Parametric Functions
-When to variables depend upon a common variable (parameter)
Example:
Eliminating the Parameter:
Step 1: Solve one equation for a variable (like t--time)
Step 2: Substitute the variable into the second equation
Step 3: Fully solve the second equation
Step 4: Answer--the parameter will be eliminated
h=1/2at^2+Vot+ho (Physics Equation that finds the height now)
d=Vot (Physics Equation that finds the distance traveled)
If air resistance is ignored, the object accelerates at 10 m/s---since the ball is dropping, the ball will accelerate a -10 m/s
Velocity in the beginning=0
h=(-10)t^2+(0)*t=100
h=-5t^2+100
d=Vot
Velocity to start=5 m/s
d=5t
To eliminate t=Time: Solve for T and Substitute
h=-5t^2+100 d=5t
h=-5(d/5)^2+100
Answer: h=-d/5^2+100
-able to find out the height of the ball with the distance it traved and without time
Step 1: Solve one equation for a variable (like t--time)
Step 2: Substitute the variable into the second equation
Step 3: Fully solve the second equation
Step 4: Answer--the parameter will be eliminated
h=1/2at^2+Vot+ho (Physics Equation that finds the height now)
d=Vot (Physics Equation that finds the distance traveled)
If air resistance is ignored, the object accelerates at 10 m/s---since the ball is dropping, the ball will accelerate a -10 m/s
Velocity in the beginning=0
h=(-10)t^2+(0)*t=100
h=-5t^2+100
d=Vot
Velocity to start=5 m/s
d=5t
To eliminate t=Time: Solve for T and Substitute
h=-5t^2+100 d=5t
h=-5(d/5)^2+100
Answer: h=-d/5^2+100
-able to find out the height of the ball with the distance it traved and without time
Example:
x=1-2t, y=2-t negative infinity<t< positive infinity
For the first equation: Solve for t
x=1-2t
2t=1-x
t=1/2(1-x)
Substitute t from the first equation into the second equation
y=2-t
y=2-1/2(1-x)
Answer: y=0.5x+1.5
The equation is a line!
Example #2:
y= 3+6t 2t=x-2
Substitute and Solve for the parameter:
For the first equation: Solve for t
x=1-2t
2t=1-x
t=1/2(1-x)
Substitute t from the first equation into the second equation
y=2-t
y=2-1/2(1-x)
Answer: y=0.5x+1.5
The equation is a line!
Example #2:
y= 3+6t 2t=x-2
Substitute and Solve for the parameter:
Another Example:
Questions:
#1 Eliminate the parameter. State what type of function it is: x=t y=t^3-2t+3
#2: Eliminate the parameter: x=2(sin)t y=3(cos)t
#3: Eliminate the parameter: x= 2t-3 y=9-4t
#2: Eliminate the parameter: x=2(sin)t y=3(cos)t
#3: Eliminate the parameter: x= 2t-3 y=9-4t
Answers:
Question #1
Answer above.
Step 1: Substitute x=t into the 2nd equation
Step 2: Cubic-3rd Degree
Question #2
Step 1: Substitute x=t into the 2nd equation
Step 2: Cubic-3rd Degree
Question #2
Answer above.
Step 1: Since-- sin^2t+cos^2t=1
Step 2: Multiplication
Step 3: Answer
Question #3
Step 1: Since-- sin^2t+cos^2t=1
Step 2: Multiplication
Step 3: Answer
Question #3
Answer above.
Step 1: Solve the 1st equation for t
Step 2: Substitute t into the 2nd equation
Step 3: Solve
Step 4: Answer
Step 1: Solve the 1st equation for t
Step 2: Substitute t into the 2nd equation
Step 3: Solve
Step 4: Answer
Common Core:
Eliminate the parameter: x=3t-7 y=-4.9t^2+15t+74
Answers:
Answers:
Answer above.
Step 1: Solve the 1st equation for t
Step 2: Substitute t into the 2nd equation
Step 3: Solve
Step 4: Answer
Step 1: Solve the 1st equation for t
Step 2: Substitute t into the 2nd equation
Step 3: Solve
Step 4: Answer