Problem 85 - College Algebra

Question

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The parabola whose equation is \(x=2 y-y^{2}+5\) opens to the right.

Step-by-Step Solution

Answer

The statement is False. The correct statement is: The parabola whose equation is \(x = 2y - y^2 + 5\) opens to the left.

Step 1: Analyze the given equation

The given equation of the parabola is \(x = 2y - y^2 + 5\). Here, by rearranging, the equation can be written in standard form: \(x = -(y^2) + 2y + 5\). Now, the coefficient of \(y^2\) is -1.

Step 2: Determine opening direction

As discussed, if the squares term is on 'y' in the equation, parabola will open rightwards for a > 0, and leftwards for a < 0. Here, since the coefficient of \(y^2\) is -1 (a < 0), the parabola opens to the left.

Step 3: Correct the statement

The statement in the exercise is false. We're asked to correct the statement when it is false. The correct statement should be: 'The parabola whose equation is \(x = 2y - y^2 + 5\) opens to the left.'

Key Concepts

Equation of a Parabola

A parabola is a U-shaped curve on a plane, defined by a quadratic equation. Understanding its equation is key to predicting its shape and orientation. The equation of a parabola can have different forms, but the standard form for those opening sideways is

• For parabolas that open horizontally: \[ x = ay^2 + by + c \]Here, \(a\), \(b\), and \(c\) are constants.
• For parabolas that open vertically: \[ y = ax^2 + bx + c \]

The term with the square indicates whether it opens up/down or left/right. In our case, the given equation, \( x = 2y - y^2 + 5 \), can be rewritten in standard form as \( x = -(y^2) + 2y + 5 \). This setup points to a parabola that opens horizontally, since the squared term is \(y^2\). This is crucial for understanding its orientation.

Parabola Orientation

The orientation of a parabola refers to the direction in which it opens, which depends on the coefficient of the squared term. Here's the key to figure it out:

• If the squared term involves \(y\) and the coefficient is positive, the parabola opens to the right.
• If the squared term involves \(y\) and the coefficient is negative, it opens to the left.
• For equations involving \(x^2\):
  • Positive coefficients mean it opens upwards.
  • Negative coefficients mean it opens downwards.

In our exercise, the key term is \(-y^2\). Since the coefficient is \(-1\), the parabola opens to the left. Recognizing this helps correct any false statements about its direction.

Coordinate Geometry

Coordinate geometry, or analytic geometry, links geometry and algebra through graphs and equations. This field is essential for understanding parabolas and their equations.

• By using coordinates, the position and orientation of the parabola can be visualized easily.
• It uses Cartesian coordinates (x, y) to express geometric shapes and is pivotal in solving real-world problems.
• A parabola is characterized by its vertex, axis of symmetry, and direction it opens.

In our scenario, coordinate geometry helps determine the equation form, the direction of opening, and how to graphically represent the parabola. By working with equations like \(x = 2y - y^2 + 5\), you begin to see the strong interplay between algebraic manipulation and geometric visualization.