Week 1
Linear Functions * As you hop into a taxicab in Kuala Lumpur, the meter will immediately read RM3.30; this is the “drop” charge made when the taximeter is activated. After that initial fee, the taximeter will add RM2.40 for each kilometer the taxi drives. In this scenario, the total taxi fare depends upon the number of kilometer ridden in the taxi, and we can ask whether it is possible to model this type of scenario with a function.

As you hop into a taxicab in Kuala Lumpur, the meter will immediately read RM3.30; this is the “drop” charge made when the taximeter is activated. After that initial fee, the taximeter will add RM2.40 for each kilometer the taxi drives. In this scenario, the total taxi fare depends upon the number of kilometer ridden in the taxi, and we can ask whether it is possible to model this type of scenario with a function.

An equation whose graph is a straight line is called a linear function (y = mx + c). * Consider this Example 1:

* Using descriptive variables, we choose k for kilometers and R for Cost in Ringgit Malaysia as a function of miles: R(k). * We know for certain that R(0) = 3.30, since the RM3.30 drop charge is assessed regardless of how many kilometers are driven. * Since RM2.40 is added for each kilometer driven, then: R(1) = 3.30 + 2.40 = 5.70. * If we then drove a second kilometer, another RM2.40 would be added to the cost: R(2) = 3.30 + 2.40 +2.40 = 8.10. * If we drove a third mile, another RM2.40 would be added to the cost:
R(3) = 3.30 + 2.40 +2.40 + 2.40 = 10.50 * From this we might observe the pattern, and conclude that if k kilometers are driven,
R(k) = 3.30 + 2.40k because we start with a RM3.30 drop fee and then for each kilometer increase we add RM2.40. * Notice this equation R(k) = 3.30 + 2.40k consisted of two quantities. * The first is the fixed RM3.30 charge which does not change based on the value of the input. * The second is the RM2.40 dollars per kilometer value, which is a rate of change. In the equation this rate of change is multiplied by the input value. * Looking at this same problem in table format we can also see the cost changes by RM2.40 for every 1 kilometer increase.

k | | | | | R(k) | | | | |

* It is important here to note that in this equation, the rate of change is constant; over any interval, the rate of change is the same. * Graphing this equation R(k) = 3.30 + 2.40k, we see the shape is a line, which is how these functions get their name: linear functions
R(k)
R(k)

k k * When the number of miles is zero the cost is RM3.30, giving the point (0, 3.30) on the graph. * This is the vertical or R(k) intercept. * The graph is increasing in a straight line from left to right because for each kilometer the cost goes up by RM2.40; this rate remains consistent.

* Consider this Example 2:
Malik currently owns 200 songs in his android phone collection. Every month, he adds 15 new songs. Write a formula for the number of songs, N, in his android phone collection as a function of the number of months, m. How many songs will he own in a year?

Malik currently owns 200 songs in his android phone collection. Every month, he adds 15 new songs. Write a formula for the number of songs, N, in his android phone collection as a function of the number of months, m. How many songs will he own in a year?

* The initial value for this function is 200, since he currently owns 200 songs, so
N(0) = 200. * The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. * With this information, we can write the formula:
N(m) = 200 + 15m * N(m) is an increasing linear function * With this formula we can predict how many songs he will have in 1 year (12 months):
N(12) = ___________________ = ______ * Try this Exercise 1:
If you earn RM24,000 per year and you spend RM23,000 per year write an equation for the amount of money you save after y years, if you start with nothing.
If you earn RM24,000 per year and you spend RM23,000 per year write an equation for the amount of money you save after y years, if you start with nothing.

* Write you answer below:

“The most important thing, spend less than you earn!”
CALCULATING RATE OF CHANGE * Given two values for the input,x1 and x2, and two corresponding values for the output,y1 and y2 , or a set of points, (x1,y1) and (x2,y2) * If we wish to find a linear function that contains both points we can calculate the rate of change, m:

m=change in outputchange in input=ΔyΔx=y1-y2x1-x2

* Rate of change of a linear function is also called the slope of the line. * Note in function notation, y1 = f(x1) and y2 = f(x2), so we could equivalently write:

m=fx2-f(x1)x2-x1

* Consider Example 3:
The population of a city increased from 23,400 to 27,800 between 2002 and 2006. Find the rate of change of the population during this time span.
The population of a city increased from 23,400 to 27,800 between 2002 and 2006. Find the rate of change of the population during this time span.

* The rate of change will relate the change in population to the change in time. The population increased by 27,800 – 23,400 = 4,400 people over the 4 year time interval. * To find the rate of change, the number of people per year the population changed by: m= 4,400 people4 years=1,100 people per year * Notice that we knew the population was increasing, so we would expect our value for m to be positive. This is a quick way to check to see if your value is reasonable. * Consider Example 4:
If f(x) is a linear function, f(3) = -2 and f(8) = 1, find the rate of change.
If f(x) is a linear function, f(3) = -2 and f(8) = 1, find the rate of change.

* f(3) = -2 tells us that the input 3 corresponds with the output -2 * f(8) = 1 tells us that the input 8 corresponds with the output 1 * To find the rate of change, we divide the change in output by the change in input: m= change in outputchange in input=1-(-2)8-3=35 * If desired we could also write this as m = 0.6 * Note that it is not important which pair of values comes first in the subtractions so long as the first output value used corresponds with the first input value use * Try this Exercise 2:
Given the two points (2, 3) and (0, 4), find the rate of change. Is this function increasing or decreasing?
Given the two points (2, 3) and (0, 4), find the rate of change. Is this function increasing or decreasing?

* Consider Example 5:
Write an equation for the linear function graphed below.

Write an equation for the linear function graphed below.

* Looking at the graph, we might notice that it passes through the points (0, 7) and (4, 4). * From the first value, we know the initial value of the function is c = 7. * So in this case we will only need to calculate the rate of change: m=4-74-0= -34 * This allows us to write the equation: fx=7-34x

Finding Horizontal Intercept (y-axis) * The horizontal intercept of the function is where the graph crosses the horizontal axis. * If a function has a horizontal intercept, you can always find it by solving f(x) = 0. * Consider Example 6:
If f(x) is a linear function, f(3) = -2 and f(8) = 1, find an equation for the function
If f(x) is a linear function, f(3) = -2 and f(8) = 1, find an equation for the function

* In example 4, we computed the rate of change to be m = 35. * In this case, we do not know the initial value f(0), so we will have to solve for it. * Using the rate of change, we know the equation will have the form fx= 35x+c * Since we know when x = 3 the value of the function equal to -2, we can evaluate the function at 3. * f3= 35(3)+c (Since we know that f(3) = -2, we can substitute -2 on the left side ) * -2= 35(3)+c (This leaves us with an equation we can solve for the initial c) * c= -2-353=-195 * Combining this with the value for the rate of change, we can now write a formula for this function: fx= 35x-195
Graphs of Linear Functions * When graphing a linear function, there are two basic ways to graph it: * By plotting points (at least 2) and drawing a line through the points * Using the initial value (output when x = 0) and rate of change (slope) * Consider Example 7:
Graph by plotting points f(x) = 5 - 23x
Graph by plotting points f(x) = 5 - 23x

Plotting Points * In general, we evaluate the function at two or more inputs to find at least two points on the graph. * Usually it is best to pick input values that will “work nicely” in the equation. * In this equation, multiples of 3 will work nicely due to the 23 in the equation, and of course using x = 0 to get the vertical intercept. * Evaluating f(x) at x = 0, 3 and 6: f0=5- 230=5 f3=5- 233=3 f6=5- 236=1 * These evaluations tell us that the points (0,5) , (3,3), and (6,1) lie on the graph of the line. * Plotting these points and drawing a line through them gives us the graph
(3,3)

(3,3)

(6,1)

(6,1)

(0,5)

(0,5)

Initial value (output when x = 0) and rate of change (slope) * Remember the initial value of the function is the output when the input is zero. * So in the equation f(x) = 5 - 23x, the graph includes the point (0, 5) * On the graph, this is the vertical intercept – the point where the graph crosses the vertical axis (y-axis). * For the rate of change, it is helpful to recall that we calculated this value as: m= change in outputchange in input * From a graph of a line, this tells us that if we divide the vertical difference, or rise, of the function outputs by the horizontal difference, or run, of the inputs, we will obtain the rate of change, also called slope of the line. m= change in outputchange in input=riserun * Notice that this ratio is the same regardless of which two points we use.

m = 2/3 m = 2/3 run = 3

run = 3

rise = 2

rise = 2

Vertical intercept at (0,5)
Vertical intercept at (0,5)

Parallel and Perpendicular Lines * When two lines are graphed together, the lines will be parallel if they are increasing at the same rate. * In this case, the graphs will never cross (unless they’re the same line). * In other words, two lines are parallel if the slopes are equal (or, if both lines are vertical). * Given two linear equations f(x) = m1x + c1 and g(x) = m2x + c2. The lines will be parallel if and only if m1 = m2 * Consider Example 8:
Find a line parallel to f(x) = 6 +3x that passes through the point (3, 0)
Find a line parallel to f(x) = 6 +3x that passes through the point (3, 0)

* We know the line we’re looking for will have the same slope as the given line, m = 3. * Using this and the given point, we can solve for the new line’s vertical intercept, c: g(x) = 3x + c * Substitute point (3,0) into the g(x): g(3) = 3(3)+ c where g(3) = 0 0 = 9 + c then solve for c -9 = c * Therefore, the line we’re looking for is g(x) = 3x – 9

* If two lines are not parallel, one other interesting possibility is that the lines are perpendicular, which means the lines form a right angle (90 degree angle – a square corner) where they meet. * In this case, the slopes when multiplied together will equal -1. * Given two linear equations f(x) = m1x + c1 and g(x) = m2x + c2. * The lines will be perpendicular if m1m2 = -1, and so m1 = -1m2 * Find a line perpendicular to f(x) = 6 +3x that passes through the point (3, 0)
Find a line perpendicular to f(x) = 6 +3x that passes through the point (3, 0)
Consider Example 9:

* The given line has slope m = 3. The perpendicular line will have slope m = -13. * Using this and the given point, we can find the equation for the line: g(x) = -13x + c * Substitute point (3,0) into the g(x): g(3) = -13(3)+ c where g(3) = 0 0 = -1 + c then solve for c 1 = c * Therefore, the line we’re looking for is g(x) = -13x + 1 * Try this Exercise 3
A line passes through the points (-2, 6) and (4, 5). Find the equation of a perpendicular line that passes through the point (4, 5).
A line passes through the points (-2, 6) and (4, 5). Find the equation of a perpendicular line that passes through the point (4, 5).

Intersections of Lines * The graphs of two lines will intersect if they are not parallel. They will intersect at the point that satisfies both equations. * To find this point when the equations are given as functions, we can solve for an input value so that f(x) = g(x) * In other words, we can set the formulas for the lines equal, and solve for the input that satisfies the equation. * Find the intersection of the lines f(x) = 3x - 4 and g(x) = 5 - x
Find the intersection of the lines f(x) = 3x - 4 and g(x) = 5 - x
Consider Example 10:

* Setting f(x) = g(x) * 3x – 4 = 5 – x (Put all the ‘x’ on LHS and others on RHS) * 3x + x = 5 +4 (Solve both LHS and RHS) * 4x = 9 (Solve for x) * x = 94 * This tells us the lines intersect when the input is 94 * We can then find the output value of the intersection point by evaluating either function at this input g94=5-94=114 * These lines intersect at the point (94,114)

(94,114)
(94,114)
g(x) g(x) f(x) f(x) Tutorial 1 1. A town's population has been growing linearly. In 2003, the population was 45,000, and the population has been growing by 1700 people each year. Write an equation P(t),for the population t years after 2003.

2. A town's population has been growing linearly. In 2005, the population was 69,000, and the population has been growing by 2500 people each year. Write an equation P(t),for the population t years after 2005

3. Suria is currently 10 kilometers from home, and is walking further away at 2 kilometers per hour. Write an equation for her distance from home t hours from now.

4. A boat is 100 kilometers away from the marina, sailing directly towards it at 10 kilometers per hour. Write an equation for the distance of the boat from the marina after t hours.

5. Tariq goes to the fair with RM40. Each ride costs RM2. How much money will he have left after riding n rides?

6. At noon, a waitress notices she has RM20 in her tip jar. If she makes an average of RM0.50 from each customer, how much will she have in her tip jar if she serves n more customers during her shift?
Find the slope of the line that passes through the two given points 7. (2, 4) and (4, 10) 8. (-1,4) and (5, 2) 9. (6,11) and (-4,3) 10. (1, 5) and (4, 11) 11. (-2, 8) and (4, 6) 12. (9,10) and (-6,-12)

Find the slope of the lines graphed 13.

14.

15. Shima is walking home from a friend’s house. After 2 minutes she is 1.4 kilometers from home. Twelve minutes after leaving, she is 0.9 kilometers from home. What is her rate?

16. A gym membership with two personal training sessions costs RM125, while gym membership with 5 personal training sessions costs RM260. What is the rate for personal training sessions?

17. A city's population in the year 1960 was 287,500. In 1989 the population was 275,900. Compute the slope of the population growth (or decline) and make a statement about the population rate of change in people per year.

18. A city's population in the year 1958 was 2,113,000. In 1991 the population was 2,099,800. Compute the slope of the population growth (or decline) and make a statement about the population rate of change in people per year.

19. A phone company charges for service according to the formula: C(n) = 24 + 0.1n, where n is the number of minutes talked, and C(n) is the monthly charge, in Ringgit Malaysia. Find and interpret the rate of change and initial value.

20. Terry is skiing down a steep hill. Terry's elevation, E(t), in meters after t seconds is given by E(t) =3000 – 70t. Write a complete sentence describing Terry’s starting elevation and how it is changing over time.
Given each set of information, find a linear equation satisfying the conditions, if possible 21. f(-5)= -4, and f(5) = 2 22. f(-1) = 4 and f(5) = 1 23. Passes through (2, 4) and (4, 10) 24. Passes through (1, 5) and (4, 11) 25. Passes through (-1,4) and (5, 2) 26. Passes through (-2, 8) and (4, 6) 27. x intercept at (-2, 0) and y intercept at (0, -3) 28. x intercept at (-5, 0) and y intercept at (0, 4)
Find an equation for the function graphed 29.

30. 31.

32.

33. A clothing business finds there is a linear relationship between the number of shirts, n, it can sell and the price, p, it can charge per shirt. In particular, historical data shows that 1000 shirts can be sold at a price of RM30, while 3000 shirts can be sold at a price of RM22. Find a linear equation in the form p = mn + b that gives the price p they can charge for n shirts.

34. Which of the following tables could represent a linear function? For each that could be linear, find a linear equation models the data.

Match each linear equation with its graph 35. f(x) = -x -1 36. f(x) = -2x – 1 37. f(x) = -12x – 1 38. f(x) = 2 39. f(x) = 2 + x 40. f(x) = 3x + 2

F
F
E
E
D
D
C
C
B
B
A
A

Sketch a line with the given features 41. An x-intercept of (-4, 0) and y-intercept of (0, -2) 42. An x-intercept of (-2, 0) and y-intercept of (0, 4) 43. A vertical intercept of (0, 7) and slope - −32 44. A vertical intercept of (0, 3) and slope 25 45. Passing through the points (-6,-2) and (6,-6) 46. Passing through the points (-3,-4) and (3,0)

Write the equation of the line shown 47.

48.

49.

50.

Given below are descriptions of two lines. Find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular or neither? 51. Line 1: Passes through (0,6) and (3,24)
Line 2: Passes through (1,19) and (8,71)

52. Line 1: Passes through (8, 55) and (10, 89)
Line 2: Passes through (9, 44) and (4,14)

53. Line 1: Passes through (2,3) and (4,1
Line 2: Passes through (6,3) and (8,5)

54. Line 1: Passes through (1, 7) and (5,5)
Line 2: Passes through (1,3) and (1,1)

55. Line 1: Passes through (0, 5) and (3,3)
Line 2: Passes through (1,5) and (3,2)

56. Line 1: Passes through (2,5) and (5,1)
Line 2: Passes through (3,7) and (3,5)

57. Write an equation for a line parallel to f(x)= - 5x - 3and passing through the point (2,-12)

58. Write an equation for a line parallel to g(x) = 3x - 1and passing through the point (4,9)

59. Write an equation for a line perpendicular to h(t) = - 2t + 4 and passing through the point (-4,-1)

60. Write an equation for a line perpendicular to p(t) = 3t + 4and passing through the point (3,1)

61. Find the point at which the line f(x) = - 2x – 1 intersects the line g(x) = - x

62. Find the point at which the line f(x) = 2x + 5 intersects the line g(x) = - 3x – 5

63. A car rental company offers two plans for renting a car.
Plan A: RM30 per day and RM0.18 per kilometer
Plan B: RM50 per day with free unlimited kilometers.
How many miles would you need to drive for plan B to save you money?

64. A cell phone company offers two data options for its prepaid phones
Pay per use: RM0.002 per Kilobyte (KB) used
Data Package: RM5 for 5 Megabytes (5120 Kilobytes) + RM0.002 per addition KB
Assuming you will use less than 5 Megabytes, under what circumstances will the data package save you money?