WARM UP What is a function. How are they used.

FUNCTIONS

OBJECTIVES Work with functions that are defined algebraically, numerically, or verbally Make connections among the algebraic equation for a function, its name, and its graph. Review functions learned in Algebra 1 and Algebra 2. Graph functions

IMPORTANT TERMS & CONCEPTS Function Expressing mathematical ideas graphically, algebraically, numerically & verbally. Mathematical model Dependent variable Domain Range Asymptote

MATHEMATICAL OVERVIEW In Algebra 1 and 2 you studied linear, quadratic, exponential, power and other important functions. In this course, you will learn general properties that apply to all types of functions. In particular, you will learn how to transform a function so that its graph fits real-world data. You will learn this in four ways: You will learn this in four ways: GraphicallyAlgebraicallyNumericallyVerbally

THE FOUR WAYS GRAPHICALLY: This is the graph of a quadratic function. The y-variable could represent the height of an arrow at various times, x, after its release into the air. For larger time values, the quadratic function shows y is negative. These values may or may not be reasonable in the real world. ALGEBRAICALLY: The equation of the function is NUMERICALLY: The table shows corresponding x- and y- values that satisfy the equation of the function: x(s)y(m) VERBALLY: When the variables in a function stand for things in the real world, the function is being used as a mathematical model. The coefficients in the equation of the function have a real- world meaning. For example, the coefficient -4.9 is a constant that is a result of the gravitational acceleration, 20 is the initial velocity and 5 reflects the initial height of the arrow.

FUNCTIONS If you pour a cup of coffee, it cools more rapidly at first, than less rapidly, finally approaching room temperature. Since there is one and only one temperature at any given time, the temperature is called a function of time. You can show the relationship between coffee temperature and time graphically. The graph shows the temperature y, as function of time x. At x = 0, the coffee has just been poured.

FUNCTIONS The graph shows that as time goes on, the temperature levels off, until it is so close to room temperature, 20 degree centigrade, that you can’t tell the difference. This graph might have come from an algebraic equation, From the equation, you can find numerical information. If you enter the equation into your grapher, then use the table feature, you can find these temperatures rounded to 0.1 degrees. x (min)y (°C)

MATHEMATICAL MODELS Functions that are used to make predictions and interpretations about something in the real world are called mathematical models. Temperature is the dependent variable because the temperature of the coffee depends on the time is has been cooling. Time is the independent variable. You cannot change time simply by changing coffee temperature. Always plot the independent variable on the horizontal axis and dependent variable on the vertical axis.

GRAPHING TERMS The set of values the independent variable of a function can have are called domain. In the cup of coffee example, the domain is the set of non-negative numbers or x > 0. The set of values of the dependent variable corresponding to the domain is called the range of the function. If you don’t drink the coffee (which would end the domain) the range is the set of temperatures between 20 C and 90 C, including 90 degrees centigrade or 20 < x < 90. The horizontal line at 20 is called the asymptote. The graph gets arbitrarily close to the asymptote but never touches it.

DEFINITION OF FUNCTION If you plot the function, you get a graph that rises and then falls. For any x value you pick there is only one y-value. This is not the case for all graphs. In graph 2, there are places where the graph has more than one y-value for the same x-value. Although the two variables are related, the relation is not a function.

f(x) TERMINOLOGY You should recall f (x) notation from previous courses. It is used for y, the dependent variable of a function. With it, you show what value you substitute for x, the independent variable. For instance, to substitute 4 for x in the quadratic function f (x) = x 2 + 5x + 3, you would write f (4) = (4) + 3 = 39 The symbol f (4) is pronounced “f of 4” or sometimes “f at 4.” You must recognize that the parentheses mean substitution and not multiplication.

NAMES OF FUNCTIONS Functions are named for the operation performed on the independent variable. The different types of functions are: PolynomialLinear Direct variation PowerExponential Indirect Variation Rational Algebraic In the following examples, the letters a, b, c, m, and stand for constants. The symbols x and f(x) stand for variables, x for the independent variable and f(x) for dependent variable.

QUADRATIC FUNCTION General equation: f (x) = ax 2 + bx + c, where n is a nonnegative integer. Verbally: f(x) varies quadratically with x, or f(x) is a quadratic function. Features: The graph changes direction at its one vertex. The domain is in all numbers.

LINEAR FUNCTION General equation: f (x) = ax + b (or f(x) = mx + b) Verbally: f(x) varies linearly with x, or f(x) is a linear function of x. Features: The straight-line graph, f(x) changes at a constant rate as x changes. The domain is all real numbers.

DIRECT VARIATION FUNCTION General equation: f (x) = ax (or f (x) = mx + 0, or f (x) = ax 1 ) Verbally: f(x) varies directly with x, or f(x) is directly proportional to x. Features: The straight-line graph goes through the origin. The domain is x > 0 (as shown) for most real-world applications.

POWER FUNCTION General equation: f (x) = ax (a variable with a constant exponent) Verbally: f(x) varies directly with bth power of x, or (x) is directly proportional to the bth power of x. Features: The graph contains the origin if b is positive. In most real-world applications, the domain is nonnegative real numbers if b is positive and positive real numbers if b is negative.

EXPONENTIAL FUNCTION General equation: (x) = a · b (a constant with a variable exponent) Verbally: f(x) varies exponentially with x, or f(x) is an exponential function of x. Features: The graph crosses the y-axis at f(0) = a and has the x-axis as an asymptote.

INVERSE VARIATION General equation: or f(x) = ax or or f(x) = ax or f(x) = ax Verbally: f(x) varies inversely with x (or with the nth power of x). Alternatively f(x) is inversely proportional to x (or to the nth power of x.) Features: Both of the axes are asymptotes. The domain is x ≠ 0. For most real-world application, the domain is x > 0.

RATIONAL ALGEBRAIC General equation:, where p and q are polynomial functions. Verbally: is a rational function of x. Features: A rational function has a discontinuity (asymptote or missing point) where the denominator is 0; it may have horizontal or other asymptotes.

ACTIVITIES PAPER CUP ANALYSIS TEXTBOOK: pg. 5 #2 & pg #2, 6, 12, 22, 24, & 29 Using a Graphing Calculator