Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Focus is \(\left(0,-\frac{1}{9}\right)\)

Step-by-Step Solution

Answer

The standard equation of the parabola is \(x^2 = -\frac{4}{9}y\).

Step 1: Understand the Parabola's Orientation

The focus of the parabola is located at \(\left(0,-\frac{1}{9}\right)\). Since the focus is at a negative \(y\)-coordinate and the \(x\)-coordinate is 0, the parabola opens downwards with a vertical axis of symmetry.

Step 2: Standard Equation for Downward Opening Parabola

For a parabola opening downwards with vertex at the origin \((0,0)\), the standard equation is \(x^2 = -4py\), where \(p\) is the distance from the vertex to the focus.

Step 3: Find the Value of \(p\)

The distance \(p\) from the vertex to the focus is given by the \(y\)-coordinate of the focus, which is \(\frac{1}{9}\). This makes \(p = \frac{1}{9}\).

Step 4: Substitute \(p\) in the Equation

Substitute \(p = \frac{1}{9}\) into the equation \(x^2 = -4py\). This results in \(x^2 = -4\left(\frac{1}{9}\right)y\).

Step 5: Simplify the Equation

Simplify the equation: \(x^2 = -\frac{4}{9}y\). This is the standard equation of the parabola with the given focus.

Key Concepts

Vertex

The vertex of a parabola is a crucial point on its curve. It represents the highest or lowest point depending on how the parabola opens. In this exercise, the vertex is conveniently located at the origin - (0,0). For parabolas that open upwards or downwards, the vertex represents a maximum or minimum. Since the vertex is at the origin, it simplifies finding the equation because we do not need to translate the parabola horizontally or vertically. By knowing the vertex is at the origin, we can focus on the standard form equation based solely on the parabola's orientation and the position of the focus.

Focus

The focus of a parabola is another critical component that determines its shape. It is a point through which all points on a parabola are equidistant to the corresponding directrix. In this exercise, the focus is at the point - (0,-\frac{1}{9}). This position signifies that the parabola opens downwards because the focus is located below the vertex along the y-axis. The distance from the vertex to the focus, denoted as \(p\), is crucial for constructing the standard equation of the parabola. For our example, \(p\) is \(\frac{1}{9}\). This indicates how deep or wide the parabola opens based on the value of \(p\). The smaller the \(p\), the closer the focus is to the vertex, influencing the narrowness of the parabola's opening.

Axis of Symmetry

The axis of symmetry of a parabola is an imaginary line that runs through the vertex and the focus, dividing the parabola into two mirror-image halves. This line helps us visualize and understand how the parabola is oriented. In our exercise, because the vertex and focus have the same x-coordinate (zero), the axis of symmetry is the y-axis or x = 0. - Knowing the axis of symmetry makes it easier to graph the parabola and understand its structure. Since the parabola opens downwards, the axis of symmetry guides us in confirming that the parabola is perfectly balanced on either side of this vertical line.

Standard Equation

Understanding the standard equation of a parabola is essential for analyzing its characteristics. When a parabola opens downwards and the vertex is at the origin \((0,0)\), its standard equation is given by:- \(x^2 = -4py\). Here, \(-4p\) decides how "stretched" or "compressed" the parabola is vertically. From the exercise, we determined \(p = \frac{1}{9}\), which means the equation becomes:- \(x^2 = -\frac{4}{9}y\). This equation provides all the information needed to sketch and analyze the parabola's shape and orientation. It emphasizes that for every \(x^2\), \(y\) moves "-\frac{4}{9}\" units. Using this equation, you can predict how the parabola will look and behaves when graphed.