Analyze families of functions Objective: Analyze families of functions J Reasons Thursday, January 03, 2019

f (x) = mx + b m and b are real numbers Linear Functions f (x) = mx + b m and b are real numbers Domain: All real numbers Range: All real numbers Graph is a non-vertical line with slope m and y-intercept b J Reasons Thursday, January 03, 2019

Linear Functions d) Increasing if m > 0 decreasing if m < 0 constant if m = 0 J Reasons Thursday, January 03, 2019

f (x) = b b is a real number Constant Function f (x) = b b is a real number Domain: All real numbers Range: b Graph is a horizontal line with y-intercept b An even function whose graph is constant over its domain. J Reasons Thursday, January 03, 2019

y b x f (x) = b J Reasons Thursday, January 03, 2019

Domain : All real numbers Range: All real numbers Identity Function f (x) = x Domain : All real numbers Range: All real numbers Graph is a line with slope 1 and y-intercept 0 An odd function whose graph is increasing over its domain. J Reasons Thursday, January 03, 2019

y b (1, 1) x (-1, -1) f (x) = x J Reasons Thursday, January 03, 2019

Domain: All real numbers Range: Nonnegative real numbers Polynomial Functions Quadratic Function: Domain: All real numbers Range: Nonnegative real numbers Graph is a parabola whose intercept is (0, 0) An even function whose graph is decreasing for x < 0 and increasing for x > 0. f (x) = x2 J Reasons Thursday, January 03, 2019

J Reasons Thursday, January 03, 2019

Domain: All real numbers Range: All real numbers Cubic Function f (x) = x3 Domain: All real numbers Range: All real numbers The intercept of the graph is (0, 0) An odd function whose graph is increasing over its domain J Reasons Thursday, January 03, 2019

J Reasons Thursday, January 03, 2019

Domain: All nonnegative real numbers Square Root Function Domain: All nonnegative real numbers Range: All nonnegative real numbers The intercept of the graph is (0, 0) Neither even nor odd function whose graph is increasing for x > 0 J Reasons Thursday, January 03, 2019

J Reasons Thursday, January 03, 2019

Domain: All real numbers Range: All nonnegative real numbers Absolute Value Function Domain: All real numbers Range: All nonnegative real numbers The intercept of the graph is (0, 0) An even function whose graph is decreasing for x < 0 and increasing for x > 0 J Reasons Thursday, January 03, 2019

J Reasons Thursday, January 03, 2019

Domain: All real numbers Range: The set of integers Greatest-integer Function Domain: All real numbers Range: The set of integers y-intercept: 0; x-intercepts: [0, 1) Neither even nor odd function whose graph is constant [k, k + 1) for k an integer. J Reasons Thursday, January 03, 2019

y 3 x -2 2 J Reasons Thursday, January 03, 2019

6) Rational Function: R(x) = where p and q are polynomial functions and q is not the zero polynomial. Domain is all real numbers except those for which the denominator q is 0 If, as x   or as x  -, the values of R(x) approach some fixed number L, then the line y = L is a horizontal asymptote of the graph of R. If, as x approaches some number c, the values |R(x)|  , then the line x = c is a vertical asymptote of the graph of R. If an asymptote is neither horizontal nor vertical it is called oblique (slant). J Reasons Thursday, January 03, 2019

Domain is all real numbers. 7) Exponential Function y = bx where base b is a positive real number and the exponent is a variable. For b> 1 Domain is all real numbers. Range is all real numbers greater than zero. The x-intercept is none The y-intercept is (0,1) The behavior is continuous, one-to-one, and increasing. Horizontal asymptote is y = 0 (negative x-axis). Vertical asymptote is none J Reasons Thursday, January 03, 2019

For 0<b<1 Domain is all real numbers Exponential Function y = bx where base b is a positive real number and the exponent is a variable. For 0<b<1 Domain is all real numbers Range is all real numbers greater than zero The x-intercept is none The y-intercept is (0,1) The behavior is continuous, one-to-one, and increasing. Horizontal asymptote is y = 0 (positive x-axis). Vertical asymptote is none. J Reasons Thursday, January 03, 2019

8) Logarithmic Function y = logb x where b > 0 and b ≠1 Domain is all real numbers greater than zero. Range is all real numbers. The x-intercept is (1,0). The y-intercept is none Horizontal asymptote is none Vertical asymptote is x = 0 The behavior is continuous, one-to-one, and increasing for x >0 J Reasons Thursday, January 03, 2019

Natural logarithmic functions y = ln x Domain is all real numbers greater than zero Range is all real numbers The x-intercept is (1, 0). The y-intercept is none Horizontal asymptote is none. Vertical asymptote is x = 0 (y-axis). The behavior continuous, one-to-one, and increasing for x >0. J Reasons Thursday, January 03, 2019

Trigonometric Functions Trigonometric Functions: sine, cosine, tangent, cotangent, secant, and cosecant.  J Reasons Thursday, January 03, 2019

Piecewise functions Piecewise functions is a function f(x) defined piecewise, that is f(x) is given by different expressions on various intervals. Thursday, January 03, 2019 J Reasons

Family of functions (graphs) Group of graphs that displays one or more similar characteristics. y = x 2 y =3 x 2 y = x 2 + 2x + 3 J Reasons Thursday, January 03, 2019

Parent graph basic graph that is transformed to create other members in a family of graphs. y = x 2 y =3 x 2 y = x 2 + 2x + 3 J Reasons Thursday, January 03, 2019