Back to Exercise: Fit a function to data

Exercises: Fit a Function to Data

Work through each section in order. When choosing a model family, justify
from BOTH the scatter plot's form and the context's mechanism. When using a
fitted function, substitute or solve carefully and state every answer with
units. Remember: a fitted model captures the overall trend — it need not pass
through every point — and predictions far outside the data range
(extrapolation) are not as trustworthy as predictions inside it.

Grade 10·23 problems·~35 min·Common Core Math - HS Statistics and Probability·standard·hss-id-b-6a

Printable layout

Warm-Up: Function Families and Evaluation

These problems review skills you already have from Algebra 1.

A regression on real data produces the equation $y = 2.1x + 15$ , and it tracks the scatter plot's trend closely. Which statement best describes what this fitted function is?

A useful approximation of the relationship over the observed data range.

The exact, true law that the data must obey at every point.

A rule guaranteeing every data point lies exactly on the line.

A probability that describes the chance of each outcome.

Which function family produces a graph that rises to a turning point and then falls (a U-shape or upside-down U)?

Quadratic

Linear

Exponential

Constant

A context supplies the linear function $y = 0.10x + 20$ for the monthly cost $y$ (in dollars) of a phone plan after $x$ minutes of calls. Evaluate the function at $x = 200$ minutes. Give the cost in dollars.

Fluency Practice: Choosing the Model Family

For each scatter plot, decide which model family fits its form.

$A scatter plot whose points rise in a roughly straight line from lower left to upper right.$

The scatter plot below shows data that increases by roughly the same amount for each equal step in $x$ — the points fall close to a straight line. Which model family best fits this data?

Linear

Quadratic

Exponential

No function can fit this data.

$A scatter plot whose points rise to a peak near the middle and then fall, forming an upside-down U.$

The scatter plot below shows a ball's height versus time after a throw: the points rise to a peak and then fall back down. Which model family best fits this data?

Quadratic

Linear

Exponential

Linear, because height versus time is always a line.

$A scatter plot whose points stay low at first then curve steeply upward to the right.$

The scatter plot below shows a bacteria count versus hours: the points stay low at first, then curve steeply upward, roughly multiplying each hour. Which model family best fits this data?

Exponential

Linear

Quadratic

No model fits, because the curve misses some of the points.

A bacteria population is modeled by the exponential function $y = 50 \cdot 2^{x}$ , where $x$ is the number of hours and $y$ is the count. Predict the count after $x = 3$ hours.

Using the fitted phone-plan function $y = 0.10x + 20$ (cost $y$ in dollars for $x$ minutes), how many minutes $x$ give a bill of $y = 35$ dollars? Solve for $x$ .

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