What is a function? Quite simply, a function is a rule which takes certain values as input values and assigns to each input value exactly one output value. When a function is defined, we say that the output value is a function of the input value. Example. A certain child’s height (in inches) on his birthday is a function of his age (in years): The inputs and outputs of a function are called variables. Age 1 2 3 4 5 Height 24 30 37 41 48

Four ways to represent a function Each batch of 5 dozen sugar cookies requires 2 and one-half cups of flour (description in words). We may also use a table: We may use a graph: Or we may use a formula: F = 2.5B No. of Batches (B) 1 2 3 No. of Cups Flour (F) 2.5 5.0 7.5 F 7.5 5.0 2.5 1 2 3 B

To indicate that a quantity Q is a function of a quantity t, we abbreviate to: Q equals f of t and, using function notation, to: Here, Q is the dependent variable and t is the independent variable. In the previous cookie example, F is the dependent variable and B is the independent variable, and we can write F = f(B) = 2.5B Note that we could use another letter instead of f. How about c for cookie? Then F = c(B) = 2.5B

Graphing a function By tradition, the independent variable is always plotted on the horizontal axis, and the dependent variable is always plotted on the vertical axis. In the cookie example, the independent variable is B, while the dependent variable is F. Thus, the graph of the function is: F 7.5 5.0 2.5 1 2 3 B

Example of quantities which are related, but neither quantity is a function of the other. Both F and R are functions of t. However, F is not a function of R, and R is not a function of F (do you see why?). In other words, if we know which month is being discussed, we can determine the values of F and R uniquely. However, if we only know the value of F, then the value of R may not be determined uniquely. Similarly, if we only know the value of R, then F may not be uniquely determined. t, month 1 2 3 4 5 6 7 R, no.rabbits 750 567 500 567 750 1000 1250 F, no.foxes 143 125 100 75 57 50 57

How to tell if a graph represents a function: Vertical Line Test The vertical line test for a graph states: if there is a vertical line which intersects a graph in more than one point, then the graph does not represent a function. In which of the graphs below could y be a function of x? Clearly, the one on the right fails the vertical line test, so this graph does not represent y as a function of x. y y x x

Three related concepts, where Q = f(t) The average rate of change or rate of change of Q with respect to t over an interval is: f is an increasing function of t if the values of f increase as t increases. f is a decreasing function of t if the values of f decrease as t increases. The way these concepts are related is given on the next slide.

Assume that Q = f(t). If f is an increasing function, then the average rate of change of f with respect to t is positive on every interval. If f is a decreasing function, then the average rate of change of f with respect to t is negative on every interval. Q Q t t

Karim’s excellent trip The graph of Karim’s distance vs. time is shown next. Karim’s average speed for is given by distance (miles) Note: Average speed is average rate of change of distance with respect to time. Karim’s average speed is different for different time intervals! 60 50 40 (2,35) 30 (1,20) 15 miles 20 1 hr 10 1 2 3 4 5 time (hours)

Amanda’s excellent trip The graph of Amanda’s distance vs. time is shown next. Amanda’s average speed for is given by distance (miles) 60 Note: Amanda’s average speed is the same for different time intervals! 50 40 30 (2,24) 20 (1,12) 12 miles 10 1 hr 1 2 3 4 5 time (hours)

Function Notation for Average Rate of change Suppose we want to find the average rate of change of a function Q = f(t) over the interval On this interval, t = b – a, while Q = f(b) – f(a). Using function notation, we express the average rate of change of Q = f(t) over the interval as: Note that the average rate of change can vary from one interval to another (recall Karim’s trip).

More terminology associated with average rate of change. In the figure below, we visualize the average rate of change as a slope, that is, as the ratio rise/run. Q slope = average rate of change t

Calculate the average rates of change for f(x) = x2. Between x = 1 and x = 3. Between x = –2 and x = 1. f(x) = x2 (3,9) Slope = 4 (-2,4) Slope = –1 (1,1) x

What makes a function linear? If a function has the same average rate of change over every interval, then we say the function is linear. We say that a linear function changes at a constant rate, and we talk about the rate of change of a linear function. It turns out that the graph of any linear function is a straight line. The rate of change of the function is the slope of this line. Problem. The table below gives the fine r = f(v) imposed on a motorist for speeding, where v is the motorist’s speed and 55 mph is the speed limit. Is f a linear function? Why? v (mph) 60 65 70 75 80 85 r ($) 75 100 125 150 175 200

The graph of f appears below (assuming fines are prorated): Problem, continued. The function is linear. What does its rate of change represent in practical terms for the motorist? The graph of f appears below (assuming fines are prorated): r ($) 200 175 150 125 100 75 v (mph) 60 65 70 75 80 85

Usage of duplicating paper at Lee High School At present there are 400 packages of duplicating paper available. Each week 12 packages are used. A table is shown next which gives the number of packages left, L, versus the number of weeks from now, w. It is clear that L = f(w) is a linear function. Such a linear function can be given a formula of the type: where m is the slope or rate of change and b is the vertical intercept. Can you tell what the values of m and b are in this situation? What is the significance of the horizontal intercept? w 1 2 3 4 5 6 7 L 388 376 364 352 340 328 316

The slope-intercept form when y is a linear function of x The slope-intercept form is where m is the slope and b is the y-intercept. The y-intercept, b, tells us where the line crosses the y-axis. If the slope, m, is positive, the line climbs from left to right. If the slope, m, is negative, the line falls from left to right. The larger the value of m (either positive or negative), the steeper the graph of the linear function f(x) = b + mx. If (x0, y0) and (x1, y1) are two distinct points on the graph of f, then

Equation of a linear function from a table Suppose the value of a Batman comic book is increasing as follows: If we let year 2000 correspond to t = 0, the table becomes: Since is constant, we let m = and the equation is: Year 2000 2002 2004 2006 Value ($) 65 90 115 140 t (year) 2 4 6 v ($) 65 90 115 140

Graphing a function The independent variable is always associated with the horizontal axis. The dependent variable is always associated with the vertical axis. Along with the variable name, a variable’s units should be listed, if they are available. Example. Let v = f(t) be the value of the Batman comic book from the previous slide.

Weekly Profits at a Theater Suppose the theater manager knows that weekly profits are a linear function of the number of patrons. One week the profit was $11,328 when 1324 patrons attended. Another week 1529 patrons produced a profit of $13,275.50. (a) Find a formula for weekly profit, y, as a function of the number of patrons, x. (b) Interpret the slope and the y-intercept. (c) Find the break-even point (the number of patrons for which there is zero profit). (d) Find a formula for the number of patrons as a function of profit. (e) If the weekly profit was $17,759.50, how many patrons attended the theater?

Other equations for y as a linear function of x The point-slope form is where m is the slope and (x0, y0) is a point on the line. The standard form is where A, B, and C are constants. Note that this form only gives y as a function of x when

Example. Use the point-slope form of the line to derive the equation which converts temperature in degrees Celsius, C, to degrees Fahrenheit, F. We are given that the slope is 9/5 and that (20,68) is a point on the line. That is, C0 = 20 and F0 = 68. Using the point-slope form, we have: The equation may be rewritten in slope-intercept form as:

Use of Maple to graph temperature conversion > plot((9/5)*C+32,C=0..40,color=black,labels=["C","F"]);

Intersection of Two Lines To find the point at which two lines intersect, notice that the (x, y)-coordinates of such a point must satisfy the equations for both lines. To find the point of intersection algebraically, we solve the equations simultaneously. Of course, this only works if the lines are not parallel. Example. Find the point of intersection of the lines: Set the y-values for the two lines equal and solve the resulting equation for x. Thus, we must solve Complete the solution to get x = 42/13. Now substitute this back into either equation to get the y-value. What is it?

Janna and Wanda and their piggy banks Janna is spending money while her sister Wanda is saving it. At present, Janna has $65 but she spends $2 more than her allowance each week. Wanda has $40 now, but she is saving $3 each week. How can we determine when Janna and Wanda will have the same amount of money? A (dollars) Wanda: A= 40 + 3w 70 60 50 40 Janna: A= 65 – 2w 30 20 10 w (weeks) 1 2 3 4 5 6 7 8 9

Solution of Janna and Wanda and their piggy banks If we set the formula for Janna’s amount equal to the formula for Wanda’s amount, we will have an equation with a single unknown, and we can solve it for the number of weeks. We have:

Use of Maple to solve Janna/Wanda problem > plot({40+3*w,65-2*w},w=4.5..5.5,54.5..55.5,color=black); > evalf(solve(40+3*w=65-2*w,w)); 5.

Useful facts about equations of lines For any constant k: The graph of the equation y = k is a horizontal line and its slope is zero. The graph of the equation x = k is a vertical line and its slope is undefined. Let L1 and L2 be two lines having slopes m1 and m2, respectively. Then: These lines are parallel if and only if m1 = m2. These lines are perpendicular if and only if

Use of Maple to plot perpendicular lines > plot({2*x,-(1/2)*x+5},x=0..4,3..5,color=black,scaling=constrained);

Summary—Linear Functions and Change Important terms: function, variable, independent variable, dependent variable, vertical line test. Four ways to represent a function: describe it in words, use a table, graph it, or use a formula. For a function f, there is a relation between the sign of the avg. rate of change and whether f is increasing or decreasing. The avg. rate of change of a function is a certain slope. If a function has the same average rate of change over every interval, then the function is linear. The graph of a linear function is a straight line and the slope of this line is the rate of change of the linear function. There are three forms for the equation of a straight line: slope-intercept form, point-slope form, and standard form. The graph of y = k is a horizontal line with slope zero.

Summary, continued The graph of x = k is a vertical line with undefined slope. Two lines having slopes are parallel iff their slopes are equal. Two lines having slopes are perpendicular iff the slope of one of these lines is the negative reciprocal of the slope of the other. To find the point of intersection of two lines, set the formulas for their y values equal, and solve the resulting equation for x. Once the x-value has been found, insert it into the formula for either line to get the y-value of the point of intersection.