How do you graph the curve whose parametric equations are given and show its orientation given #x = sqrt{t} + 4#, #y = sqrt{t} - 4#, where #t=0#?

Question

How do you graph the curve whose parametric equations are given and show its orientation given #x = sqrt{t} + 4#, #y = sqrt{t} - 4#, where #t>=0#?

Isaiah Allen · Accepted Answer

Pleasesee below.

For a parametric equation #f)(x,y)=(x(t),y(t))#, you put different values of #t#, ofcourse where #t>=0#, to get different pairs of values for #x# and #y# to get sets of points, joining which, we get the desired curve. Here we have #x=sqrtt+4# and #y=sqrtt-4#. Let us consider #t=0,1,4,9,16,25,36# - note that we have intentionally selected square numbers, so that getting pair of values of #x# and #y# is easy. We get #(4,-4),(5,-3),(6,-2),(7,-1),(8,0),(9,1),(10,2)# and joining them we get the following graph nand this is a striaght line, shown below. Also observe that given #x=sqrtt+4# and #y=sqrtt-4#, subtracting latter from former eliminates #t# and we get equation of line #x-y=8#. What does #sqrtt# does to this? It just restricts line to #x>=4# or #y>=-4#. graph{x-y=8 [4, 24, -7, 3]}

Cameron Alexander · Answer

undefined 1. **Plot Initial Points**: Start by substituting various values of $t \geq 0$ (like $t = 0, 1, 4, 9$) into the parametric equations $x = \sqrt{t} + 4$ and $y = \sqrt{t} - 4$. Calculate the corresponding $x$ and $y$ values for these $t$ values.

2. **Graph Points and Curve**:
   - For $t = 0$, $x = 4$ and $y = -4$.
   - Increase $t$ gradually. As $t$ increases, both $x$ and $y$ increase since both are functions of $\sqrt{t}$, but $x$ will always be 8 units more than $y$ due to their respective constants.
   - Plot these points on the coordinate plane.

3. **Orientation**: The curve's orientation (direction of increasing $t$) starts from the point (4, -4) and moves upwards and to the right, as both $x$ and $y$ increase with $t$.

4. **Shape of the Curve**: The curve will be a part of the graph where $x > 4$ and $y > -4$, showing a diagonal movement away from the origin as $t$ increases, following the line $y = x - 8$, but with a curve that softens as it moves away from the point (4, -4) due to the square root function's decreasing rate of increase.

5. **Drawing**: On graph paper, plot the calculated points, draw the curve through these points, and indicate the orientation with an arrow showing the direction in which $t$ increases.

How do you graph the curve whose parametric equations are given and show its orientation given #x = sqrt{t} + 4#, #y = sqrt{t} - 4#, where #t&gt;=0#?

How do you graph the curve whose parametric equations are given and show its orientation given #x = sqrt{t} + 4#, #y = sqrt{t} - 4#, where #t>=0#?