Question

Consider the following parametric equations. a. Make a brief table of values of \(t, x,\) and \(y.\) b. Plot the \((x, y)\) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing \(t\)). c. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) d. Describe the curve. $$x=t^{2}+2, y=4 t ;-4 \leq t \leq 4$$

Short answer

Question: Describe the shape of the curve represented by the parametric equations $$x=t^{2}+2, y=4t$$ and its corresponding Cartesian equation $$y=\pm 4\sqrt{x-2}$$. Answer: The parametric curve has a hyperbolic shape, consisting of two separate branches. The positive branch is a vertically stretched and horizontally translated square root function, and the negative branch is a vertically reflected, stretched, and horizontally translated square root function. The two branches share a common vertex at (2,0).

Step-by-step solution

Create a table of values for x and y

To create a table of values, we need to substitute different t-values from the given range \(-4 \leq t \leq 4\) into our parametric equations: $$x=t^{2}+2, y=4t$$ Select a few values for \(t\) and find the corresponding \((x,y)\) as below: | t | x | y | |-----|-----|-----| |-4 |18 |-16 | |-2 |6 |-8 | |0 |2 |0 | |2 |6 |8 | |4 |18 |16 |

Plot (x, y) pairs on the coordinate plane

Using the table of values from Step 1, plot the (x, y) pairs on the coordinate plane. Positive orientation (direction of increasing t) should be indicated with arrows going from left to right since \(x\) depends on \(t^2\). Then, draw the complete curve connecting all the points.

Eliminate the parameter t from the parametric equations

To eliminate the parameter t, we will perform the following steps: 1. Rewrite the \(y\) equation in terms of \(t\): $$t=\frac{y}{4}$$ 2. Substitute this equation into the \(x\) equation: $$x=(\frac{y}{4})^2 + 2$$ 3. Solve for \(y\) in terms of \(x\): $$x=y^2/16 + 2$$ 4. Manipulate the equation to get \(y\): $$y=\pm 4\sqrt{x-2}$$ So the Cartesian equation of the parametric curve in terms of x and y is $$y=\pm 4\sqrt{x-2}$$.

Describe the curve

The equation obtained in Step 3 $$y=\pm 4\sqrt{x-2}$$ describes two separate curves. The curve with positive y values is a vertically stretched and horizontally translated square root function. The positive branch goes through the points \((6,8)\) and \((2,0)\). Similarly, the curve with negative y values is a vertically reflected, stretched, and horizontally translated square root function. The negative branch goes through the points \((6,-8)\) and \((2,0)\). Both curves share a common endpoint at \((2,0)\), which is the vertex joining the two branches. Overall, the parametric curve describes a hyperbolic shape.

Key concepts

Plotting Parametric Curves

Plotting parametric curves involves creating a visual representation of equations where the x and y coordinates are expressed in terms of a third variable, typically denoted as 't'. To begin plotting, first, construct a table by selecting a range of values for 't' and calculating the corresponding x and y values using the parametric equations.

Next, transfer the (x, y) pairs onto a coordinate plane, marking each point. This process is akin to plotting points in a typical Cartesian coordinate system, but here the 't' values can affect the trajectory and orientation of the curve. Positive orientation is indicated by arrows, hinting at an increase in the 't' value. The completed curve is then drawn by connecting these plotted points smoothly, reflecting the continuous nature and the path of the curve as 't' increases. Visualization is key to understanding the dynamics of parametric curves.

Eliminate Parameter

The process of eliminating the parameter in parametric equations allows us to re-express the relationship between x and y without the intermediary variable 't'. This is useful for interpreting the curve in a more conventional y=f(x) form. To do this, isolate 't' in one of the parametric equations and substitute this into the other, resulting in an equation only in terms of x and y.

Using our example exercise, we rewrote the equation for y as a function of 't' then substituted this into the x equation. The goal is to solve for 'y' in terms of 'x' which usually requires algebraic manipulation. The resulting equation, in this case, depicts a hyperbolic shape and illustrates how parametric equations can map complex relationships that may be difficult to express otherwise.

Curve Sketching

Curve sketching is the art of drawing a curve based on its mathematical expression. When it comes to parametric curves, curve sketching gives us insight into the shape and behavior of the curve without requiring extensive calculations. After plotting points and eliminating the parameter, we can deduce certain aspects, like symmetry, direction, and bounds, of the curve. For instance, a parametric curve may resemble familiar shapes such as circles, ellipses, or hyperbolas; these insights are crucial for sketching.

In our example, the eliminated parameter yielded the equation of two mirrored branches of a hyperbola, allowing us to sketch a hyperbolic shape. Key points from the table, such as where the curve intersects axes or changes direction, help to anchor the sketch accurately. Curve sketching is a potent tool for visual learners and serves as a bridge between abstract equations and tangible representation.