Question 1

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = t , \quad y = 3 t ^ { 2 } , \quad t \in ( - \infty , \infty )$$

Accepted Answer

Since $x = t$, we can directly substitute $t$ with $x$ in the equation for $y$, giving $y = 3x^2$. The direction is left to right because as $t$ (or $x$) increases, $y$ also increases.

Question 2

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 1 - t , \quad y = - 2 t ^ { 2 } , \quad t \geq 0$$

Accepted Answer

To eliminate the parameter $t$, we solve the first equation for $t$: $t = 1 - x$. Substituting this into the second equation gives $y = -2(1 - x)^2 = -2(x - 1)^2$. Since $t \geq 0$, and $t = 1 - x$, it follows that $x \leq 1$. The direction is right to left because as $t$ increases, $x$ decreases.

Question 3

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 3 t + 1 , \quad y = 9 t ^ { 2 } - 2 , \quad t \in ( - \infty , \infty )$$

Accepted Answer

To eliminate the parameter $t$, we solve the equation $x = 3t + 1$ for $t$, getting $t = \frac{x - 1}{3}$. Substituting this into the equation for $y$, we get $y = 9\left(\frac{x - 1}{3}\right)^2 - 2$, which simplifies to $y = (x - 1)^2 - 2$. Since $t$ increases as $x$ increases (from the equation $x = 3t + 1$), the direction of the curve is from left to right.

Question 4

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = - 2 t , \quad y = 6 t ^ { 2 } - 1 , \quad t \in [ - 2,3 ]$$

Accepted Answer

To eliminate the parameter $t$, we solve the equation $x = -2t$ for $t$, getting $t = -\frac{x}{2}$. Substituting this into the equation for $y$, we get $y = 6\left(-\frac{x}{2}\right)^2 - 1 = \frac{3}{2}x^2 - 1$. The domain of $x$ is determined by the given range of $t$, which translates to $x = -2(-2) = 4$ and $x = -2(3) = -6$, so $x$ ranges from $-6$ to $4$. The direction is right to left because as $t$ increases, $x$ decreases.

Question 5

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 2 t + 1 , \quad y = 3 t , \quad t \in [ - 1,3 ]$$

Accepted Answer

To eliminate the parameter $t$, solve the first equation for $t$: $t = \frac{x - 1}{2}$. Substitute this into the second equation to get $y = 3\left(\frac{x - 1}{2}\right)$, which simplifies to $y = \frac{3}{2}(x - 1)$. The range of $x$ is found by substituting the limits of $t$ into $x = 2t + 1$, giving $x$ values from $-1$ to $7$. The direction is left to right since as $t$ increases, $x$ increases.

Question 6

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = t - 1 , \quad y = 2 t + 5 , \quad t \in ( - \infty , \infty )$$

Accepted Answer

The answer of Find an equation of y as a...

Question 7

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = - 4 t , \quad y = - 3 t + 1 , \quad t \in [ - 2,1 ]$$

Accepted Answer

To eliminate the parameter $t$, we solve one of the equations for $t$ and substitute into the other. From $x = -4t$, we get $t = -\frac{x}{4}$. Substituting this into $y = -3t + 1$ gives $y = -3\left(-\frac{x}{4}\right) + 1 = \frac{3}{4}x + 1$. The range of $x$ is found by substituting the $t$ values into $x = -4t$, giving $x \in [-8, 4]$, not $[-4, 8]$ as stated in all options, which seems to be a mistake in the question. The direction is from right to left since as $t$ increases, $x$ decreases.

Question 8

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = t + 1 , \quad y = e ^ { t } , \quad t \in ( - \infty , \infty )$$

Accepted Answer

To eliminate the parameter $t$, we solve the first equation for $t$: $t = x - 1$. Substituting this into the second equation gives $y = e^{x-1}$. The direction is left to right because as $t$ increases, $x$ increases.

Question 9

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 2 \sin t , \quad y = 2 \cos t , \quad t \in [ 0,2 \pi ]$$

Accepted Answer

The answer of Find an equation of y as a...

Question 10

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 3 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$$

Accepted Answer

Using the Pythagorean identity $\sin^2 t + \cos^2 t = 1$, and given $x = 3\cos t$ and $y = 3\sin t$, squaring both equations and adding them yields $x^2 + y^2 = 9\cos^2 t + 9\sin^2 t = 9$. The direction is counterclockwise because as $t$ increases from $0$ to $2\pi$, the motion follows the standard direction of rotation on the unit circle, starting from $(3,0)$ and moving counterclockwise back to the same point.

Question 11

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = - 2 t - 1 , \quad y = 3 t ^ { 2 } + 2 , \quad t \in ( - \infty , \infty )$$

Accepted Answer

To eliminate the parameter $t$, we solve the equation $x = -2t - 1$ for $t$: $t = -\frac{1}{2}x - \frac{1}{2}$. Substituting this into the equation for $y$, we get $y = 3\left(-\frac{1}{2}x - \frac{1}{2}\right)^2 + 2$, which simplifies to $y = \frac{3}{4}(x + 1)^2 + 2$. Since $x$ decreases as $t$ increases (given by the negative coefficient of $t$ in $x = -2t - 1$), the curve goes from right to left.

Question 12

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 3 \sin t , \quad y = 4 \cos t , \quad t \in [ 0,2 \pi ]$$

Accepted Answer

The answer of Find an equation of y as a...

Question 13

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 4 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$$

Accepted Answer

The answer of Find an equation of y as a...

Question 14

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = \sin t , \quad y = \cos 2 t , \quad t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$$

Accepted Answer

Using the double angle formula for cosine, $y = \cos 2t = 1 - 2\sin^2 t$. Since $x = \sin t$, we substitute to get $y = 1 - 2x^2$. The range of $x = \sin t$ for $t \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$ is $[-1, 1]$. As $t$ increases, $x$ moves from left to right.

Question 15

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = \cos t , \quad y = \cos 2 t , \quad t \in [ 0 , \pi ]$$

Accepted Answer

The parametric equations are $x = \cos t$ and $y = \cos 2t = 2\cos^2 t - 1$. Substituting $x = \cos t$ into the equation for $y$, we get $y = 2x^2 - 1$. As $t$ increases from 0 to $\pi$, $x$ decreases from 1 to -1, indicating the direction is right to left.

Question 16

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve.x = e²^t, y = t, t ? 0

Accepted Answer

Given $x = e^{2t}$ and $y = t$, solving for $t$ in terms of $x$ gives $t = \frac{1}{2}\ln(x)$, thus $y = \frac{1}{2}\ln(x)$, with $x \geq 1$ because $e^{2t}$ is always positive and increases as $t$ increases, indicating a left to right direction.

Question 17

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = \cos ^ { 2 } t , \quad y = 3 \sin ^ { 2 } t , \quad t \in \left[ 0 , \frac { \pi } { 2 } \right]$$

Accepted Answer

The answer of Find an equation of y as a...

Question 18

Find the equation of the tangent line to the parametric curve: $$x = 2 t + 1 , \quad y = 3 t + 4 , \quad \text { at } t = - 1$$

Accepted Answer

The answer of Find the equation of the tangent line...

Question 19

Find the equation of the tangent line to the parametric curve: $$x = t + 1 , \quad y = e ^ { 2 t } , \quad \text { at } t = 0$$

Accepted Answer

To find the tangent line, calculate the derivatives dx/dt = 1 and dy/dt = 2e^(2t). At t = 0, dy/dx = (2e^0)/(1) = 2. The point is (1,1). Using point-slope form: y - 1 = 2(x - 1), which simplifies to y = 2x - 1.

Question 20

Find the equation of the tangent line to the parametric curve: $$x = t ^ { 3 } , \quad y = ( 3 - t ) ^ { 2 } , \quad \text { at } t = 1$$

Accepted Answer

To find the equation of the tangent line to the given parametric curve at $t = 1$, we first find the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$, and then use these to find the slope of the tangent line, $\frac{dy}{dx}$, at $t = 1$. Given $x = t^3$ and $y = (3 - t)^2$, we have:$$\frac{dx}{dt} = 3t^2$$$$\frac{dy}{dt} = 2(3 - t)(-1) = -2(3 - t)$$At $t = 1$, $\frac{dx}{dt} = 3(1)^2 = 3$ and $\frac{dy}{dt} = -2(3 - 1) = -4$. Thus, the slope of the tangent line is:$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-4}{3}$$The point on the curve at $t = 1$ is given by $x(1) = 1^3 = 1$ and $y(1) = (3 - 1)^2 = 4$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line, we have:$$y - 4 = -\frac{4}{3}(x - 1)$$Simplifying, we get:$$y = -\frac{4}{3}x + \frac{4}{3} + 4 = -\frac{4}{3}x + \frac{16}{3}$$Therefore, the correct equation of the tangent line is $y = -\frac{4}{3}x + \frac{16}{3}$.

Quiz 9: Parametric Equations Polar Coordinates and Conic Sections

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = t , \quad y = 3 t ^ { 2 } , \quad t \in ( - \infty , \infty )$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 1 - t , \quad y = - 2 t ^ { 2 } , \quad t \geq 0$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 3 t + 1 , \quad y = 9 t ^ { 2 } - 2 , \quad t \in ( - \infty , \infty )$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = - 2 t , \quad y = 6 t ^ { 2 } - 1 , \quad t \in [ - 2,3 ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 2 t + 1 , \quad y = 3 t , \quad t \in [ - 1,3 ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = t - 1 , \quad y = 2 t + 5 , \quad t \in ( - \infty , \infty )$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = - 4 t , \quad y = - 3 t + 1 , \quad t \in [ - 2,1 ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = t + 1 , \quad y = e ^ { t } , \quad t \in ( - \infty , \infty )$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 2 \sin t , \quad y = 2 \cos t , \quad t \in [ 0,2 \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 3 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = - 2 t - 1 , \quad y = 3 t ^ { 2 } + 2 , \quad t \in ( - \infty , \infty )$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 3 \sin t , \quad y = 4 \cos t , \quad t \in [ 0,2 \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 4 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = \sin t , \quad y = \cos 2 t , \quad t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = \cos t , \quad y = \cos 2 t , \quad t \in [ 0 , \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve.x = e²^t, y = t, t ? 0

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = \cos ^ { 2 } t , \quad y = 3 \sin ^ { 2 } t , \quad t \in \left[ 0 , \frac { \pi } { 2 } \right]$

Find the equation of the tangent line to the parametric curve: $x = 2 t + 1 , \quad y = 3 t + 4 , \quad \text { at } t = - 1$

Find the equation of the tangent line to the parametric curve: $x = t + 1 , \quad y = e ^ { 2 t } , \quad \text { at } t = 0$

Find the equation of the tangent line to the parametric curve: $x = t ^ { 3 } , \quad y = ( 3 - t ) ^ { 2 } , \quad \text { at } t = 1$

Quiz 9: Parametric Equations Polar Coordinates and Conic Sections

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=t,y=3t2,t∈(−∞,∞) x = t , \quad y = 3 t ^ { 2 } , \quad t \in ( - \infty , \infty ) x=t,y=3t2,t∈(−∞,∞)

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=1−t,y=−2t2,t≥0x = 1 - t , \quad y = - 2 t ^ { 2 } , \quad t \geq 0x=1−t,y=−2t2,t≥0

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=3t+1,y=9t2−2,t∈(−∞,∞) x = 3 t + 1 , \quad y = 9 t ^ { 2 } - 2 , \quad t \in ( - \infty , \infty ) x=3t+1,y=9t2−2,t∈(−∞,∞)

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=−2t,y=6t2−1,t∈[−2,3]x = - 2 t , \quad y = 6 t ^ { 2 } - 1 , \quad t \in [ - 2,3 ]x=−2t,y=6t2−1,t∈[−2,3]

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=2t+1,y=3t,t∈[−1,3]x = 2 t + 1 , \quad y = 3 t , \quad t \in [ - 1,3 ]x=2t+1,y=3t,t∈[−1,3]

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=t−1,y=2t+5,t∈(−∞,∞) x = t - 1 , \quad y = 2 t + 5 , \quad t \in ( - \infty , \infty ) x=t−1,y=2t+5,t∈(−∞,∞)

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=−4t,y=−3t+1,t∈[−2,1]x = - 4 t , \quad y = - 3 t + 1 , \quad t \in [ - 2,1 ]x=−4t,y=−3t+1,t∈[−2,1]

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=t+1,y=et,t∈(−∞,∞) x = t + 1 , \quad y = e ^ { t } , \quad t \in ( - \infty , \infty ) x=t+1,y=et,t∈(−∞,∞)

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=2sin⁡t,y=2cos⁡t,t∈[0,2π]x = 2 \sin t , \quad y = 2 \cos t , \quad t \in [ 0,2 \pi ]x=2sint,y=2cost,t∈[0,2π]

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=3cos⁡t,y=3sin⁡t,t∈[0,2π]x = 3 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]x=3cost,y=3sint,t∈[0,2π]

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=3sin⁡t,y=4cos⁡t,t∈[0,2π]x = 3 \sin t , \quad y = 4 \cos t , \quad t \in [ 0,2 \pi ]x=3sint,y=4cost,t∈[0,2π]

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=4cos⁡t,y=3sin⁡t,t∈[0,2π]x = 4 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]x=4cost,y=3sint,t∈[0,2π]

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=cos⁡t,y=cos⁡2t,t∈[0,π]x = \cos t , \quad y = \cos 2 t , \quad t \in [ 0 , \pi ]x=cost,y=cos2t,t∈[0,π]

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve.x = e2t, y = t, t ? 0

Find the equation of the tangent line to the parametric curve: x=2t+1,y=3t+4, at t=−1x = 2 t + 1 , \quad y = 3 t + 4 , \quad \text { at } t = - 1x=2t+1,y=3t+4, at t=−1

Find the equation of the tangent line to the parametric curve: x=t+1,y=e2t, at t=0x = t + 1 , \quad y = e ^ { 2 t } , \quad \text { at } t = 0x=t+1,y=e2t, at t=0

Find the equation of the tangent line to the parametric curve: x=t3,y=(3−t) 2, at t=1x = t ^ { 3 } , \quad y = ( 3 - t ) ^ { 2 } , \quad \text { at } t = 1x=t3,y=(3−t) 2, at t=1

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = t , \quad y = 3 t ^ { 2 } , \quad t \in ( - \infty , \infty )$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 1 - t , \quad y = - 2 t ^ { 2 } , \quad t \geq 0$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 3 t + 1 , \quad y = 9 t ^ { 2 } - 2 , \quad t \in ( - \infty , \infty )$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = - 2 t , \quad y = 6 t ^ { 2 } - 1 , \quad t \in [ - 2,3 ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 2 t + 1 , \quad y = 3 t , \quad t \in [ - 1,3 ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = t - 1 , \quad y = 2 t + 5 , \quad t \in ( - \infty , \infty )$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = - 4 t , \quad y = - 3 t + 1 , \quad t \in [ - 2,1 ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = t + 1 , \quad y = e ^ { t } , \quad t \in ( - \infty , \infty )$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 2 \sin t , \quad y = 2 \cos t , \quad t \in [ 0,2 \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 3 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 3 \sin t , \quad y = 4 \cos t , \quad t \in [ 0,2 \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 4 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = \cos t , \quad y = \cos 2 t , \quad t \in [ 0 , \pi ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve.x = e²^t, y = t, t ? 0

Find the equation of the tangent line to the parametric curve: $x = 2 t + 1 , \quad y = 3 t + 4 , \quad \text { at } t = - 1$

Find the equation of the tangent line to the parametric curve: $x = t + 1 , \quad y = e ^ { 2 t } , \quad \text { at } t = 0$

Find the equation of the tangent line to the parametric curve: $x = t ^ { 3 } , \quad y = ( 3 - t ) ^ { 2 } , \quad \text { at } t = 1$