Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

10. Parametric Equations

Graphing Parametric Equations

Trigonometry

10. Parametric Equations

Graphing Parametric Equations

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1:27 minutes

Problem 55

Textbook Question

In Exercises 53–56, find two different sets of parametric equations for each rectangular equation.y = x² + 4

Verified step by step guidance

Step 1: Recognize that the given equation is a parabola in the form \( y = x^2 + 4 \). We need to express both \( x \) and \( y \) in terms of a parameter, typically \( t \).

Step 2: For the first set of parametric equations, choose \( x = t \). Then substitute \( x = t \) into the rectangular equation to find \( y \): \( y = t^2 + 4 \). Thus, the first set of parametric equations is \( x = t \) and \( y = t^2 + 4 \).

Step 3: For the second set of parametric equations, choose a different expression for \( x \). For example, let \( x = t + 1 \). Substitute \( x = t + 1 \) into the rectangular equation to find \( y \): \( y = (t + 1)^2 + 4 \).

Step 4: Expand \( (t + 1)^2 \) to get \( t^2 + 2t + 1 \). Substitute this into the equation for \( y \) to get \( y = t^2 + 2t + 1 + 4 \), which simplifies to \( y = t^2 + 2t + 5 \).

Step 5: Thus, the second set of parametric equations is \( x = t + 1 \) and \( y = t^2 + 2t + 5 \).

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Key Concepts

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, typically denoted as 't'. Instead of defining y directly in terms of x, both x and y are defined in terms of 't', allowing for a more flexible representation of curves. For example, for the equation y = x² + 4, one could set x = t and y = t² + 4, or use a different parameterization such as x = 2t and y = (2t)² + 4.

Rectangular Equations

A rectangular equation relates the x and y coordinates of points in a Cartesian plane without involving a third variable. In this case, the equation y = x² + 4 describes a parabola that opens upwards, with its vertex at the point (0, 4). Understanding how to convert this rectangular form into parametric equations is essential for exploring the curve's properties and behavior.

Graphing Parabolas

Graphing parabolas involves understanding their shape, direction, and key features such as the vertex and axis of symmetry. The equation y = x² + 4 indicates that the parabola opens upwards, with the vertex located at (0, 4). Recognizing these characteristics helps in creating accurate parametric equations that reflect the same geometric properties as the original rectangular equation.