Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine how the plane curves differ from each other. (a) \(x=t\) \(y=2 t+1\) (b) \(x=\cos \theta\) \(y=2 \cos \theta+1\) (c) \(x=e^{-t}\) \(y=2 e^{-t}+1\) (d) \(x=e^{t}\) \(y=2 e^{t}+1\)

Step-by-Step Solution

Answer

Curve (a) is a straight line with a positive linear relationship between x and y. Curve (b) is a circle centered at (0,1). Curve (c) is a declining exponential graph with y-values always larger than the x-values. Curve (d) is a rising exponential graph with both x and y increasing as t increases and y-values larger than the x-values.

Step 1: Plot curve (a)

We're given \(x = t\) and \(y = 2t + 1\). This describes a straight line. If a graph of this curve is plotted, we can see that it slopes upward, indicating a positive linear relationship between x and y.

Step 2: Plot curve (b)

We're given \(x = \cos\theta\) and \(y = 2\cos\theta + 1\). These equations make a circle on a plane about the origin. When plotted it gives a circle of radius 1 centered at (0,1).

Step 3: Plot curve (c)

The equations for curve c are \(x = e^{-t}\) and \(y = 2e^{-t} + 1\). This will give a declining exponential curve in a plane. The y-values will be always larger than the x-values.

Step 4: Plot curve (d)

Finally, for curve d, we have \(x = e^{t}\) and \(y = 2e^{t} + 1\). This will represent a rising exponential curve. The y-values will always be larger than the x-values but in this case, unlike graph c, both x and y increase as t increases.