Ch 1.3 - Graphs of Functions
Ch 1.4 - Slope and Rate of Change
Output variabl
Dependent Variable
2500
Ch 1.3 Graphs of Functions (Pg 39)( Ex 1)
Reading Function Values from a Graph
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
P (15, 2412)
2400
Dow Jones Industrial Average
f(15) = 2412
f(20) = 1726
2300
2200
2100
2000
1900
1800
Q (20, 1726)
12 13 14 15 16 19 20 21 22 23
October 1987
Time Independent Variable
Input 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
Graph the equation
Y = -2.6x – 5.4
X min = -5, X max = 4.4,
Y min = - 20 , Y max = 15
Press Y1 enter
Press 2nd 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
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
One point
Function
Two points
Not a function
Go through all example 4 ( pg 43- 44)
Graphical Solution of Inequalities (Pg – 45, 46)
Consider the inequality 285 – 15x > 150
x
285 – 15x
0
2
4
6
8
10
12
285
255
225
195
165
135
105
y = 285 – 15x
300
200
150
100
5
9
10
25
- 100
The solution is x< 9
f
Ex 1.3
No 4 (pg 49)
Highest point
(3, 6)
y-intercept
6
(0,5) 5
f(f)
(4, 5)
4
(-1,3)
3
2
x- intercept
-5
-4
(- 4, -1)
(-2, 0)
-3
-2
1
-1
1
2
3
4
5
t
Lowest point
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 )
No. 13. Make a table of values and sketch a graph
( Use calculator) Pg 51
Enter Y1
Enter the values in window
Hit Graph
Hit 2nd and table
No. 35 ( Pg 55)
Graph y1 = 0.5x3 – 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
Enter Y1
Enter Window
Hit Graph
( -1.6, 4.352)
( 1.6, 4.352)
Turning points are approximately ( -1.6, 4.352) and ( 1.6, - 4.352)
And equations are
F ( - 1.6) = 4.352
F(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
5
4
3
2
1
A
0
2
B
3
4
Slope = Change in y-coordinate = 5 - 4 = 1
Change in x- coordinate 4 – 2 2
Notation for Slope (Pg 60)
y
Change in y coordinate
x
Change in x coordinate
Slope of a line is given by
m=
, where
y
x
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
Significance of the slope (Ex 6, Pg 63)
Distance in miles traveled
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 ?
250
H (4, 200)
200
D = 100
150
G (2, 100)
100
t=2
50
1
2
3
4
5
t
No of hours
Slope m =
D Change in distance
100miles
=
=
= 50 miles per hour
T Change in time
2 hours
The slope represents the trucker’s average speed or velocity
Formula for Slope two point slope form (Pg 64)
y
x
m=
=
y2 – y1
x2 = x
1
x2 – x1
The slope of the line passing through the points P1(x1, y1) and P2 ( x2, y2) is given by
Slope Formula
m=
y 2 – y1
x2 – x1
10
9 -(-6)
=
2-7
-15
=
5
-3
=
P1 (2, 9)
5
10
-5
P2 (7, -6)
Slope formula in Function Notation ( Pg 64 )
m = y2 – y1
x2 – x1
f(x2) – f(x1) , x2 = x1
= x -x
2
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 – 0 -3
x- Intercept
(-3, 0)
-4
-2
2
-2
-4
-6
y-intercept
-8
(0, -9)
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
200
150
( 10, 150)
100
50
(0, 25)
5
10
15
20
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
x+2
a) a = 0, b = 6
b) a = -1, b = 2
( -1, 4)
( 0, 2)
h(a) = h(0) =
(6, ½)
4
=2
0+2
h(b) = h(6) = 4 = 1
6+2
2
m = h(b) – h(a) = ½ - 2 = -3/2 = -1/4
b–a
6–0
6
(2, 1)
h(a) = h(-1) = 4
=4
(-1) + 2
h(b) = h(2) = 4 = 1
2+2
m = h(b) – h(a) = 1 - 4 = -3 = - 1
b–a
2 – (-1) 3