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Explain when to use a linear function model.
Use for situations with a constant rate of change.
Explain when to use a quadratic function model.
Use for situations with a changing rate of change or symmetrical shapes with a max/min.
Explain when to use polynomial functions.
Use for complex scenarios with multiple zeros, maxima, or minima.
Explain when to use piecewise functions.
Use for situations with varying behavior over different intervals.
Why are assumptions important in function modeling?
Models assume certain conditions are consistent, affecting the model's accuracy.
Why are domain restrictions important in function modeling?
They ensure the model is valid within real-world or mathematical constraints.
Why are range restrictions important in function modeling?
They ensure the model's output is realistic and meaningful.
What does the degree of a polynomial tell you?
The degree indicates the complexity and behavior of the polynomial, including the number of possible turning points.
What real-world scenarios can be modeled by quadratic functions?
Projectile motion, roller coaster heights, parabolic antennas, stock prices, pendulum motion, stress distribution, crop yield, and reaction rates.
What are some common assumptions made when using function models?
Conservation of energy, direct proportionality between quantities, and consistent environmental conditions.
How do you find the linear model given two data points?
1. Calculate the slope (m). 2. Use one point and the slope to find the y-intercept (b). 3. Write the equation y = mx + b.
How do you determine the domain restriction for y = 1/x?
Recognize that division by zero is undefined, therefore x ≠ 0.
How do you determine the fixed cost from the cost function C(x) = 0.1x² + 5x + 100?
Evaluate C(0) to find the cost when no items are produced. In this case, C(0) = 100.
How do you find the value of a piecewise function at a specific point?
Determine which interval the point falls into, then use the corresponding equation for that interval.
What is the general form of a linear function?
$y = mx + b$
What is the general form of a quadratic function?
$y = ax^2 + bx + c$
What is the general form of a polynomial function?
$y = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$
How do you calculate the slope (m) given two points (x₁, y₁) and (x₂, y₂)?
$m = \frac{y_2 - y_1}{x_2 - x_1}$