1. Chapter 2 Linear Functions and Models

2. 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

3. …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

4. Representation of Function x (yar ds 1234567 y(fe et) 36912151821 0 4 8 12 16 20 24 20 16 12 8 4 y= 3x Table of Values Graph x y Numerically Graphically

5. Diagrammatic Representation (pg 76) Function Not a function 123123 3 6 9 1212 456456 (1, 3), (2, 6), (3, 9) x y 123123 4545 (1,4), (2, 4), (3, 5) (1, 4), (2, 5), (2, 6)

6. Domain and Range Graphically (Pg 80) -3 -2 -1 0 1 2 3 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 3 2 1 Range R includes all y – values satisfying 0 < y < 3 x Domain D includes all x values Satisfying –3 < x < 3 y

7. Vertical Line Test ( pg 83) -4 -3 -2 -1 0 1 2 3 4 5 4 3 2 1 -2 -3 -4 -5 (-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

8. …Continued -3 -2 -1 0 1 2 3 (-1, 1) (1, -1) 4 3 2 1 -2 -3 Not a function

9. Using Technology [ - 10, 10, 1] by [ - 10, 10, 1] Hit Y and enter 2x - 1 x y -3 0 11 23 Graph of y = 2x - 1 Hit 2 nd and hit table and enter data

10. 2.2 Linear Function A function f represented by f(x) = ax + b, where a and b are constants, is a linear function. 0 1 2 3 4 5 6 100 90 80 70 60 100 90 80 70 60 Scatter Plot A Linear Function f(x) = 2x + 80

11. Modeling data with Linear Functions Pg ( 97) Example 7 1500 1250 1000 750 500 250 0 48 12 16 20 x Credits Cost (dollars) Symbolic Representation f(x) = 80x + 50 Numerical representation 4 8 12 16 $ 370 $ 690 $1010 $1330

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

13. 2.3 The Slope of a line 1 2 3 4 5 6 x Gasoline (gallons ) Cost of Gasoline Every 2 gallons purchased the cost increases by $3 8765432187654321 Run = 2 Rise = 3 Slope = Rise = 3 Run 2 Y Cost (dollars)

14. 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

15. m = - ½ < 0 m = 2 > 0 m = 0 m is undefined Positive slope Negative slope Zero slope Undefined slope -4 -2 1 2 3 4 4 3 2 1 -2 -3 2 4 3 2 1 0 -2 - 4 -2 1 2 2 (Pg 107)

16. - 4 - 3 - 2 1 0 1 2 3 4 4 3 2 1 - 2 -3 - 4 ( 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

17. 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

18. Example – 4 (pg 109) -3 -2 -1 1 2 y = ½ x + 2 y = ½ x y = ½ x - 2 3 2 1 -2 -3

19. Analyzing Growth in Walmart Example 10 0 1999 2003 2007 3.0 2.5 2.0 1.5 1.0 0.5 m1 = 1.1 – 0.7 = 0.2 m2 = 1.4 - 1.1 = 0.1 and 1999 – 1997 2002 – 1999 m3 = 2.2 - 1.4 = 0.16 2007 - 2002 Years Employees (millions) Year 1997 1999 2002 2007 Employees 0.7 1.1 1.4 2.2 m1 m2 m3 Average increase rate

20. 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)

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

22. …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

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