Ch Graphs of Functions Ch Slope and Rate of Change

Ch 1.3 Graphs of Functions (Pg 39)( Ex 1) Reading Function Values from a Graph October 1987 Dow Jones Industrial Average Dependent Variable P (15, 2412) Q (20, 1726) Time Independent Variable f(15) = 2412 f(20) = 1726 The Dow-Jones Industrial Average value of the stock prices is given as a function of time during 8 days from October 15 to October 22 Input variable Output variable

Graph of a function ( pg 39) The point ( a, b) lies on the graph of the function f if and only if f (a) = b Functions and coordinates Each point on the graph of the function f has coordinates ( x, f(x)) for some value of x

Finding Coordinates with a Graphing Calculator Pg = 41 Press Y 1 enter Press 2 nd and Table Press Graph and then press, Trace and enter “Bug” begins flashing on the display. The coordinates of the bug appear at the bottom of the display.Use the left and right arrows to move the bug along the graph Graph the equation Y = -2.6x – 5.4 X min = -5, X max = 4.4, Y min = - 20, Y max = 15

Vertical Line Test ( pg - 43) A graph represents a function if and only if every vertical line intersects the graph in at most one point Function Not a function Go through all example 4 ( pg ) Two points One point

Graphical Solution of Inequalities (Pg – 45, 46) Consider the inequality 285 – 15x > 150 x – 15x y = 285 – 15x The solution is x< 9

t f(f) a) Find f(-1) and f(3) The points (-1,3) and (3,6) lie on the graph so f(-1) = 3 and f(3) = 6 b) For what value(s) of t is f(t) = 5? The points (0,5) and (4,5) lie on the graph so f(t) = 5 when t = 0 and t = 4 c) Find the intercepts of the graph. List the function values given by the intercepts The t-intercept is ( -2, 0) and the f- intercept is ( 0, 5) ; f(-2) = 0, f(0) = 5 d) Find the maximum and minimum values of f(t) The highest point is (3, 6) and the lowest is ( -4, -1), so f(t) has a maximum value of 6 and a Minimum value of – 1 e) For what value(s) of t does f take on its maximum and minimum values? The maximum occurs for t = 3 The minimum occurs for t = - 4 f) On what intervals is the function increasing ? Decreasing ? The function increasing on the interval ( - 4, 3 ) and decreasing on the interval ( 3, 5 ) Ex 1.3 No 4 (pg 49) (-1,3) (3, 6) (0,5) (4, 5) (-2, 0) f Highest point (- 4, -1) Lowest point x- intercept y-intercept

No. 13. Make a table of values and sketch a graph ( Use calculator) Pg 51 Enter Y 1 Enter the values in window Hit 2 nd and table Hit Graph

No. 35 ( Pg 55) Graph y 1 = 0.5x 3 – 4x Estimate the coordinates of the turning point ( Increasing and decreasing or vice versa and write equation of the form F(a) = b for each turning point Turning points are approximately ( -1.6, 4.352) and ( 1.6, ) And equations are F ( - 1.6) = F(1.6) = Enter Y1 Enter Window Hit Graph ( -1.6, 4.352) ( 1.6, 4.352)

1.4 Measuring Steepness ( pg 57) Which path is more strenuous ? 5 ft 2 ft Steepness measures how sharply the altitude increases.. To compare the steepness of two inclined paths, we compute the ratio of change in horizontal distance for each path

1.4 Slope (Pg 59) Definition of Slope: The slope of a line is the ratio Change in y- Coordinate Change in x- coordinate A B Slope = Change in y-coordinate = = 1 Change in x- coordinate 4 – 2 2

Notation for Slope (Pg 60) x y y x m =, where x is not equal to zero The slope of line measures the rate of change of the output variable with respect to the input variable Slope of a line is given by Change in y coordinate Change in x coordinate

Significance of the slope (Ex 6, Pg 63) The distance in miles traveled by a big-rig truck driver after t hours on the road. Compute the slope and what does the slope tell us ? t t = 2 D = 100 H (4, 200) G (2, 100) D Change in distance 100miles = = = 50 miles per hour T Change in time 2 hours Slope m = No of hours Distance in miles traveled The slope represents the trucker’s average speed or velocity

Formula for Slope two point slope form (Pg 64) Slope Formula m = y 2 – y 1 9 -(-6) x 2 – x 1 = = 5 = m = = y x P 1 (2, 9) P 2 (7, -6) The slope of the line passing through the points P 1 (x 1, y 1) and P 2 ( x 2, y 2 ) is given by y 2 – y 1 x 2 – x 1 x2x2 = x 1

Slope formula in Function Notation ( Pg 64 ) m = y 2 – y 1 f(x 2 ) – f(x 1 ), x 2 = x 1 x 2 – x 1 = x 2 - x 1

Ex 1.4 ( Pg = 67) No 11. a) Graph each line by the intercept method b) Use the intercepts to compute the slope 2y + 6x = -18 Set x = 0 2y + 6(0) = -18 2y = -18 y = -9 The y-intercept is ( 0,-9) Set y = 0 2( 0) + 6(x) = -18 6x = -18 x = -3 The x- intercept is ( -3, 0) b) Slope m = 0 –(-9) = 9 = - 3 (Use Slope formula ) -3 – (-3, 0) (0, -9) x- Intercept y-intercept

No. 34 The graph shows the amount of garbage, G (in tons), that has been deposited at a dump site t years after new regulations go into effect a) Choose two points and compute the slope of the graph ( including units ) b) Explain what the slope measures in the context of the problem (0, 25) ( 10, 150) Slope m = 150 – 25 = 125 = 12.5 tons per year 10 – 0 10

Evaluate the function at x = a and x = b, and find the slope of the line segment joining the two corresponding points on the graph, illustrate the line segment on a graph of the function No 55 h(x)= 4 a) a = 0, b = 6 b) a = -1, b = 2 x + 2 h(a) = h(0) = 4 = h(b) = h(6) = 4 = m = h(b) – h(a) = ½ - 2 = -3/2 = -1/4 b – a 6 – 0 6 ( 0, 2) (6, ½) h(a) = h(-1) = 4 = 4 (-1) + 2 h(b) = h(2) = 4 = m = h(b) – h(a) = = -3 = - 1 b – a 2 – (-1) 3 ( -1, 4) (2, 1)