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Calculus 2 : Parametric Form

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Example Questions

Example Question #37 : Parametric, Polar, And Vector

If $x=\frac{2}{t}+6$ and y=12t-5 , what is in terms of ?

Possible Answers:

$y=\frac{24}{x-6}+5$

None of the above

$y=\frac{24}{x+6}+6$

$y=\frac{24}{x-6}-5$

$y=\frac{24}{x+6}-5$

Correct answer:

$y=\frac{24}{x-6}-5$

Explanation:

Given $x=\frac{2}{t}+6$ and y=12t-5 , let's solve both equations for :

$x=\frac{2}{t}+6\rightarrow t=\frac{2}{x-6}$

$y=12t-5\rightarrow t=\frac{y+5}{12}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y+5}{12}=\frac{2}{x-6}$

$y+5=\frac{24}{x-6}$

$y=\frac{24}{x-6}-5$

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Example Question #31 : Parametric Form

Convert the following function from parametric to rectangular coordinates:

x=t^2+4 , y=t+2

Possible Answers:

$y=2\pm \sqrt{x+4}$

$y=2+\sqrt{x-4}$

$y=2-\sqrt{x-4}$

$y=2\pm \sqrt{x-4}$

Correct answer:

$y=2\pm \sqrt{x-4}$

Explanation:

To convert to rectangular coordinates, eliminate the parameter by setting one of the functions equal to t:

$t=\pm \sqrt{x-4}$

To finish, substitute this into the equation for y:

$y=2\pm \sqrt{x-4}$

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Example Question #39 : Parametric, Polar, And Vector

Given x=4t+14 and y=5t-2 , what is in terms (rectangular form)?

Possible Answers:

$y=\frac{5}{4}(x-14)-2$

$y=\frac{5}{4}(x+14)-2$

$y=\frac{5}{4}(x+14)+2$

None of the above

$y=\frac{5}{4}(x-14)+2$

Correct answer:

$y=\frac{5}{4}(x-14)-2$

Explanation:

Given x=4t+14 and y=5t-2 , let's solve both equations for :

$x=4t+14\rightarrow t=\frac{x-14}{4}$

$y=5t-2\rightarrow t=\frac{y+2}{5}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y+2}{5}=\frac{x-14}{4}$

$y+2=5(\frac{x-14}{4})$

$y=\frac{5}{4}(x-14)-2$

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Example Question #40 : Parametric, Polar, And Vector

Given x=t-5 and y=2t+11 , what is in terms (rectangular form)?

Possible Answers:

y=2x-21

None of the above

y=2x+1

y=2x-1

y=2x+21

Correct answer:

y=2x+21

Explanation:

Given x=t-5 and y=2t+11 , let's solve both equations for :

$x=t-5\rightarrow t=x+5$

$y=2t+11\rightarrow t=\frac{y-11}{2}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-11}{2}=x+5$

y-11=2(x+5)

y=2(x+5)+11

y=2x+10+11

y=2x+21

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Example Question #41 : Parametric, Polar, And Vector

Given x=3t+7 and y=7t+3 , what is in terms (rectangular form)?

Possible Answers:

$y=\frac{7}{3}(x-7)-3$

$y=\frac{7}{3}(x-7)+3$

$y=\frac{3}{7}(x+7)+3$

$y=\frac{7}{3}(x+7)+3$

$y=\frac{3}{7}(x-7)+3$

Correct answer:

$y=\frac{7}{3}(x-7)+3$

Explanation:

Given x=3t+7 and y=7t+3 , let's solve both equations for :

$x=3t+7\rightarrow t=\frac{x-7}{3}$

$y=7t+3\rightarrow t=\frac{y-3}{7}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-3}{7}=\frac{x-7}{3}$

$y-3=7(\frac{x-7}{3})$

$y=\frac{7}{3}(x-7)+3$

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Example Question #42 : Parametric

Given x=7t+5 and y=11t+2 , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{11}{7}(x-5)-2$

None of the above

$y=\frac{11}{7}(x+5)-2$

$y=\frac{11}{7}(x-5)+2$

$y=\frac{11}{7}(x+5)+2$

Correct answer:

$y=\frac{11}{7}(x-5)+2$

Explanation:

Given x=7t+5 and y=11t+2 , let's solve both equations for :

$x=7t+5\rightarrow t=\frac{x-5}{7}$

$y=11t+2\rightarrow t=\frac{y-2}{11}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-2}{11}=\frac{x-5}{7}$

$y-2=11(\frac{x-5}{7})$

$y=\frac{11}{7}(x-5)+2$

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Example Question #31 : Parametric Form

Given x=-6t+11 and y=2t-10 , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{x-11}{3}-10$

None of the above

$y=-\frac{x-11}{3}+10$

$y=\frac{x-11}{3}+10$

$y=-\frac{x-11}{3}-10$

Correct answer:

$y=-\frac{x-11}{3}-10$

Explanation:

Given x=-6t+11 and y=2t-10 let's solve both equations for :

$x=-6t+11\rightarrow t=-\frac{x-11}{6}$

$y=2t-10\rightarrow t=\frac{y+10}{2}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y+10}{2}=-\frac{x-11}{6}$

$y+10=-2(\frac{x-11}{6})$

$y=-\frac{x-11}{3}-10$

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Example Question #32 : Parametric Form

Convert the following equation from parametric to rectangular coordinates:

x=2t^2, y=10t

Possible Answers:

$y=10\sqrt{\frac{x}{2}}$

$y=\pm10\sqrt{\frac{x}{2}}$

$y=-10\sqrt{\frac{x}{2}}$

y=5x

Correct answer:

$y=\pm10\sqrt{\frac{x}{2}}$

Explanation:

To convert the equation from parametric form to rectangular form, we must eliminate the parameter. To do this, we can solve for t with respect to x:

$t=\pm \sqrt{\frac{x}{2}}$

The plus or minus is important to remember because a square root was taken.

Now, simply plug this into the equation for y:

$y=\pm10\sqrt{\frac{x}{2}}$

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Example Question #33 : Parametric Form

Convert the following to rectangular form from parametric form:

x=2t^3+5 , y=t

Possible Answers:

y=x

$y=\sqrt[3]{\left(\frac{x-5}{2}\right)}$

$y=\frac{x-5}{2}$

$y=\left(\frac{x-5}{2}\right)^{\frac{3}{2}}$

Correct answer:

$y=\sqrt[3]{\left(\frac{x-5}{2}\right)}$

Explanation:

To convert a parametric equation into a rectangular equation, we must eliminate the parameter. We were already given an equation where t was in terms of just a variable:

t=y

Next, substitute this into the equation for x which contains t:

x=2(y^3)+5

Finally, rearrage and solve for y:

$y=\sqrt[3]{\left(\frac{x-5}{2}\right)}$

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Example Question #33 : Parametric Form

Convert the following equation into rectanglar form:

$x=\cos(t), y=cos^2 (t)+3$

Possible Answers:

y=-x^2+3

y=x^2+3

$y=\cos(x)+3$

$y=\arccos(x)+3$

Correct answer:

y=x^2+3

Explanation:

To convert the given parametric equation into rectangualr coordinates, we must eliminate the parameter by solving for t:

$t=\arccos (x)$

Now replace t with the new term in the equation for y:

$y= (\cos(\textup{arccos}(x)^2)+3=x^2+3$

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Calculus 2 : Parametric Form

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #37 : Parametric, Polar, And Vector

Example Question #31 : Parametric Form

Example Question #39 : Parametric, Polar, And Vector

Example Question #40 : Parametric, Polar, And Vector

Example Question #41 : Parametric, Polar, And Vector

Example Question #42 : Parametric

Example Question #31 : Parametric Form

Example Question #32 : Parametric Form

Example Question #33 : Parametric Form

Example Question #33 : Parametric Form

All Calculus 2 Resources

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