Okay I have a scatterplot, so I have points on the graph. I need to use these points (without using regression on my TI-83) to discover a cubic function that would fit the data (as a sort of trend line). How do I go about this?
I looked up the graphing form of the cubic function:
y=a((x-h)^3)+k
But have no idea how to progress from here. Guidence, please?
To fit a cubic function to your scatterplot data, you'll need to follow these steps:
Step 1: Plot your scatterplot points on a coordinate grid. Make sure you have enough data points to provide a reliable representation of the trend.
Step 2: Choose four points from your scatterplot that approximately lie on your desired trend line. These four points should be spread out evenly across the range of your data.
Step 3: Assign variables to each of the four points. Let's call the x-values x1, x2, x3, and x4, and the corresponding y-values y1, y2, y3, and y4.
Step 4: Set up a system of equations using the cubic function form you mentioned: y = a((x - h)^3) + k.
For each of the four points, substitute the x and y values into the equation:
For point 1 (x1, y1): y1 = a((x1 - h)^3) + k
For point 2 (x2, y2): y2 = a((x2 - h)^3) + k
For point 3 (x3, y3): y3 = a((x3 - h)^3) + k
For point 4 (x4, y4): y4 = a((x4 - h)^3) + k
You now have four equations with four unknowns (a, h, k). We need to solve this system of equations to find their values.
Step 5: Expand each of the equations to simplify them and convert them to a standard system of linear equations. You'll have four equations with four unknowns, which might look something like this:
y1 = ah^3 - 3ah^2x1 + 3ahx1^2 - ax1^3 + k
y2 = ah^3 - 3ah^2x2 + 3ahx2^2 - ax2^3 + k
y3 = ah^3 - 3ah^2x3 + 3ahx3^2 - ax3^3 + k
y4 = ah^3 - 3ah^2x4 + 3ahx4^2 - ax4^3 + k
Step 6: Rearrange the equations to isolate the desired unknown variables on one side and the known values on the other side. The equations should be in a form that allows you to solve for a, h, and k.
Step 7: Once you have solved the system of equations and found the values of a, h, and k, substitute these values back into the cubic function form: y = a((x - h)^3) + k.
This cubic function should approximate the trend line of your scatterplot data.
Remember, since the scatterplot may not perfectly fit a cubic function, the resulting function may not be an exact fit, but it should provide a reasonable representation of the trend.
To find a cubic function that fits your scatterplot, you can follow these steps:
Step 1: Plot the points on a graph.
Step 2: Try to identify any apparent trends or patterns in the data. This will help you make an initial guess about the position of the function.
Step 3: Choose an arbitrary point on or near your scatterplot. This point will act as the vertex of your cubic function. Assign the x and y coordinates of this point as (h, k), where h represents the horizontal shift and k represents the vertical shift of the cubic function.
Step 4: Select another point from the scatterplot and substitute the values of (x, y) into the cubic function equation: y = a((x - h)^3) + k
Step 5: Solve the equation by substituting the x and y values of the second point into the equation obtained in Step 4.
Step 6: Repeat Steps 4 and 5 using additional points from the scatterplot. This will give you a system of equations involving the unknown variable 'a'.
Step 7: Using algebraic techniques like substitution or elimination, solve the system of equations to find the value of 'a'.
Step 8: With the value of 'a' determined, substitute it back into the cubic function equation: y = a((x - h)^3) + k.
By completing these steps, you will find the values of 'a', 'h', and 'k', which will allow you to write the cubic function that best fits your scatterplot.
Remember, this method is an approximation that relies on finding the best cubic function that fits the data points. It may not be an exact fit for all data points, but it will provide a good estimate of the trend.