A curve is defined in parametric form by $x = 2 + t$ and $y = 3 - 2t^2$ for $-1 \leq t \leq 0$ - HSC - SSCE Mathematics Extension 1 - Question 5 - 2022 - Paper 1
Question 5
A curve is defined in parametric form by $x = 2 + t$ and $y = 3 - 2t^2$ for $-1 \leq t \leq 0$.
Which diagram best represents this curve?
Worked Solution & Example Answer:A curve is defined in parametric form by $x = 2 + t$ and $y = 3 - 2t^2$ for $-1 \leq t \leq 0$ - HSC - SSCE Mathematics Extension 1 - Question 5 - 2022 - Paper 1
Step 1
$x = 2 + t$ and $y = 3 - 2t^2$: Determine Points
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Answer
To analyze the curve, we can start by evaluating the equations at the endpoints of the parameter t.
For t=−1:
x=2−1=1
y=3−2(−1)2=3−2=1
Thus, we have the point (1,1).
For t=0:
x=2+0=2
y=3−2(0)2=3−0=3
Thus, the point is (2,3).
Step 2
Graph the Points and Behavior
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Answer
Plotting these points on a Cartesian coordinate system, we notice the following:
The curve starts at (1,1) when t=−1 and ends at (2,3) when t=0.
As t changes from −1 to 0, the value of y decreases initially and then increases as t approaches 0.
This indicates that the curve will have a shape that initially decreases then rises, corresponding to a parabolic arc.
Step 3
Select the Correct Diagram
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Answer
Among the provided diagrams, option B best represents the curve. In this diagram, the curve moves from (1,1) to (2,3), capturing the behavior of the function as described by the parametric equations.
