A parabola is graphed on the x y-plane.The parabola opens to the right. The point left parenthesis 0 comma negative 2 right-parenthesis is labeled t equals 0. The point left-parenthesis negative 3 comma negative 1 right-parenthesis is labeled t equals 1. The point left-parenthesis negative 4 comma 0 right-parenthesis is labeled t equals 2. The point left-parenthesis 5 comma 3 right-parenthesis is labeled t equals 5.

Question

Question
Use the graph to answer the question.
What is the orientation of the plane curve?

x equals t squared minus 4 t line break y equals t minus 2
(1 point)
Responses

down to the left and then down to the right
down to the left and then down to the right

up to the left and then up to the right
up to the left and then up to the right

down from the left and up from the left at the same time
down from the left and up from the left at the same time

up to the right and down to the right at the same time
up to the right and down to the right at the same time
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Answer 1

To determine the orientation of the curve represented by the parametric equations \( x = t^2 - 4t \) and \( y = t - 2 \), we can analyze the behavior of the curve as the parameter \( t \) changes.

1. Find the important points based on the provided \( t \) values:

   • At \( t = 0 \):

     • \( x(0) = 0^2 - 4(0) = 0 \)
     • \( y(0) = 0 - 2 = -2 \)
     • Point: \( (0, -2) \)

   • At \( t = 1 \):

     • \( x(1) = 1^2 - 4(1) = 1 - 4 = -3 \)
     • \( y(1) = 1 - 2 = -1 \)
     • Point: \( (-3, -1) \)

   • At \( t = 2 \):

     • \( x(2) = 2^2 - 4(2) = 4 - 8 = -4 \)
     • \( y(2) = 2 - 2 = 0 \)
     • Point: \( (-4, 0) \)

   • At \( t = 5 \):

     • \( x(5) = 5^2 - 4(5) = 25 - 20 = 5 \)
     • \( y(5) = 5 - 2 = 3 \)
     • Point: \( (5, 3) \)

2. Plotting the points:

   • As \( t \) increases from \( 0 \) to \( 1 \), the curve moves from \( (0, -2) \) to \( (-3, -1) \), which is towards the left and slightly upwards.
   • As \( t \) increases from \( 1 \) to \( 2 \), the curve moves from \( (-3, -1) \) to \( (-4, 0) \), which is again towards the left and ups slightly.
   • As \( t \) increases from \( 2 \) to \( 5 \), the curve moves from \( (-4, 0) \) to \( (5, 3) \), moving considerably to the right and upwards.

3. Orientation Analysis:

   • The curve starts at \( t = 0 \) going left to point \( (0, -2) \) and then goes up to \( (-4, 0) \) as \( t \) increases, and finally makes a significant move to the right while going up toward \( (5, 3) \).
   • Overall, as \( t \) increases, the curve consistently goes up while moving from left to right.

Based on this analysis, the curve has the following orientation:

Response: up to the left and then up to the right.