Use an algebraic method to eliminate the parameter and identify the graph of the parametric curve.
x=5-3t
y=2+t
-1<-x-<3
t = y-2
x = 5-3t = 5-3(y-2)
It is clearly a line.
To eliminate the parameter and identify the graph of the parametric curve, we can solve the first equation for t and substitute it into the second equation.
Given:
x = 5 - 3t
y = 2 + t
-1 ≤ -x ≤ 3
Step 1: Solve the first equation for t
Rearrange the equation: x = 5 - 3t
Subtract 5 from both sides: x - 5 = -3t
Divide by -3: (x - 5)/(-3) = t
Simplify: t = (5 - x)/3
Step 2: Substitute t into the second equation
Replace t in the second equation: y = 2 + t
With the expression for t: y = 2 + (5 - x)/3
Step 3: Simplify the equation for y
Multiply through by 3 to eliminate the fraction: 3y = 6 + 5 - x
Combine like terms: 3y = 11 - x
Rearrange: x + 3y = 11
Step 4: Find the range of x for the given inequality
Given: -1 ≤ -x ≤ 3
Multiply all sides by -1 (since it changes the inequality signs): 1 ≥ x ≥ -3
Step 5: Identify the graph of the parametric curve
The equation x + 3y = 11 represents a line with a slope of -1/3 and a y-intercept of 11/3.
The range of x for the given inequality is 1 ≥ x ≥ -3. This means the line segment on the line x + 3y = 11 that lies between -3 and 1 is the graph of the parametric curve.
Therefore, the graph of the parametric curve is a line segment from the point (-3, 4) to the point (1, -2).
To eliminate the parameter, we need to express x in terms of y and vice versa. We can do this by solving the first equation for t in terms of x.
Given x = 5 - 3t, we can first isolate t by subtracting 5 from both sides:
x - 5 = -3t
Now divide both sides by -3 to solve for t:
t = (x - 5) / -3
Next, substitute this expression for t into the second equation:
y = 2 + [(x - 5) / -3]
Simplifying this equation gives:
y = 2 - (x - 5) / 3
To identify the graph of the parametric curve, we can plot several points.
Let's start by picking some values for x and calculating the corresponding y-values:
When x = -1:
y = 2 - (-1 - 5) / 3
y = 2 + 6/3
y = 4
So one point on the graph is (-1, 4).
When x = 0:
y = 2 - (0 - 5) / 3
y = 2 + 5/3
y = 3 2/3
Another point on the graph is (0, 3 2/3).
When x = 3:
y = 2 - (3 - 5) / 3
y = 2 + 2/3
y = 2 2/3
Another point on the graph is (3, 2 2/3).
Connecting these points, we can see that the parametric curve represents a line segment starting from point (-1, 4), passing through (0, 3 2/3), and ending at (3, 2 2/3). The line segment is bounded between the values of x satisfying -1 ≤ x ≤ 3.
Graphically, the parametric curve represents a straight line segment on the xy-plane.