Warm-up (8 min.) Find the domain and range of of f(x) = . State whether the function is odd, even, or neither. Confirm algebraically. f(x) = 2x4 b. g(x) = 2x3 – 3x c. h(x) = Find all horizontal and vertical asymptotes.

1.3 Twelve Basic Functions 1.5 Transformations of Functions After today’s lesson you will be able to: Recognize graphs of twelve basic functions Determine domains of functions related to the twelve basic functions Combine the twelve basic functions in various ways to create new functions To transform the basic functions to describe other functions

The Linear (Identity) Function f(x) = x Domain and Range: All real numbers (-,) Always continuous and Always increasing (-,) Unbounded Symmetry through the origin (odd function) No asymptotes nor extrema End behavior: How can you the identity function to describe g(x) = -x?

The Quadratic (Squaring) Function f(x) = x2 Note: Graph called parabola. Domain: All real numbers (-,) Range: All non-negative real numbers [0,) Always continuous (-,) Decreasing: (-, 0) Increasing: (0,) Bounded below at (0,0). Symmetry through the y-axis (even function) No asymptotes End behavior: Absolute minimum at (0,0) How can you use the squaring function to describe g(x) = x2 – 2?

The Cubic (Cubing) Function f(x) = x3 Note: The graph changes curvature at the origin which is called a point of inflection. Domain and Range: All real numbers (-,) Always continuous (-,) Always Increasing: (-,) Unbounded Symmetry through the origin (odd function) No asymptotes and no extrema End behavior: Describe the graph of h(x) = (x+3)3 using f(x)?

The Square Root Function f(x) = x Graph is top-half of a sideways parabola. Domain & range: All non-negative real numbers [0,) Always continuous and always increasing: [0,) Bounded below at (0,0). No symmetry and No asymptotes End behavior: Absolute minimum at (0,0) Use the square root function to describe g(x) = x-2

The Common Logarithmic Function f(x) = log x Domain: All positive real numbers (0,) Range: All real numbers (-,) Always continuous and always increasing: (-,) Bounded on the left by x = 0 (y-axis) Note: Inverse of exponential function f(x) =10x. No symmetry and no extrema End behavior: How could you use the log x function to describe g(x) = log (x – 2) + 3?

The Natural Logarithmic Function f(x) = ln x Used to describe many real-life phenomena included intensity of earthquakes (Richter scale). Domain: All positive real numbers (0,) Range: All real numbers (-,) Always continuous and always increasing: (-,) Bounded on the left by x = 0 (y-axis) Note: Inverse of exponential function f(x) =ex. No symmetry and no extrema End behavior: Where are the vertical and horizontal asymptotes of g(x) = ln (x + 1) -2?

The Exponential Function f(x) = ex Graph is half of a hyperbola. Value of e  2.718. Domain: All real numbers (-,) Range: All positive real numbers (0,) Always continuous and always increasing: (-,) Bounded below by y = 0 (x-axis) Note: Inverse of exponential function f(x) = ln x. No symmetry and no extrema End behavior: Use the exponential function to describe g(x) = -ex.

The Absolute Value Function f(x) = |x| Note: V-shaped graph Domain: All real numbers (-,) Range: All non-negative real numbers [0,) Always continuous (-,) Decreasing: (-, 0) Increasing: (0,) Bounded below at (0,0). Symmetry through the y-axis (even function) No asymptotes End behavior: Absolute minimum at (0,0) Use the graph of the absolute value function to describe the graph of g(x) = |x | + 2 .

The Reciprocal Function f(x) = 1/x Note: Graph called a hyperbola Domain & Range: All real numbers  0. (-,0)(0,) Infinite discontinuity at x = 0 Unbounded Symmetry through the origin (odd function) V.A.: x = 0 (y-axis) H.A. y = 0 (x-axis) End behavior: No extrema Write an equation for g(x) which transforms the reciprocal function 2 units to the left?

The Greatest Integer (Step) function f(x) = [x] = int (x) Also called the “step” function. Domain: All real numbers (-,) Range: All integers Discontinuous at each integer (jump discontinuity) Constant between integer values Unbounded and no symmetry No asymptotes and no extrema End behavior:

The Sine Function f(x) = sin x Note: Graph is a continuous wave Domain: All real numbers (-,) Range: [-1,1] Always continuous (-,) Increasing: (0, p/2)(3p/2, 2p) Decreasing: (p/2,3p/2) Note: Periodic function (repeats every 2p units). Bounded No asymptotes but symmetry through the origin (odd function) Local Maximum: p/2 + kp, where k is an odd integer Local Minimum: 3p/2 + kp, where k is an integer Describe the change in the graph of f(x) = sin x to create the graph of g(x) = 2 sin x.

The Cosine Function f(x) = cos x Note: Graph is a continuous wave Domain: All real numbers (-,) Range: [-1,1] Always continuous (-,) Decreasing: (0, p) Increasing: (p,2p) Note: Periodic function (repeats every 2p units). Bounded No asymptotes but symmetry through the y-axis (even function) Local Maximum: kp, where k is an even integer Local Minimum: kp/2, where k is an odd integer Write the equation of a cosine function with maximum of 5 and minimum of -5 with no horizontal shift.

The Logistic Function f(x) = Note: Model for many applications of biology and business. Domain: All real numbers (-,) Range: (0,1) Always continuous and always increasing (-,) Bounded (below with x-axis (y = 0 and above by y = 1) Symmetry through the point (0,1/2) No extrema H.A.: y = 0 and y = 1 End behavior

Infinite Discontinuity Ex1 Which of the basic functions are continuous? For the functions that are discontinuous, identify as infinite or jump. Continuous Jump Discontinuity Infinite Discontinuity

Ex 2 Which of the basic functions have symmetry? Describe each. Symmetry in x-axis y-axis (even) Symmetry in origin (odd)

Exploration: Looking for Asymptotes Two of the basic functions have vertical asymptotes at x = 0. Which two? Form a new function by adding these functions together. Does the new function have a vertical asymptote at x = 0? Three of the basic functions have horizontal asymptotes at y = 0. which three? Form a new function by adding these functions together. Does the new function have a horizontal asymptote y = 0? Graph f(x) = 1/x, g(x) = 1/(2x2-x), and h(x) = f(x) + g(x). Does h(x) have a vertical asymptote at x = 0? Explain.

Piecewise-defined Functions Which of the twelve basic functions has the following piecewise definition over separate intervals of its domain? x if x  0 f(x) = -x if x < 0

Ex 3 Use the basic functions from this lesson to construct a piecewise definition for the function shown. Is your function continuous?

Ex4 To what basic functions does w(x) = x3 – 2x2 + x relate? Describe the behavior of the function above.

Tonight’s Assignment Complete 1.3 & 1.5 Assignments from Unit 1 outline Unit 1 Test will be Friday, Feb. 5th