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Wize High School Grade 9 Math Textbook > Investigating Relationships

Curve of Best Fit

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Curve of Best Fit

Sometimes the data points in a scatter plot can follow a pattern/trend that is not a straight line. In these situations, it might be more appropriate to draw a smooth curve that follows the pattern of the data points -- this is called the curve of best fit.

Note
In this case, the two variables do not follow a linear relation!


Wize Tip
When sketching the curve of best fit, the rule for discrete vs. continuous data still applies!
  • For continuous data: sketch the curve of best fit using a solid line because all values in between data points are achievable
  • For discrete data: sketch the curve of best fit using a dotted line because not all values in between data points are achievable

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Example: Curve or Line of Best Fit


a) Construct a scatter plot representing the data from the table above.
b) Sketch a line of best fit for these data points.
c) Sketch a curve of best fit for these data points.
d) Would a line of best fit or curve of best fit be better for approximating the yy value when x=60x=60? Explain.
e) Would a line of best fit or curve of best fit be better for approximating the yy value when x=15x=15? Explain.
f) Is y=5y=5 a good estimate for when the value of x=25x=25?
g) Is y=100y=100 a good estimate for when the value of x=60x=60?

Parts a), b), and c)



Parts d) and e)

In either case, the curve of best fit gives us a better approximation because it follows the data points more closely.

Part f)

According to the curve of best fit, when x=25x=25, the yy value should be somewhere close to the middle between 1 and 9. So, y=5y=5 is a good estimate.

Part g)

According to the curve of best fit, when x=60x=60, the yy value should be much larger than the 81, which is the value when x=50x=50. Although y=100y=100 is larger than 81, it's not much larger. So, y=100y=100 is NOT a good estimate.

*Consider the "y-gap" between x=40x=40 and x=50x=50, the different in yy values is 81−27=5481-27=54. Seeing how the curve of best fit curves up steeper and steeper as xx increases, we know that the yy value when x=60x=60 will be larger than 81+54=13581+54=135.

Practice: Curve of Best Fit for the Height of a Ball

Kathy is trying to shoot a basketball through a hoop, and Leigh is recording the height of the basketball at various points in time.


a) What is the independent variable?

b) What is the dependent variable?

c) Draw a scatter plot to represent these data points.

d) Decide if a line or curve of best fit is better for approximating the height of this basketball. Then draw in the line or curve of best fit.

e) True or False? The height of the ball at time t=1.5t=1.5 seconds is approximately 12.1 ft.

f) True or False? A reasonable estimate for the time when the ball's height reaches h=10h=10ft is 0.7s or 3.3s.

Bonus
g) If you extend the curve of best fit to the left passed the vertical axis, we see that the curve passes through the point (−0.23,0)(-0.23, 0). Interpret the real-world meaning of this point.

Practice: Curve of Best Fit for the Growth of Bacteria


Josh is growing 2 different types of bacteria -- bacteria A and bacteria B. He recorded the number (in thousands) of each type of bacteria in the table below.



a) Create a scatter plot for both types of bacteria, then sketch the curve or line of best fit for both types of bacteria.

b) According to the curves of best fit, which type of bacteria will reach 1 million first?

c) Using the curves of best fit, estimate the point in time when the number of bacteria A is approximately the same as the number of bacteria B. How many bacteria of each type are there at this point?