# Relations and Functions

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Relations and Functions
Relation: A set of pairs of input and output values (input, output) or (x,y) How Can we Represent a Relation? Four Ways: Ordered Pairs: Ex: (x,y): (-3,4), (3,-1),(4,-1) Mapping Diagram: input -> output: -3->4, 3->-1,4->-1 Table of values: Graph: The output goes on the vertical axis. x y -3 4 3 -1

Ex: Skydivers undergoing Freefall
A skydiver falls freely from a plane at an initial height of 10,000 feet. His heights at various times are measured to be 9744 ft. 4 seconds later, 8976 feet 4 seconds after that, 7696 feet 4 seconds after that, and 5904 feet 4 seconds after that. Represent the relationship using A mapping diagram A set of ordered pairs A Table of values A graph Based on your studies of Physics, what algebraic equation do you expect the height to follow?

Domain and Range The set of inputs in a relation is called the domain.
The set of outputs in a relation is called the range. Example: What are the domain and range of the time height relation example?

Function A function is a relation in which there is one and only one output for each input. Look at page 62, Problem 3: Which of these relations are functions? Vertical Line Test: Since the output is plotted on the vertical axis of a graph, no vertical line drawn can intersect the graph more than once for the relation to be a function. Look at page 63, Problem 4. Which relations are functions? Ex: What functions have you seen in Physics? Give the inputs and outputs.

Function Rule and Function Notation
A function rule is an algebraic equation, by which we define what we do to an input (or independent variable) to obtain our output (or dependent variable). Function Notations: y=output, x = input y= 3x+2 f(x) = 3x+2 read as “f of x” f(1) = 3(1)+2 read as “f of 1”.

Direct Variation When the output (y) can be expressed in terms of the input (x) in the form: y = kx where k is a constant number, we say “y varies directly” with x. k = the “constant of variation”. If y varies directly with x, does y have a linear relationship with x? What is the difference between saying “y varies directly with x” and “y is linear with x”?

Linear Relationships Linear Function:
A function whose graph is a line. Characterized by a slope and vertical intercept: y = mx+b m = slope, b = y-intercept. Forms include: Slope-Intercept form: y = mx + b Point-Slope Form: y-y1 = m(x-x1) where (x1, y1) is any point on the line. Standard Form: Ax+By+C = 0 where A,B,C, are real numbers and A and B cannot both be zero.

How are parallel and perpendicular lines related to each other?
The slopes of parallel lines are equal. (m1=m2) The slopes of perpendicular lines are negative reciprocals of each other. (m1= -1/m2) (We proved this earlier on). What is the equation of the line parallel to y=6x-2 that runs through the point (1,-3)? What is the equation of the line perpendicular to y=-4x+(2/3) that runs through (8,5)?

Standard Form for a Line
Ax + By = C where A, B, and C are coefficients (constants). A and B cannot both be zero. Example: Express y=(3/4)x-5 in standard form. Use integer coefficients. Standard Form is useful to quickly graph a line by plotting its x and y intercepts. Example: Graph: 3x + 5y = 15 Get x intercept by letting y = 0 Get y intercept by letting x = 0.

Piecewise Functions Piecewise functions are functions that behave differently depending on the input. Examples: Graph the following functions: f(x) = |x| f(x) = x2 if x<2 6 if x=2 10 – x if x > 2 and x ≤ 6 f(x) = [x] = the greatest integer less than or equal to x. (Called the “greatest integer function”) (see pages 90-91)

A Linear Model (see page 84)
The office manager of a small office ordered 140 packs of printer paper. Based on average daily use, she knows that the paper will last about 80 days. Write a linear equation that gives the number of packs of paper left after a certain number of days. Define all variables using “Let”. Graph the equation. What is the equation in standard form? How many packs of paper will they have after 30 days?

Use of Models Modeling Data: We may gather data from a situation and use the data to analyze the situation and predict its future. Example: Suppose you open a gas station. It costs money to stay open each hour paying workers and utilities. Should you stay open 24 hours? When should you open the gas station and when should you close to maximize your profits? How could you predict your income over the years?

Using Linear Models In the simplest case, we try to model real world data with a linear relationship. First make a scatterplot: Plot your data as ordered pairs with the dependent variable on the vertical axis and the independent variable on the horizontal axis. The closer the data looks to forming a line, the stronger the relationship, or correlation. The correlation can form a positive pattern (slope) or negative pattern (slope).

Pictorial Examples of Correlation

Types of correlation Strong negative correlation
Weak negative correlation No correlation Weak positive correlation Strong positive correlation Draw an example of each.

Creating and Using a Linear Model Ex: “Got it” Problem on page 94
Given the data table of median home prices: Make a Scatterplot: Median Price (Y) vs. Year (X) Sketch a trend line Measure and calculate the slope using two points on the line. (DO NOT USE ANY ORIGINAL DATA POINTS) Create the equation of the line: y=mx+b Use the line to predict prices in desired years. Year 1940 1950 1960 1970 1980 1990 2000 Price 36700 57900 74400 88700 167300 249800 211500

Using the TI-nspire to create a Linear Model

Families of Functions We split up functions into “families” that share a basic dependence on the independent variable, Examples: f(x) = x (“parent function”) g(x) = 4 f(x) = 4x h(x) = f(x) + 4 = x+4 f(x) = x2 (“parent function”) g(x) = 4 f(x)= 4 x2 h(x) = 4f(x) + 3 = 4 x Other functions within a family are formed by performing a “transformation” on the parent function.

Types of Transformation f(x) = original function g(x) = transformed function
Translations (k and h are positive constants) Vertical Shift: g(x) = f(x) ± k Horizontal Shift: g(x) = f(x + h), left shift g(x) = f(x – h), right shift Plot each of the following f(x), g(x) pairs on the same graph: f(x) = x; g(x) = x-2; h(x) = x+2 f(x) = x2; g(x) = f(x)+2; h(x)= f(x)-2 f(x) = x2; g(x) = f(x+2); h(x) = f(x-2)

More Transformations Reflection In the y-axis: g(x) = f(-x)
In the x- axis: g(x) = -f(x) On the same graph, plot: f(x) = 3x+3, g(x) = f(-x), h(x) = -f(x) On the same graph, plot f(x) = x2 , g(x) = f(-x), h(x) = -f(x) (Was it difficult to plot g(x))?

More Transformations 3. Vertical Stretching or Compressing (Also called “Scaling”) g(x) = a f(x) If a > 1 : We’re stretching If 0< a < 1 : We’re compressing. (What if a < 0?) Plot on the same graph f(x) = x2, g(x) = ½ f(x), h(x) = 2f(x)

We can combine transformations of an original function.
Examples: what series of transformations would turn f(x) into g(x) for: f(x) = 4x and g(x) = -4x + 2 f(x) = 4x and g(x) = 2x-3 f(x) = 2x2 and g(x) = 6x2 -1

How do we apply transformations to absolute value functions?
Vertical Translation (k>0): y=|x| + k (up), y=|x| - k (down) Horizontal Translation (h>0): y =|x-h|(right), y=|x+h|(left) Vertical Stretch: y = a|x| with a>1 Compression: y=a|x| with 0<a<1 Reflection: In the x-axis: y=-|x|, In the y-axis: y=|-x|

General Form of the Absolute Value Function
We can combine the equations for stretches, compressions and translations to write a general form for absolute value equations: y = a|x-h|= k has vertex at (h,k) and axis of symmetry x=h. It is stretched or compressed by the factor |a|. y = a |-x-h| + k has vertex at (-h,k) because it is reflected in the y-axis with axis of symmetry: x=-h.

y=3|x-2|+4 y = -1/3|x+1|+4 y = 2 |x+4|-3 y = |-x-3|+9
For the following, find the vertex and axis of symmetry and the transformations from |x|: y=3|x-2|+4 y = -1/3|x+1|+4 y = 2 |x+4|-3 y = |-x-3|+9

How can we form an absolute value equation by looking at its graph?
Follow these steps: Identify the vertex: (h,k) Identify the a = slope of right branch (that is, if graph is “upside-down” then a is negative due to reflection in x-axis.) Form the equation: y = a|x-h|+k Try “Got it? Problem 5 on page 110 and problems 29 and 30 on page 111.

Graphing Two Variable Inequalities
When we say: y = mx + b We’re talking about a set of points that make up a line. If we say: y > mx + b or y < mx + b , we’re talking about a set of points that make up a region (“half-plane”) on one side or the other of a line, but not including the line. And if we say: y≥ mx + b or y ≤ mx + b, the line is included in the region. The line forms the boundary between the > (above the line half-plane) and the < (below the line half-plane).

Some Examples Try this: y > 3x – 1 Now try:
First: Graph the line y = 3x -1; Use a dashed line if the line isn’t included, solid if it is included. Second: To be safe, test a point in one of the half-planes separated by the line. If it satisfies the inequality condition, shade it in as your solution. Otherwise shade in the other half-plane. Now try: y ≤ -2x +1

A Clean Day at the Fair At a county fair, you can buy tickets and redeem them for rides. A small ride requires 3 tickets and a large ride requires 5 tickets. You are constrained to spend \$15 at most with which you can buy 60 tickets. What are the small ride/large ride combinations you can buy? Let y = the number of large rides Let x = the number of small rides. Form an appropriate inequality Graph the linear equation (Hint: use standard form) Only present practical (discrete, non-negative) solutions. How would you change the set of solutions if you added the constraint that you always throw up during your sixth large ride?

Graphing Absolute Value Inequalities
Example: 1-y < |x+2| Get y alone on left hand side of inequality. This may result in a change from < to > or vice versa. Graph the corresponding equality. Shade the region above the graph if > or the region below the graph if <. As before, the graph is dashed if > or < and solid if ≥ or ≤.

Graph : y-4 ≥ -|x-1| Page 117: problem 4 and “Got it” 4.
Try These Graph : y-4 ≥ -|x-1| Page 117: problem 4 and “Got it” 4.