Ch. 11 - Parametric Equations and Polar Coordinates

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 11 - Parametric Equations and Polar Coordinates

Problem 11.6.2

Chapter 11, Problem 11.6.2

Identifying Graphs

Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x

Then find each parabola's focus and directrix.

Verified step by step guidance

Step 1: Identify the orientation of the parabola from the graph. This parabola opens to the right, which suggests it is of the form \(y^2 = 4px\) or \(y^2 = -4px\).

Step 2: Compare the given equations to the form \(y^2 = 4px\). The equations with \(y^2\) are \(y^2 = 8x\) and \(y^2 = -4x\). Since the parabola opens to the right, the coefficient of \(x\) should be positive, so the equation is \(y^2 = 8x\).

Step 3: From the equation \(y^2 = 8x\), identify \(4p = 8\), so \(p = 2\). This means the focus is at \((p, 0)\), or \((2, 0)\).

Step 4: The directrix is the vertical line \(x = -p\), so the directrix is \(x = -2\).

Step 5: Summarize: The parabola corresponds to the equation \(y^2 = 8x\), with focus at \((2, 0)\) and directrix \(x = -2\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of Parabolas

Parabolas can be expressed in standard forms such as x² = 4py or y² = 4px, where p represents the distance from the vertex to the focus. The sign and variable squared determine the parabola's orientation (up, down, left, or right). Recognizing these forms helps match equations to their graphs.

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola, and the directrix is a line outside it. Every point on the parabola is equidistant from the focus and the directrix. The value of p in the standard form determines the location of the focus and directrix relative to the vertex.

Graph Interpretation and Matching

Analyzing the graph's orientation and shape helps identify which equation corresponds to which parabola. For example, a parabola opening right or left corresponds to y² = 4px, while one opening up or down corresponds to x² = 4py. The sign of p indicates direction (positive for right/up, negative for left/down).

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