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
Determine the Parametric Equations
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Answer
The parametric equations given are:
x=2+t
y=3−2t2
Step 2
Evaluate Endpoints
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Answer
Evaluate the equations at the endpoints of t:
For t=−1:
x(−1)=2+(−1)=1
y(−1)=3−2(−1)2=3−2=1
So, the point is (1,1).
For t=0:
x(0)=2+0=2
y(0)=3−2(0)2=3
So, the point is (2,3).
Step 3
Analyze the Behavior of the Curve
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Answer
As t moves from −1 to 0, x increases from 1 to 2 while y decreases from 1 to 3. This indicates that the curve moves from point (1,1) to (2,3) while rising.
Step 4
Select the Correct Diagram
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Answer
Based on the evaluated points and the behavior of the curve, the correct diagram representing this motion is option B, as it shows the curve bending upwards from (1,1) to (2,3).
