Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Match each equation with the appropriate description . Do not use a calculator. $$x=2 y^{2}$$ A. Circle; center \((3,-4) ;\) radius 5 B. Parabola; opens left C. Parabola; opens upward D. Circle; center \((-3,4)\); radius 5 E. Parabola; opens right F. Circle; center \((0,0) ;\) radius \(\sqrt{5}\) G. No points on its graph H. Parabola; opens downward

Step-by-Step Solution

Answer

E. Parabola; opens right

Step 1: Identify the Type of Equation

The given equation is \(x = 2y^2\). This equation is in the form of \(x = ay^2\), which is characteristic of a parabola that opens horizontally.

Step 2: Determine the Direction

For the equation \(x = ay^2\), if \(a > 0\) the parabola opens towards the positive x-direction (right), and if \(a < 0\) it opens towards the negative x-direction (left). Here, \(a = 2\) is positive, so the parabola opens to the right.

Step 3: Match with Description

Based on the analysis, the equation \(x = 2y^2\) describes a parabola that opens to the right. Therefore, it matches with Description E: 'Parabola; opens right.'

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that can be represented graphically by a quadratic equation. It is a special curve, primarily recognized in the study of conics in algebra. Parabolas are prevalent in both mathematics and physics as they depict a wide range of natural phenomena, including the trajectory of projectiles. The standard equation of a parabola typically takes one of the forms:

• Vertically oriented: \( y = ax^2 + bx + c \)
• Horizontally oriented: \( x = ay^2 + by + c \)

The orientation depends on which variable is squared. The essential characteristic of a parabola is that all points on it are equidistant from a point called the "focus" and a line called the "directrix." When graphed, parabolas can open either upward, downward, left, or right, which depends on the given form of the equation and the constants involved.

Graphing Parabolas

Graphing a parabola involves plotting points that satisfy its quadratic equation and observing the shape that emerges. The vertex is the most critical point on a parabola; it represents the minimum or maximum point depending on the parabola's orientation. To graph a parabola:

• Identify the vertex from the equation, which serves as the starting point for plotting.
• Determine the axis of symmetry, which is a line running through the vertex. Vertically oriented parabolas have a vertical axis, while horizontally oriented parabolas have a horizontal axis.
• Select a few values for one variable and calculate the corresponding values for the other, creating coordinate points.
• Plot the calculated points on a graph and draw a smooth curve through these points, making sure it reflects the symmetrical nature around the axis of symmetry.

Graphing helps visualize the parabola's path and is a critical step in understanding the behavior of these curves.

Opening Direction of Parabolas

The opening direction of a parabola indicates whether the curve is facing up, down, left, or right. Understanding this is crucial for correctly interpreting quadratic relations and their graph representations.

• If the parabola takes the form \( y = ax^2 + bx + c \):
  • If \( a > 0 \), it opens upward.
  • If \( a < 0 \), it opens downward.
• In the form \( x = ay^2 + by + c \):
  • When \( a > 0 \), the parabola opens to the right.
  • When \( a < 0 \), it opens to the left.

These directions are dictated by the sign of the leading coefficient (\(a\)) in the equation. By analyzing the value of \(a\), you can predict the direction of the parabola, assisting in graphing and matching these shapes to their real-world applications or textbook descriptions.