Explain how you would graph the following set of parametric equations by plotting points and describing the orientation.
x=3t and y=t^2
just as you would for any graph on the x-y axes.
Pick a value of t, calculate x and y, and then plot (x,y)
for t=1, plot (3,1)
and so on
Well, to graph these parametric equations, we will need to plot some points. But don't worry, I promise not to make any "square" jokes! Let's get started.
First, let's choose some values for t and calculate the corresponding x and y values:
When t = -2, x = 3(-2) = -6 and y = (-2)^2 = 4.
When t = -1, x = 3(-1) = -3 and y = (-1)^2 = 1.
When t = 0, x = 3(0) = 0 and y = (0)^2 = 0.
When t = 1, x = 3(1) = 3 and y = (1)^2 = 1.
When t = 2, x = 3(2) = 6 and y = (2)^2 = 4.
Now, let's plot these points on a graph. We have (-6, 4), (-3, 1), (0, 0), (3, 1), and (6, 4).
The orientation of this graph can be described as a "U-shaped" curve opening upwards, which seems to be symmetric along the y-axis. So, it kind of looks like a parabola, but be careful with your "square" expectations!
To graph the set of parametric equations x = 3t and y = t^2, we will plot points by substituting different values of t and describe the orientation of the resulting plot.
Let's first choose some arbitrary values for t and substitute them into the equations to find corresponding values of x and y.
- When t = -2, x = 3*(-2) = -6 and y = (-2)^2 = 4. So, we have the point (-6, 4).
- When t = -1, x = 3*(-1) = -3 and y = (-1)^2 = 1. The point is (-3, 1).
- When t = 0, x = 3*0 = 0 and y = 0^2 = 0. The point is (0, 0).
- When t = 1, x = 3*1 = 3 and y = 1^2 = 1. The point is (3, 1).
- When t = 2, x = 3*2 = 6 and y = 2^2 = 4. The point is (6, 4).
Now, let's plot these points on a coordinate plane. We have (-6, 4), (-3, 1), (0, 0), (3, 1), and (6, 4).
The orientation of the plot can be described as follows:
- The points form a parabolic shape that opens upwards and is symmetric about the y-axis.
- The curve starts at the point (-6, 4), goes through the point (0, 0) as the vertex, and ends at the point (6, 4).
- As t increases, the x-values increase by 3 and the y-values increase by a greater amount due to the squared term in the equation.
To graph the set of parametric equations x = 3t and y = t^2, we can follow these steps:
1. Choose a range of values for the parameter t.
Let's say, for example, t ranges from -2 to 2.
2. Substitute different values of t into the parametric equations to find corresponding values of x and y.
For instance, when t = -2, x = 3(-2) = -6 and y = (-2)^2 = 4.
Similarly, for t = -1, x = 3(-1) = -3 and y = (-1)^2 = 1.
3. Plot the points obtained from the substitution in step 2 on a coordinate system.
In this case, we would plot the points (-6, 4) and (-3, 1), and so on, until we have a set of points that represents the curve.
4. Connect the plotted points with a smooth, continuous line.
This line will represent the graph of the parametric equations x = 3t and y = t^2.
In terms of orientation, the equations describe a parabolic curve that opens upwards. This can be observed from the fact that as t increases, the values of y increase because t^2 is always positive. The curve extends infinitely in both directions along the x-axis because t can take on any real value.