Is this an open or closed interval?
Is this an open or closed interval?
Essential Question:How can we solve and interpret absolute value inequalities to find the range of possible solutions?
Learning Target: Students will be able to solve absolute value inequalities and represent their solutions on a number line, understanding how the inequality affects the direction and type of solution set (e.g., union of intervals or a bounded interval).
Show your work for credit.
Solve, graph, and write the solutions to the following inequalities in interval notation.
Solve, graph, and write the solutions to the following inequalities in interval notation.
Solve, graph, and write the solutions to the following inequalities in interval notation.
Solve, graph, and write the solutions to the following inequalities in interval notation.
Solve, graph, and write the solutions to the following inequalities in interval notation.
Reasoning: Explain why the absolute value equation |3x| + 8 = 5 has no solution.
Compare and Contrast: Explain the similarities and differences in solving these inequalities |x - 1| ≤ 2 and |x - 1| ≥ 2.
Compare these two inequalities. (How are they alike?)
Contrast these two inequalities. (How are they different?)
Is this an open or closed interval?
Is this an open or closed interval?
Is this an open or closed interval?
Is this an open or closed interval?
Use interval notation to describe the range of this function.
Use interval notation to describe the domain of this function.
Write the following in interval notation.
Write the following in interval notatin.
Match each inequality, absolute value, and interval notation with the correct graph. (Not every choice will be used)
(-∞,-2)∪(2,∞)
|x|<2
|x|≥2
-2≤x≤2
|x|≤2
(-2,2)
|x|>2
-2<x<2
x<-2 or x>2
Match each inequality, absolute value, and interval notation with the correct graph. (Not every choice will be used)
(-3,13)
|x-5|<8
|x|>4
x<-4 or x>4
-3≤x≤13
x≤-4 or x≥4
[-3,13]
|x|<4
(-∞,-4)∪(4,∞)
Match each inequality, absolute value, and interval notation with the correct graph. (Not every choice will be used)
x<1 or x≥4
1<x≤4
(-∞,3)∪(5,∞)
(1,4]
|x-2.5|<1.5
3<x<5
|x-4|>1
x<3 or x>5
1>x≤4
Match each inequality, absolute value, and interval notation with the correct graph. (Not every choice will be used)
(-∞,5]∪(9,∞)
5<x≤9
(5,9]
x<5 or x≥9
(5,9)
x≤5 or x>9
(-∞,5)∪[9,∞)
Solve and graph the inequality
Solve and graph the inequality:
Solve and graph the inequality:
Solve and graph.
Solve and graph the inequality:
Use a set of x values to graph the parent function y=|x|
Use a set of x values to graph the function y=|x|+ 2
Compare the graph y=|x|to y=|x|+ 2
How are they the same?
How are they different?
Use a set of x values to graph the function y=|x - 2|
Compare the graph y=|x| to y=|x - 2|
How are they the same?
How are they different?
Make a prediction. What will the graph of y=|x - 2|+ 2 look like?
Use a set of x values to graph the function y=|x - 2|+ 2
Notes: Critical Values of Absolute Value Functions
1)What are the critical values of this absolute value function:
y=|x-1|+2
Opens (upward or downward)
Axis of Symmetry
Vertex
Slope
2) Use the critical values of this equation to graph it.
What are the critical values of this absolute value function:
y=2|x+2|+2
Opens (upward or downward)
Axis of Symmetry
Vertex
Slope
2) Use the critical values of this equation to graph it.
What are the critical values of this absolute value function:
Opens (upward or downward)
Axis of Symmetry
Vertex
Slope
2) Use the critical values of this equation to graph it.
What are the critical values of this absolute value function:
Opens (upward or downward)
Axis of Symmetry
Vertex
Slope
2) Use the critical values of this equation to graph it.
What are the critical values of this absolute value function:
Opens (upward or downward)
Axis of Symmetry
Vertex
Slope
2) Use the critical values of this equation to graph it.
How is the absolute function below different than the parent function y=|x|:
y=|x-1|+2
Opens (Upward or Downward)
Horizontal Shift (write none if there is none)
Vertical Shift (write none if there is none)
Stretched (0<|a|<1), Compressed (|a|>1), or None
How is the absolute function below different than the parent function y=|x|:
y=2|x+6|-2
Opens (Upward or Downward)
Horizontal Shift (write none if there is none)
Vertical Shift (write none if there is none)
Stretched (0<|a|<1), Compressed (|a|>1), or None
How is the absolute function below different than the parent function y=|x|:
y=-3|x-4|-9
Opens (Upward or Downward)
Horizontal Shift (write none if there is none)
Vertical Shift (write none if there is none)
Stretched (0<|a|<1), Compressed (|a|>1), or None
How is the absolute function below different than the parent function y=|x|:
y=1/2|x+3|+1
Opens (Upward or Downward)
Horizontal Shift (write none if there is none)
Vertical Shift (write none if there is none)
Stretched (0<|a|<1), Compressed (|a|>1), or None
How is the absolute function below different than the parent function y=|x|:
y=-1/3|x-5|-5
Opens (Upward or Downward)
Horizontal Shift (write none if there is none)
Vertical Shift (write none if there is none)
Stretched (0<|a|<1), Compressed (|a|>1), or None
How is the absolute function below different than the parent function y=|x|:
y=.5|x-8|+3
Opens (Upward or Downward)
Horizontal Shift (write none if there is none)
Vertical Shift (write none if there is none)
Stretched (0<|a|<1), Compressed (|a|>1), or None
Simplify each radical
Simplify each radical
Simplify this expression. Your answer should not have negative exponents.
Simplify this expression. Your answer should not have negative exponents.
Simplify this expression. Your answer should not have negative exponents.
Sam is sitting in a boat on a lake. She can get burned by the sunlight that hits her directly and by sunlight that reflects off the water. Sunlight reflects off the water at the point (2, 0) and hits Sam at the point (3.5, 3). Write and graph the function that shows the path of the sunlight.
1) Graph the function
2) Write the function that shows the path of the sunlight.
A rainstorm begins as a drizzle, builds up to a heavy rain, and then drops back to a drizzle. The rate r (in inches per hour) at which it rains is given by the function r = −0.5 ⎜t − 1⎟ + 0.5, where t is the time (in hours). Graph the function. Determine for how long it rains and when it rains the hardest.
Opens (upward or downward)
Axis of Symmetry
Vertex
Slope
2) Use the critical values of this equation to graph it.
3) Determine for how long it rains
and when it rains the hardest.
While playing pool, a player tries to shoot the eight ball into the corner pocket as shown. Imagine that a coordinate plane is placed over the pool table. The eight ball is at (5, 5/4) and the pocket they are aiming for is at (10, 5). The player is going to bank the ball off the side at (6, 0).
2) Write an equation for the path of the ball.
3) Did the player make the shot? How do you know?
Explain the Error Explain why the graph shown is not the graph of y = ⎜x + 3⎟ + 2. What is the correct equation shown in the graph?
