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.

Problem 1Vertical Translation How are the functions y = x and y = x + 6 related? How are their graphs related? xy = xy = x They are related because the blue graph is 6 spaces up from the brown.

5 units up What does the graph oftranslated up 5 units look like? 5 units up -2(-2)(-2)-4 =

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 f(x), 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 f(x), 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 | |.

2-8 Two Variable Absolute Value Inequalities

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 “v” shaped and facing up. It is (3, -2)…this is your (h,k). (slope: a = 1) It is dashed so use. It is shaded above and opening up so use > y > | x - 3 | - 2 It is “v” shaped and facing down. It is (-4, 3). (slope: a= -1) It is dashed so use It is shaded above and opening down so use > y > - | x +4 | + 3

19) y ≤ │x + 2│ (a=1, vertex (-2, 0) solid line, opening up and shaded below) 20) y < - │x - 4│ (a= -1, vertex (4, 0) dashed line, opening down and shaded inside) 21) y > │x + 1│ - 1 (a=1, vertex (-1, -1) dashed line, opening up and shaded above)