8.01 Comparing functions across representations

Choose which of the following functions would model Nasrin's exercise chart best: a radical function, a rational function, quadratic function, or a logarithmic function.

Graph the data on a coordinate plane and determine which regression model fits best.

In order to find the regression curve, we can input the x and y values into a table, then use the Two-Variable Statistics tool and choose a model:

The shape of the graph could lend itself to any of the given functions, since the graph shows that the number of push-ups Nasrin can do increases at a decreasing rate.

We can take key features into consideration when choosing a model. We might note that the domain and range of a square root function are always positive, so a radical function could fit our data. We could, however, restrict the domain and range of any of the functions.

The regression model for a logarithmic function follows:

The regression model for a quadratic function follows:

Another option for fitting models is adding the points to a graph and building a transformed parent function using our knowledge of transformations of functions. This is not as accurate as using a regression model, but the functions can be manipulated to follow the shape of our data.

A potential radical function follows:

A potential rational function follows:

Given the various function models, the logarithmic function fits Nasrin's charted exercise data the best.

State the number of push ups that Nasrin will be able to do after 16 weeks, according to the model you chose in part (a).

Use the equation given for the regression curve of a logarithmic function and substitute 16 for x.

The equation that best models Nasrin's push-up routine is given by y=15 + 7.2135 \ln \left( x \right). By substituting x=16, we get y=35, meaning that Nasrin will be able to do 35 push-ups in 16 weeks.

We could also follow the pattern in the given data set, which shows the weeks doubling and the push-ups increasing by 5. By continuing the pattern in the set of data, we can also conclude that Nasrin will be able to do 35 push-ups in 16 weeks. By using the equation that models the context, we are verifying that the equation is a good fit.

Determine whose ball is falling at a faster rate from 2 to 3 seconds.

Calculate the average rate of change for both functions over the interval \left[2, 3 \right], then compare.

For f \left(x \right), we know that f \left( 2 \right) = 12 and f \left(3 \right)=9, so we have \dfrac{ f\left(3 \right) - f \left( 2 \right)}{3-2} = \dfrac{9 - 12 \text{ feet}}{3-2 \text{ seconds}} = \dfrac{-3 \text{ feet}}{1\text{ second}}

For g \left( x \right) over the interval \left[2, 3 \right], we must use 14 \sin \frac{3}{4}x. So g \left( 2 \right) = 13.96 and g \left( 3 \right) = 10.89, so we have \dfrac{ g\left(3 \right) - g \left( 2 \right)}{3-2} = \dfrac{10.89 - 13.96 \text{ feet}}{3-2 \text{ seconds}} = \dfrac{-3.07 \text{ feet}}{1\text{ second}}

Based on the average rate of change, Massiel's ball is falling faster 2–3 seconds after it has been hit.

Determine which part of the function g \left(x \right) shows the ball's vertical height in the sand trap.

Graph g \left( x \right), then interpret the piecewise function graphically.

After seeing the vertical height of the ball graphically, we can break the three parts of Massiel's ball trajectory into the three functions of the piecewise function. Initially, Massiel's ball follows a path similar to Ameth's as it is hit into the air. Then, it rolls into a sand trap and back out onto the green.

The part of g \left( x \right) that shows the ball's height in the sand trap is \frac{1}{2} \left( x-6 \right) ^2.

Different representations of a given function may help us to interpret contextual situations better.

Determine whose ball goes higher.

Compare the maximum values of graphs of Ameth's ball and Massiel's ball.

Ameth's ball reaches a maximum height of 12 \text{ ft} while Massiel's ball reaches a maximum height of 14 \text{ ft}. Massiel's ball goes higher.