1.What is the absolute value of a number? 2.What is the absolute value parent function? 3.What kind of transformations can be done to a graph? 4.How do you graph a linear inequality?

2.7 Make a T-chart and graph each problem x y x y x y x y x y

A transformation changes a graph’s size, shape, position or orientation. A translation shifts a graph horizontally/vertically, but does not change its size, shape, or orientation

EXAMPLE 1 Graphing a function of the form y = | x – h | + k h moves the graph horizontally – opposite direction k moves the graph vertically (h,k) is the vertex of the graph a stretches/shrinks graph If a > 1 it gets skinnier If a < 1, it gets wider If a is negative, reflected in the x-axis (the graph flips upside down )

Graph y = | x + 4 | – 2. Compare the graph with the graph of y = | x |. First, identify and plot the vertex Plot another point on the graph by substitution, such as (–2, 0). Use symmetry to plot a third point, (– 6, 0 ) (h, k) = (– 4, – 2). The graph of y = | x + 4 | – 2 is the graph of y = | x | translated down 2 units and left 4 units.

EXAMPLE 3 Graph y = –2 x – Compare the graph with the graph of y = x. Identify and plot the vertex, (h, k) = (1, 3). y = –2 x – Flips the graph upside down Shifts the graph to the right 1 Shifts the graph up 3 Stretches the graph vertically by a factor of 2

GUIDED PRACTICE 1. y = |x – 2| + 5 (h, k) = (2, 5). The graph is translated right 2 units and up 5 units

2. y = |x| 1 4 The graph is shrunk vertically by a factor of

3. f (x) = – 3| x + 1| – 2 The graph is reflected over the x -axis, stretched by a factor of 3, translated left 1 unit and down 2 units Vertex (-1,-2)

EXAMPLE 5 The graph of a function y = f (x) is shown. a. y = 2 f (x) Use the function to sketch a new graph. Stretch the graph vertically by a factor of 2. To draw the graph, multiply the y- coordinate of each labeled point on the graph of y = f (x) by 2 and connect their images.

EXAMPLE 5 The graph of a function y = f (x) is shown. b. y = – f (x + 2) + 1 Use the function to sketch a new graph. The graph is reflected in the x-axis, then translated left 2 units and up 1 unit. To draw the graph, first reflect the labeled points and connect their images. Then translate and connect these points to form the final image.

c. y = 0.5 f (x) Use the graph to graph the given functions. (Cut your Y values in ½)

Use the graph to graph the given functions. d. y = – f (x – 2 ) – 5 Flip the graph upside down, Move to the right 2 and down 5

GUIDED PRACTICE e. y = 2 f (x + 3) – 1 Use the graph to graph the given function. Stretch your graph by a factor of 2 Move the graph to the left 3 and down 1

2.8 Graph

9 < 6 Tell whether the given ordered pair is a solution of 5x – 2y ≤ 6. Ordered pairSubstituteConclusion 1. (0, – 4) (0, – 4 ) is not a solution 2. (2, 2) 5(2) – 2(2) = (2, 2 ) is a solution 3. (– 3, 8) 5(– 3) – 2(8) = (–3, 8 ) is a solution 5(0) – 2(– 4) = 4. (– 1, – 7) (– 1, – 7 ) is not a solution 5(– 1) – 2(– 7) = To find a solution of a linear inequality in two variables, substitute the ordered pair (x, y) for the values of x and y in the inequality 8 < 6 6 < 6 –31 < 6

Greater than implies the solution includes everything above the line. Greater than or equal to implies the solution includes the line and everything above the line. Less than implies the solution lies below the line. Less than or equal to implies the solution includes the line and everything below the line.

Graph linear inequalities with one variable a. y < – 3 b. x < 2

Graph linear inequalities with two variables a. Graph y > – 2x

Graph linear inequalities with two variables b. Graph 5x – 2y ≤ – 4 Solve for y 5x – 2y ≤ – 4 – 2y ≤ – 5x – 4

GUIDED PRACTICE y > –1 Graph y > –3x

GUIDED PRACTICE x > –4 Graph y < 2x +3

x + 3y < 9 3y < -x + 9 2x – 6y > 9 – 6y > – 2x + 9

Graph an absolute value inequality y > – 2 x – 3 + 4

Graph an absolute value inequality y > – x + 3 – 2

Graph an absolute value inequality y < x – 2 + 1

Graph an absolute value inequality y < 3 x – 1 – 3

GUIDED PRACTICE HOMEWORK 2.7 p. 127 #3-14 HOMEWORK 2.8 p. 135 #7-16(EOP); 17, 18, 22-27