Chapter 2 Part 2:

Concept Byte: Piecewise Functions & Greatest Integer Function

A piecewise function has different rules (or equations) for different parts of its domain (or x-values). Example 1Graph the piecewise function x < 0f(x) = -x x ≥ 0f(x) = x

A step function pairs every number in an interval with a single value. The graph looks like steps. One step function is the greatest integer function Example 3What is the graph of the function

2-5 Using Linear Models

Essential Understanding – Sometimes it is possible to model data from a real- world situation with a linear equation. You can then use the equation to draw conclusions about the situation. A scatter plot is a graph that relates two sets of data by plotting the data as ordered pairs. You can use a scatter plot to determine the strength of the relationship, or correlation between data sets. The closer the data points fall along a line,  the stronger the relationship  the stronger the positive or negative correlation between two variables.

Problem 1Using a scatter plot UtilitiesThe table lists average monthly temperatures and electricity costs for a Texas home in The table displays the values rounded to the nearest whole number. Make a scatter plot. How would you describe the correlation? Step1: Make a scatter plot Step 2: Describe the correlation

Got it? The table shows the numbers of hours students spent online the day before a test and the scores on the test. Maker scatter plot. How would you describe the correlation?

A trend line is a line that approximates the relationship between the variables, or data sets of a scatter plot. You can use a trend line to make predictions from the data. We can use point-slope formula to find the equation. Problem 2Writing the equation of a trend line Finances The table shows the median home prices in Florida. What is the equation of a trend line that models a relationship between time and home prices? Use the equation to predict the median home price in Make a scatter plot. 2.Sketch a trend line. 3.Choose 2 points on the trend line, use slope-intercept form to write an equation for the line. 4.Use the equation to make predictions.

Got it? The table shows median home prices in California. What is an equation for a trend line that models the relationship between time and home prices?

The trend line that gives the most accurate model of related data is the line of best fit. One method for finding this is called linear regression. The correlation coefficient indicates the strength of the correlation. The closer r is to 1 or -1, the more closely the data resembles a line and the more accurate your model is likely to be. Problem 3: Finding the line of best fit Food You research the average cost of whole milk for several recent years to look for trends. The table shows your data.

Let’s try this using a graphing calculator! Please follow the instructions written in your packet on page 5… and also found at the back of your packet…

1.Open a new document and add a “Lists & Spread sheets.” 2.The year should be your “A” Column and the cost is your “B” Column, like the picture to the right. 3.Fill in the data from the table in the appropriate columns, note, 1998 = 0 4.Add a new page by selecting the / “DOC” key and select 5: Add Data & Statistics. 5.Your data should be scattered around the screen. Move your mouse to the bottom of the screen so this appears 6.Click to add year on the x-axis then move your mouse to the left to add cost to the y-axis. 7.From the menu key select 4: Analyze then 6: Regression, make sure you select the appropriate type from the list (HINT: we are working with linear functions). 8.What is the equation that you found? 9. Based on the equation that you found, how much would a gallon of whole milk cost in 2020?

Got it? The table below lists the cost of 2% milk. Repeat the process to find a linear regression for the data. Based on the equation you come up with, how much would you expect to pay for a gallon of 2% milk in 2025? Equation?Prediction?

2-8 Two Variable Inequalities Turn to page 8 in your packet

Essential Understanding – Graphing an inequality in two variables is similar to graphing a line. The graph of the linear inequality contains all the points on one side of the line and may or may not include the points on the line. A linear inequality is an inequality whose graph is the region of the coordinate plane bounded by a line. This line is the boundary of the graph. This boundary separates the coordinate plane into two half- planes, one of which consists of solutions of the inequality.

To determine which half-plane to shade, pick a test point that is NOT on the boundary. Check whether that test point satisfies the inequality…makes a TRUE inequality. If it DOES, shade the half-plane that INCLUDES this test point. If it DOESN’T satisfy the inequality…makes a FALSE inequality, shade the half-plane on the OPPOSITE side of the boundary. The origin’s coordinate (0, 0) is usually the easiest test point to use, as long as it is NOT on the boundary.

It is dashed line. It has a y-intercept at (0, -1). It has a slope of 3. Test Point  (0, 0) 0 > 3(0) -1 0 > -1 True!!! So shade where (0, 0) is. It is solid line. It has a y-intercept at (0, -1). It has a slope of 3. Test Point  (0, 0) 0 ≤ 3(0) -1 0 ≤ -1 FALSE!!! So shade opposite of where (0, 0) is.

REMEMBER… When you are working with INEQUALITIES and you divide (or multiply) by a negative number, the symbol reverses!!!

0.25 per ticket Small rides: 0.75x Large rides: 1.25y  How many total rides can you ride with $15?  What are the two variables needed to solve the problem? What do they represent?  Write a linear inequality that would represent how many small and large rides you can go on for no more than $15.  Graph this inequality. (Use the x & y intercepts) xint (#, 0) yint (0, #) 0.75x (0) ≤ (0) y ≤ x ≤ y ≤ 15 X ≤ 20y ≤ 12  Use these numbers and create an appropriate scale and label both axes. Draw your line and shade accordingly. X = cost of small ridesY = cost of large rides 0.75x y ≤ 15

It is “v” shaped and facing up. It is (3, -2). It is dashed. It is shaded above. y > | x - 3 | - 2 It is “v” shaped and facing down. It is (-4, 3). It is dashed. It is shaded above. y > - | x +4 | + 3

Chapter 2 Part 3: 2-6 Families of Functions AKA Mother Functions

Different nonvertical lines have different slopes, or y- intercepts, or both. They are graphs of different linear functions. For two such lines, you can think of one as a transformation of the other. Essential Understanding – There are sets of functions, called families, in which each function is a transformation of a special function called the parent. The linear functions form a family of functions. Each linear function is a transformation of the function y = x. In which we call the parent linear function.

A parent function is the simplest form in a set of function that form a family. Each function in the family is a transformation of the parent function. One type of transformation is a translation. A translation shifts the graph of the parent function horizontally, vertically, or both without changing the shape or orientation. For a positive constant (k) and a parent function f (x), f (x) ± k is a vertical translation. For a positive constant (h), f ( x ± h ) is a horizontal translation.

Problem 1Vertical Translation How are the functions y = x and y = x – 2 related? How are their graphs related? xy = xy = x – They are related because the blue graph is 2 spaces down from the brown.

What does the graph oftranslated up 5 units look like? 5 units up -2(-2)(-2)-1 = units up

Problem 2Horizontal Translation The graph shows the projected altitude of an airplane scheduled to depart an airport at noon. If the plane leaves two hours late, what function (equation) represents this transformation? Got it? What would the function be if it left 30 minutes early? How would that change from the original graph?

A reflection flips the graph of a function across the x- or y-axis. Each point on the graph of the reflected function is the same distance from the line of reflection as it corresponds to the point on the original function. When you reflect a graph over the y-axis, the X- VALUES CHANGE SIGNS and the Y-VALUES stay THE SAME. The original function is, the reflected function would be f ( -x ). When you reflect a graph over the x-axis, the X- VALUES stay THE SAME and the Y-VALUES CHANGE. The original function is, the reflected function would be - f ( x ).

A vertical stretch multiplies all the y-values of a function by the SAME factor greater than 1. A vertical compression multiplies all the y-values of a function by the SAME factor between 0 and 1. Compared to the original function f (x), the stretch, or compression, would look like this: y = af (x) Remember, when a > 1, you have a stretch and when 0 < a < 1, you have a compression.

Problem 4Stretching & Compressing a Function The table represents the function f (x). What are corresponding values of g (x) and graph for the transformation g (x) = 3 f (x) ? Got it? Using f (x), create a corresponding table and graph for the transformation h (x) = 1/3 f (x).

2-7 Absolute Value Functions & Graphs

The “Solve It!” above models an absolute value graph in a realistic situation. In the lesson, you will be able to identify different parts of absolute value graph and graph transformations of the absolute value parent function. Essential Understanding – Absolute value graphs are NOT linear, but they are composed of two linear parts.

The simplest example of an absolute value function is f (x) = | x |. The graph of the absolute value of a linear function in two variables is V - shaped and symmetric about a vertical line called the axis of symmetry. Such a graph has either a single maximum (or minimum) point called the vertex.

The properties you used for transformations are similar are the same as before, except parentheses are replaced with | |.