Chapter 2 Linear Functions and Models

Ch 2.1 Functions and Their Representations A function is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value. Input x Output y Function f (x, y) Input Output

…continued y is a function of x because the output y is determined by and depends on the input x. As a result, y is called the dependent variable and x is the independent variable To emphasize that y is a function of x, we use the notation y = f(x) and is called a function notation. y = f(x) Output Input 14 = f(5) A function f forms a relation between inputs x and outputs y that can be represented verbally (Words), numerically (Table of values), Symbolically (Formula), and graphically (Graph). y x

Representation of Function x (yar ds y(fe et) y= 3x Table of Values Graph x y Numerically Graphically

Diagrammatic Representation (pg 76) Function Not a function (1, 3), (2, 6), (3, 9) x y (1,4), (2, 4), (3, 5) (1, 4), (2, 5), (2, 6)

Domain and Range Graphically (Pg 80) Domain Range The domain of f is the set of all x- values, and the range of f is the set of all y-values Range R includes all y – values satisfying 0 < y < 3 x Domain D includes all x values Satisfying –3 < x < 3 y

Vertical Line Test ( pg 83) (-1, 1) (-1, -1) If each vertical line intersects the graph at most once, then it is a graph of a function Not a function

…Continued (-1, 1) (1, -1) Not a function

Using Technology [ - 10, 10, 1] by [ - 10, 10, 1] Hit Y and enter 2x - 1 x y Graph of y = 2x - 1 Hit 2 nd and hit table and enter data

2.2 Linear Function A function f represented by f(x) = ax + b, where a and b are constants, is a linear function Scatter Plot A Linear Function f(x) = 2x + 80

Modeling data with Linear Functions Pg ( 97) Example x Credits Cost (dollars) Symbolic Representation f(x) = 80x + 50 Numerical representation $ 370 $ 690 $1010 $1330

Using a graphing calculator Example 5 (pg 95) Give a numerical and graphical representation f(x) = 1 x Numerical representation Y1 =.5x – 2 starting x = -3 Graphical representation [ -10, 10, 1] by [-10, 10, 1]

2.3 The Slope of a line x Gasoline (gallons ) Cost of Gasoline Every 2 gallons purchased the cost increases by $ Run = 2 Rise = 3 Slope = Rise = 3 Run 2 Y Cost (dollars)

2.3 Slope (Pg 106) The Slope m of the line passing through the points (x 1 y 1 ) and (x 2, y 2 ) is m= y 2 –y 1 / x 2 –x 1 Where x 1 = x 2. That is, slope equals rise over run. y 2 (x 2, y 2 ) y 2 –y 1 y 1 (x 1, y 1 ) x 2 –x 1 rise y 2 - y 1 m = run = x 2 - x 1 Run Rise

m = - ½ < 0 m = 2 > 0 m = 0 m is undefined Positive slope Negative slope Zero slope Undefined slope (Pg 107)

( 3, 2) (0, 4) Example 2 - Sketch a line passing through the point (0, 4) and having slope - 2/3 y - values decrease 2 units each times x- values increase by 3 (0 + 3, 4 – 2) = (3, 2) ( 0, 4) Rise = -2

Slope-Intercept Form ( pg 109) The line with slope m and y = intercept b is given by y= mx + b The slope- Intercept form of a line

Example – 4 (pg 109) y = ½ x + 2 y = ½ x y = ½ x

Analyzing Growth in Walmart Example m1 = 1.1 – 0.7 = 0.2 m2 = = 0.1 and 1999 – – 1999 m3 = = Years Employees (millions) Year Employees m1 m2 m3 Average increase rate

2.4 Point- slope form ( pg 119) The line with slope m passing through the point (x 1, y 1 ) is given by y = m ( x - x 1 ) + y 1 Or equivalently, y – y 1 = m (x –x 1 ) The point- slope form of a line (x 1, y 1 ) (x, y) x – x 1 y – y 1 m =( y – y 1) / (x – x 1)

Horizontal and Vertical Lines (pg 125) x = h b h y= b Equation of Horizontal Line Equation of vertical line x y x y

…Continued (Pg 126 – 127) Parallel Lines Two lines with the same slope are parallel. m 1 = m 2 Perpendicular Lines Two lines with nonzero slopes m 1 and m 2 are perpendicular if m 1 m 2 = -1

Pg 127 m2 = -1 m2 = - 1/2 m2 = - 1/m1m1 = 1 m1 = 2 m1 Perpendicular Lines