Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 3
For the following exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve. $$ x=2 t+4, y=t-1 $$
Short Answer
Expert verified
The curve is a line: \( y = \frac{x - 6}{2} \), oriented left to right.
Step by step solution
01
Express t in terms of x
The given parametric equations are \( x = 2t + 4 \) and \( y = t - 1 \). Begin by solving the equation for \( x \) to express \( t \) in terms of \( x \). Rearrange the equation: \[ x - 4 = 2t \]Divide by 2:\[ t = \frac{x - 4}{2} \]
02
Substitute t in the equation for y
Now that we have \( t \) expressed as \( \frac{x - 4}{2} \), substitute this expression into the equation for \( y \):\[ y = t - 1 \]Substitute for \( t \):\[ y = \frac{x - 4}{2} - 1 \]
03
Simplify the equation for y
Now, simplify the equation \( y = \frac{x - 4}{2} - 1 \):First, simplify the expression:\[ y = \frac{x - 4}{2} - \frac{2}{2} \]Combine the fractions:\[ y = \frac{x - 4 - 2}{2} \]Simplify further:\[ y = \frac{x - 6}{2} \]
04
Identify the equation type and curve orientation
The equation \( y = \frac{x - 6}{2} \) represents a linear equation of the form \( y = mx + c \), which describes a straight line.To determine the orientation of the curve, observe how \( x \) and \( y \) change as \( t \) increases:- As \( t \) increases, \( x = 2t + 4 \) increases.- Likewise, \( y = t - 1 \) also increases.Thus, the line is oriented from left to right as \( t \) increases.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Curve Sketching
Curve sketching with parametric equations involves representing a curve on a graph by defining both x and y in terms of a third variable, often denoted as \( t \), called the parameter. This method allows for the depiction of complex curves through simpler equations.
In our given example, \( x = 2t + 4 \) and \( y = t - 1 \) describe a curve parametrically. To sketch this curve:
In our given example, \( x = 2t + 4 \) and \( y = t - 1 \) describe a curve parametrically. To sketch this curve:
- First, eliminate the parameter to obtain a single equation in terms of \( x \) and \( y \).
- Determine the type of curve by simplifying the derived equation.
- Understand the direction and flow of the curve. This is the curve's orientation, often determined by analyzing the change in x and y as t varies.
Linear Equations
Linear equations are algebraic expressions that represent straight lines on a graph. They have the general form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
In parametric terms, by eliminating the parameter \( t \) in \( x = 2t + 4 \) and \( y = t - 1 \), we derive the linear equation \( y = \frac{x-6}{2} \).
This:
In parametric terms, by eliminating the parameter \( t \) in \( x = 2t + 4 \) and \( y = t - 1 \), we derive the linear equation \( y = \frac{x-6}{2} \).
This:
- Reveals the relationship between \( x \) and \( y \) as a straight line.
- Indicates a slope of \( \frac{1}{2} \), which determines the angle of the line's inclination.
- Shows that the line passes through a shifted set of axes, as inferred from the intercept derived during transformation.
Eliminating the Parameter
Eliminating the parameter is a crucial step in translating parametric equations into a single Cartesian equation involving only \( x \) and \( y \). This process simplifies the characterization of curves by removing \( t \).
In our example, we performed the following:
In our example, we performed the following:
- Solve for \( t \) in terms of \( x \) from the first equation: \( t = \frac{x - 4}{2} \).
- Substitute this expression into the second equation to eliminate \( t \): \( y = \frac{x - 4}{2} - 1 \).
- Simplify the resulting equation to \( y = \frac{x - 6}{2} \), yielding a relation solely between \( x \) and \( y \).
Curve Orientation
Curve orientation refers to the direction in which a curve is traced as the parameter \( t \) increases. It's essential for understanding how the point on the curve moves, providing insights into the natural motion or flow of the curve.
For \( x = 2t + 4 \) and \( y = t - 1 \), as \( t \) increases:
For \( x = 2t + 4 \) and \( y = t - 1 \), as \( t \) increases:
- \( x \) increases linearly, since each increment in \( t \) results in a proportional increase in \( x \).
- Similarly, \( y \) also increases linearly because it is directly related to \( t \).
- The consistent increase in both parameters signifies that the curve's orientation is from left to right.
