Precalculus Chapter 2 Section 1 Linear & Quadratic Functions & Modeling

What You’ll Learn About…And Why Polynomial Functions Linear Functions and Their Graphs Average Rate of Change Linear Correlation and Modeling Quadratic Functions and Their Graphs Applications of Quadratic Functions Many Business and Economic Problems are Modeled by Linear Functions. Quadratics Model Particle Motion in Space.

Polynomial Functions Definition: Let n be a nonnegative integer and let a0, a1, a2, …, an-1, an be real numbers with an ≠ 0. The function given by 𝑓 𝑥 = 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +⋯+ 𝑎 2 𝑥 2 + 𝑎 1 𝑥+ 𝑎 0 is a polynomial function of degree n. The leading coefficient is 𝑎 𝑛 . The zero function 𝑓 𝑥 =0 is a polynomial function. It has no degree and no leading coefficient.

Identifying Polynomial Functions Which of the following are polynomial functions? For those that are polynomial functions, state the degree and leading coefficient. If not, why? 𝑓 𝑥 =4 𝑥 3 −5𝑥− 1 2 𝑓 𝑥 =6 𝑥 −4 +7 𝑓 𝑥 = 9 𝑥 4 +16 𝑥 2 𝑓 𝑥 =15𝑥−2 𝑥 4

Polynomial Functions of No or Low Degree Name Form Degree Zero Function 𝑓 𝑥 =0 Undefined Constant Function 𝑓 𝑥 =𝑎 (𝑎 ≠0) Linear Function 𝑓 𝑥 =𝑎𝑥+𝑏 (𝑎≠0) 1 Quadratic Function 𝑓 𝑥 =𝑎 𝑥 2 +𝑏𝑥+𝑐 (𝑎≠0) 2 Cubic Function 𝑓 𝑥 =𝑎 𝑥 3 +𝑏 𝑥 2 +𝑐𝑥+𝑑 (𝑎≠0) 3

Linear Functions and Their Graphs If we use m for the leading coefficient instead of a and let y = f(x), this equation becomes the familiar slope-intercept form of a line: 𝑦=𝑚𝑥+𝑏. Surprising Fact: Not all lines in the Cartesian Plane are graphs of linear functions. You should be able to find the equation of a line given: Two points A point and slope A slope and y-intercept

Average Rate of Change The average rate of change of a function between x = a and x = b, a ≠ b, is ∆𝑦 ∆𝑥 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦−𝑣𝑎𝑙𝑢𝑒𝑠 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥−𝑣𝑎𝑙𝑢𝑒𝑠 = 𝑓 𝑏 −𝑓(𝑎) 𝑏 −𝑎 Constant Rate of Change: A function defined on all real numbers is a linear function if and only if it has a constant nonzero average of change between any 2 points on its graph. All rates are ratios, whether expressed as miles per hour, dollars per year, or even rise over run.

Characterizing a Linear Function Point of View Characterization Verbal Polynomial of degree 1 Algebraic f(x) = mx + b (m ≠ 0) Graphical Slant line with slope m and y-intercept b Analytical Function with constant nonzero rate of change m; f is increasing if x > 0, decreasing if x < 0; Initial value of the function = f(0) = b

Modeling Linear Functions 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. What is the rate of change of the value of the building? 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. Evaluate v(0) and v(16). Solve v(t) = 39,000.

Linear Correlation and Modeling See page 175 for identifying the direction and strength of correlation. For your TI Calculators, you can turn on the Diagnostics to identify the correlation coefficient by doing the following: Go to the Catalog by Pressing 2nd then 0; scroll down to DIAGNOSTICSON, press ENTER twice. This should turn on the correlation coefficient when you do a Regression Analysis. Do the regression analysis of the data on page 175 and answer the questions on page 176.

Quadratic Functions and Their Graphs This is a polynomial function of degree 2. Its graph is a parabola. It is an upward-opening parabola for a > 0, and it is a downward-opening parabola for a < 0. The vertex is the turning point of the parabola and it determines a line of symmetry called the axis of symmetry. The vertex is always the highest or lowest point of the function. The vertex form of a quadratic is 𝒇 𝒙 =𝒂 (𝒙−𝒉) 𝟐 +𝒌 Axis x = h, where h = -b/(2a) and k = c – ah2

Characterizing a Quadratic Function Point of View Characterization Verbal Polynomial of degree 2 Algebraic f(x) = ax2 + bx + c or f(x) = a(x – h)2 + k (a ≠ 0) Graphical Parabola with vertex (h, k) and axis x = h; Opens upward if a > 0, opens downward if a < 0; Initial value = y-intercept = f(0) = c 𝑥−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎

Homework Pages 182 – 185; # 1 – 11 odds, # 15 – 50 by multiples of 5, # 53, 54, 55, 61, 63, 65, 67, 68