Today in PreCalculus Notes Polynomial Functions Linear Functions

Today in Pre-Calculus • Notes: – Polynomial Functions – Linear Functions • Go over test • Homework

Polynomial Functions Let n be a nonnegative integer and let a 0, a 1, a 2, …. , an-1, an be real numbers with an ≠ 0. The function given by f(x) = anxn + an-1 xn-1 +…+a 2 x 2 + a 1 x + a 0 is a polynomial function of degree n. The leading coefficient is an. Simply put: Exponents can’t be negative or fractions, the highest power in the polynomial is the degree of the function, the coefficient of this term is the leading coefficient.

Examples Which of the following are polynomial functions? For those that are polynomial functions, state the degree and leading coefficient. For those that are not, explain why not?

Polynomial Functions of No or Low Degree Name Zero Function Constant Function Linear Function Quadratic Function Form f(x) = 0 f(x) = a (a≠ 0) f(x) = ax + b (a≠ 0) f(x)=ax 2+bx+c (a≠ 0) Degree Undefined 0 1 2

Average Rate of Change (a. k. a slope) Theorem: A function defined on all real numbers is a linear function iff it has a constant nonzero average rate of change between any two points on its graph.

Examples 1. Find the slope of the line of f such that f(3) = 4 and f( -1) = -8 2. Find the equation of the line of f. y – 4 = 3(x – 3) y – 4 = 3 x – 9 y = 3 x – 5

Modeling Pg 173: Exploration 1: Camelot Apartments bought a $50, 000 building and for tax purposes are depreciating it $2000 per year over a 25 yr period using straight-line depreciation. 1. What is the rate of change of the value of the building? 2. Write an equation for the value v(t) of the building as a linear function of the time t since the building was placed in service. 3. Evaluate v(0) and v(16 4. Solve v(t)=39, 000

Homework • Test was out of 53 points • Pg. 182: 1 -10 all, 52, 56 • Bring textbooks tomorrow and Friday.

Modeling Pg 173: Exploration 1: . 1. What is the rate of change of the value of the building? -$2, 000 2. Write an equation for the value v(t) of the building as a linear function of the time t since the building was placed in service. v(t) = -$2, 000 t + $50, 000 3. Evaluate v(0) $50, 000 and v(16) $18, 000 4. Solve v(t)=39, 000=-$2, 000 t + $50, 000 -$11, 000 = -$2, 000 t t=5. 5 years