Functions Review

Vocabulary Independent Variable – the variable that provides the input values of a function. Changes on its own the x value Dependent Variable – the variable that provides the output values of a function. Does not change unless the independent variable changes – the y value Linear Function – A function whose graph is a line (highest exponent on the x is 1) Non Linear Function – a function whose graph is not a line or part of a line (exponent on the x will be something other than one or different things are happening to the x) Continuous – a graph that is unbroken (dots are connected) data measured over time Discrete –a graph that is composed of distinct isolated points (data that is counted)

Continuous Data – Measurements that change between data points, such as temperature, length, and weight. A trend line can be drawn on this graph. Straight Line Example: The height of a tree. Discrete Data – Involves a count of data items, such as a number of people or objects. A trend line cannot be drawn on this graph. Dotted Line Example: The number of books on the shelf in a library.

Example 1 The air temperature increased steadily for several hours and then remained constant. At the end of the day, the temperature increased slightly before dropping sharply. Choose the graph that best represents this situation. Graph C

As seen in Example 1, some graphs are connected lines or curves called continuous graphs. Some graphs are only distinct points. They are called discrete graphs The graph on theme park attendance is an example of a discrete graph. It consists of distinct points because each year is distinct and people are counted in whole numbers only. The values between whole numbers are not included, since they have no meaning for the situation.

When sketching or interpreting a graph, pay close attention to the labels on each axis. Helpful Hint

The graph is continuous. Example 3 Sketch a graph for the situation. Tell whether the graph is continuous or discrete. Henry begins to drain a water tank by opening a valve. Then he opens another valve. Then he closes the first valve. He leaves the second valve open until the tank is empty. As time passes while draining the tank (moving left to right along the x-axis) the water level (y-axis) does the following: Water tank Water Level Time • initially declines • decline more rapidly • and then the decline slows down. The graph is continuous.

Both graphs show a relationship about a child going down a slide Both graphs show a relationship about a child going down a slide. Graph A represents the child’s distance from the ground related to time. Graph B represents the child’s Speed related to time.

Function Rule y = 3x + 4 output value input value dependent independent

Independent and Dependent y = 3x – 2 Cost of three pounds of apples Hours worked, amount paid Weight of a book, number of pages The circumference of a circle, the measure of the radius The price of a single pair of shoes, the total price of 4 pairs of shoes Number of hours spent typing a paper, length of paper Time spent studying for a test, score on the test

Independent and Dependent-answers y = 3x – 2 I = x, D = y Cost of three pounds of apples I = # of apples D = Cost I = Hours worked, D = amount paid D = Weight of a book, I = number of pages D = The circumference of a circle, I = the measure of the radius I = The price of a single pair of shoes, D =the total price of 4 pairs of shoes I = Number of hours spent typing a paper, D = length of paper I = Time spent studying for a test, D = score on the test

Families of Functions You can identify what family a function belongs to by looking at its graph or equation. The equations and graphs of functions that are in the same family are alike. There are 3 families.

Linear Functions Equation: In a linear function the highest power of x is 1. Example: y = x – 2 y = 3x + 5 y = -5x Graph: The graph is a line. If the coefficient of x is positive the line slants to the right, and if it is negative the line slants to the left

Quadratic Functions Equation: In a quadratic function the highest power of x is 2. Examples: y = x² - 6 y = 2x² - 7x – 4 y = -3x² Graph: The graph is a “U” shaped curve. If the term x² is positive, the curve will open up. If the term containing x² is negative, the curve will open down. The name of this “U” shaped curve is called a parabola.

Absolute Value Function Equation: In a absolute value function there is an absolute value symbol around a variable expression Examples: y = -│2x│ y = │5x + 3│ Graph: The graph forms a “V” shape that opens up or down. The graph will open up unless there is a negative in front of the absolute value symbol (outside the absolute value bars).

What family of functions does each equation belong? Explain. 1.) y = ⅛x + 7 2.) y = -2│x│ 3.) y = -6x² + 13x – 5 4.) y = x + │3 - 5│

Identifying a function by its graph… How can you tell if an equation will graph a line? How can you tell if an equation will graph a parabola (a “U”)? How can you tell if an equation will graph a “v”?

Identifying a function by its graph… How can you tell if an equation will graph a line? The highest exponent on the “x” is a one (or no exponent at all). Ex: y = 2x - 3 How can you tell if an equation will graph a parabola (a “U”)? The highest exponent on the “x” is a 2 (squared). Ex: y = x2 - 3 How can you tell if an equation will graph a “v”? The variable is in the absolute value bars. Ex: y = |x – 3|

5.) 6.) 7.)

Linear or not linear? Linear Linear Function Non linear function

Linear or non linear… x y 4 8 12 16 –3 3 6 9 x y –4 –2 2 4 13 1 –3 4 8 12 16 –3 3 6 9 x y –4 –2 2 4 13 1 –3 {(3, 5), (5, 4), (7, 3), (9, 2), (11, 1)} 5 4 3 2 1 8 6

Graph:

Graph 5 4 3 2 1 8 6