Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic corresponding to the given equation. \(\frac{-x^{2}}{9}+\frac{y}{4}=0\)

Step-by-Step Solution

Answer

The conic is a parabola.

Step 1: Rearrange the Equation

Starting with the given equation \( \frac{-x^{2}}{9} + \frac{y}{4} = 0 \), move one of the terms to the other side. This gives: \( \frac{y}{4} = \frac{x^{2}}{9} \).

Step 2: Identify the Structure

Recognize that moving terms around results in \( y = \frac{4}{9}x^{2} \). This equation is in the standard form of a quadratic function, which is a specific case of a conic section.

Step 3: Classify the Conic

Since \( y = \frac{4}{9}x^{2} \) is a quadratic function, which can be written in the form \( y = ax^{2} \), it represents a parabola. Notice that it does not fit the forms of an ellipse or hyperbola.

Key Concepts

Parabola

A parabola is a distinctive curve formed by the set of all points that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. It is one of the four types of conic sections. Parabolas are ubiquitous in mathematics and appear naturally in various physical contexts, such as the path of projectiles under uniform gravity. In any parabola, there is a point at which the curve shifts direction, known as the vertex. For the equation of the parabola we have in step 3, which is \( y = \frac{4}{9}x^2 \), the focus and vertex both lie on the line of symmetry, which in this case is the y-axis.

• The point where the parabola intersects its axis of symmetry is the vertex.
• The vertical line that divides the parabola into mirror-image halves is its axis of symmetry.
• The minimum or maximum point on a parabola, depending on its orientation, is always the vertex.

Understanding how parabolas work is crucial because they model various real-world behaviors, from satellite dishes to headlamps. Recognizing the shape and being able to identify key properties, like the vertex and axis of symmetry, help to understand the nature of the function represented by the equation.

Quadratic Function

A quadratic function is any function that can be expressed in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. This form represents a parabola when graphed on a coordinate plane. Quadratic functions are fundamental in algebra and are used to describe a wide array of contexts, from economics to various natural processes.

• The graph of a quadratic function is always a parabola.
• If the coefficient \( a \) is positive, the parabola opens upwards. If it's negative, the parabola opens downwards.
• The value of \( c \) in the function \( y = ax^2 + bx + c \) determines the y-intercept of the graph.

In this exercise, our function is simplified to \( y = \frac{4}{9}x^2 \), meaning it has no linear term (like \( bx \)) or constant term (like \( c \)). Simplifying complex equations to this form helps easily identify important features of the parabola, such as the width, direction, and position of the vertex.

Standard Form

The standard form of a quadratic function is an essential concept because it provides a clear structure that represents the function's key features. In the context of the exercise, we addressed an equation and rearranged it into the standard form \( y = ax^2 \), which is a simpler case of the broader standard form \( y = ax^2 + bx + c \). Understanding how to manipulate and identify this form is crucial because it reveals:

• The "a" value tells us the direction of the parabola and how "stretched" or "compressed" it is.
• Lack of "b" and "c" in our case indicates the vertex is at the origin \((0, 0)\).
• This form provides a base for using formulas to find roots and help easily convert to vertex form, enhancing insights into transformations.

Knowing how to move terms around to reveal the standard form, as done in the step-by-step solution, helps not only in understanding parabola properties but also in applying quadratic functions to solve real-life problems effectively.