downwards.

Step 3: Conclusion Based on Negative p

Given \(p < 0\), we conclude that the graph of the parabola opens downwards because a negative \(p\) makes \(4py\) direct the parabola's branches towards negative infinity (the downward direction on the y-axis).

Key Concepts

Standard Form

The standard form of a parabola is a specific way to write the equation that helps us understand important features about the parabola’s graph. For a parabola that has a horizontal directrix, the equation typically looks like this: \[ x^2 = 4py \] In this form, the focus and directrix are directly related to the constant \( p \). The variable \( x \) is squared, and \( y \) appears linearly, which means the parabola will open in a vertical direction, either upwards or downwards, along the y-axis. Identifying the correct standard form is crucial for determining the behavior of the parabola, such as its orientation on the coordinate plane.

Directrix

The directrix of a parabola is a straight line that, along with the focus, helps define the parabola’s shape and position. When dealing with the standard form of a parabola with a horizontal directrix \( x^2 = 4py \), the directrix is a horizontal line. Understanding the directrix is important because as a property of the parabola, any point on the parabola is equidistant from the directrix and a fixed point called the focus. This property maintains the geometric identity of the parabola, influencing both its shape and its alignment in the coordinate plane. So, for our equation, knowing that the directrix is horizontal helps us determine that the parabola's axis of symmetry is vertical, affecting the orientation decision.

Parabola Orientation

The orientation of a parabola is essentially how it opens, or which direction the parabola's limbs extend. In the standard form equation \( x^2 = 4py \), determining the parabola's orientation depends heavily on the sign of \( p \). Because the directrix is horizontal, this type of parabola is focused on the vertical axis:

• If \( p > 0 \), the parabola opens upwards.
• If \( p < 0 \), the parabola opens downwards.

Understanding the orientation is crucial since it reveals the parabola's direction of growth and its symmetry axis. This orientation detail helps in predicting how the graph will behave as it extends out from its vertex, encompassing both its rise towards positive infinity or its descent towards negative infinity based on the value of \( p \).

Negative p Value

In the context of the equation \( x^2 = 4py \), the value of \( p \) plays a pivotal role in deciding the orientation of the parabola. When \( p \) is negative, it informs us that the parabola opens downward. This is because the term \( 4py \) points the branches of the parabola towards the negative y-direction, resulting in a downward opening:

• When \( p < 0 \), it’s like having a weight pulling the vertex downwards.
• Graphically, this means the parabola's arms stretch towards negative infinity along the y-axis.

The negative \( p \) essentially dictates the concavity of the parabola, helping in visualizing and sketching the graph correctly, as the downward orientation aligns the parabola’s geometry with real-world problems or contexts obeying such conditions.