10/10 Modeling Functions and Data Free Online FlashCards

What is a function model?

A function model is an equation that approximates the relationship between an input variable and an output variable.

When is a linear model appropriate?

Use a linear model when data show a constant rate of change.

What do polynomial degree and leading coefficient indicate?

In a polynomial model, the degree is the highest power of x, and the leading coefficient controls end behavior.

How does b classify an exponential model?

For f(x)=ab^x, b>1 indicates growth, while 0<b<1 indicates decay.

What data pattern suggests an exponential model?

An exponential model is appropriate when equal increases in the input produce approximately equal ratios in the output.

What domain restriction applies to a+b ln(x−h)?

A logarithmic model has domain x>h because its logarithm requires x−h>0.

What do A, B, and D represent in a sinusoidal model?

For f(t)=A sin(Bt−C)+D, amplitude is |A|, period is 2π/|B|, and midline is y=D.

How are midline and amplitude calculated from extrema?

The midline is D=(maximum+minimum)/2, and the amplitude is |A|=(maximum−minimum)/2.

What determines excluded inputs in a rational model?

In a rational function f(x)=p(x)/q(x), inputs satisfying q(x)=0 are excluded from the domain.

What is a piecewise model?

A piecewise function uses different formulas on different parts of its domain, usually because a rule changes at a threshold.

How is a residual calculated?

The residual is e_i=y_i−ŷ_i: observed value minus model prediction.

What does least-squares regression minimize?

Least squares minimizes SSE=Σ(y_i−ŷ_i)^2, the sum of squared residuals.