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Unit 3: Absolute
Value
Schedule of Upcoming Classes
Day 1 A 9/21 Introducing AV functions
Day 2 A 9/
Transformations & Analyzing
Graphs
Day 3 A 9/27 Review of Graphing AV functions *
Day 4 A 9/
Quiz: Graphing & Analyzing
AV Functions
Day 5 A 10/3 Solving & Graphing AV Equations
Day 6 A 10/5 Solving & Graphing AV Inequalities
Day 7 A 10/7 Unit Review
Day 8 A 10/12 Unit Test
- Skills Review due & Skills Check See Ms. Raschiatore AS SOON AS POSSIBLE to get work and any help you need. Blank copies of the notes are on my CMS page. Homework assignments are also on my CMS page if you lose your packet. Need Help? Mu Alpha Theta is available to help Monday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:15. Ms. Raschiatore will be available to answer questions in the morning beginning at 8: (room L404) Need to make up a test/quiz? Math Make Up Room is open Weds mornings and Tues/Thurs afternoons. Schedule is posted on front white board near the door. Algebra 2 Name ____________________ Unit 3 “Absolute Value Functions” Date ____________ Block ___
Day 1: Introducing… The Absolute Value Function ~ Using the calculator to graph and finding key characteristics of the graph Let’s take a look at y = x (Graph in your calculator as ) What happens if we change every negative y-value to a positive value? i.e. make the point (3, -3) become (3, +3) Does this sound familiar? What takes negative values and makes them positive? Introducing …….. the Absolute Value Function We can analyze the parent function for special points and behavior -
Use your calculator to graph:
Domain: Range: Vertex: y-intercept: zeros (roots, x-intercepts, solutions): Increasing: Decreasing: End Behavior: Slope of right branch:
Are you noticing any patterns yet? Let’s look at slope of the
right branch.
Review topic: 1.
- Graph the inverse of the Absolute Value Function (start out with the original ) Think about how you graph an inverse! Is the inverse a function?
Domain:
Range:
Vertex:
Y-intercept:
Zeros / X-intercepts:
Increasing:
Decreasing:
End Behavior:
Slope of right branch:
Domain:
Range:
Vertex:
Y-intercept:
Zeros / X-intercepts:
Increasing:
Decreasing:
End Behavior:
Slope of right branch:
Were you expecting this? Why? 2.
- Graph an absolute value function that has a removable discontinuity at (3, 4)
Day 2: Graphing Absolute Value FUNctions Using
TRANSFORMATIONS
In these notes we will Learn a new technique for graphing a function – shifting it up, down, left, right So we can …. Graph absolute value functions WITHOUT a calculator Eventually graph ANY function given its parent shape. First, let’s graph the absolute value “parent function”, y = | x | Use your calculator to graph this function in Y 1 What is the vertex of the graph?
Exploration of Transformations – Vertical Shifts
- Graph y = |x| + 2 on your calculator in Y 2. a) Sketch this graph and the “parent function”.
b) How does the graph move? (up or down) _____
c) What is the vertex of the graph? ______
- Graph y = |x| - 5 on your calculator in Y 2. a) Sketch this graph and the “parent function”.
b) How does the graph move? (up or down) _____
c) What is the vertex of the graph? ______
- Given that y = a|x – h| + k is the symbolic form of the absolute value function, what does the parameter k control?
Exploration of Transformations – Horizontal Shifts
- Graph y = |x| on your calculator in Y 1. a) Sketch a graph of the function.
b) What is the vertex of the graph? ______
- Graph y = |x - 1| on your calculator in Y 2. a) Sketch a graph of the function and the function in #1.
b) How does the graph move? Left or Right? _____
c) What was the SIGN inside the absolute value?
d) What is the vertex of the graph? ______
- Graph y = |x + 3| on your calculator in Y 2. a) Sketch a graph of the function and the function in #1.
b) How does the graph move? Left or Right? _____
c) What was the SIGN inside the absolute value?
d) What is the vertex of the graph? ______
- Graph y = |x - 5| on your calculator in Y 2. a) Sketch a graph of the function and the function in #1.
b) How does the graph move? Left or Right? _____
c) What was the SIGN inside the absolute value?
d) What is the vertex of the graph? ______
- Given that y = a|x – h| + k is the symbolic from of the absolute value function, what does the parameter h control? When we have |x – h|, what direction does the graph move? When we have |x + h|, what direction does the graph move? How is the motion related to the sign of h?
Exploration of Transformations – Vertical Stretch or Shrink
- Graph y = |x| on your calculator in Y 1.
a) What direction does the graph open? _____
b) What is the vertex of the graph? ______
c) Fill in the table to the right. These coordinates are the basic ordered pairs of the absolute value function.
- Graph y = 2|x| on your calculator in Y 2.
a) What direction does the graph open? _____
b) What is the vertex of the graph? ______
c) Fill in the table. How do these y-coordinates compare with the y-coordinates in question 1? Is the graph fatter or skinnier?
- Graph y = ½ |x| on your calculator in Y 2.
a) What direction does the graph open? _____
b) What is the vertex of the graph? ______
c) Fill in the table. How do these y-coordinates compare with the y-coordinates in question 1? Is the graph ‘taller’ or ‘shorter’?
- Graph y = -|x| on your calculator in Y 2. a) Sketch a graph of the function and the function in #1.
b) How did the graph change? __________
- Graph y = -2|x| on your calculator in Y 2. a) Sketch a graph of the function and the function in #1.
b) How did the graph change? __________
- Given that y = a|x – h| + k is the symbolic from of the absolute value function, what does the parameter a control? 1. __________________________ 2. __________________________ Examples 1 and 2 Examples 3 and 4
x y
x y
x y
x y x y x y Graph the following using your knowledge of transformations (no calculator). Verify (check your answer) by graphing on your calculator and comparing your answer to the calculators.
- y = ¼ |x + 4| - 9 2. y = - 2|x + 1| + 6 3. y = 4|x – 3| + 5 Domain: _________ Domain: _________ Domain: ________ Range: __________ Range: __________ Range: ________ Given the absolute value equation graph, write the absolute value equation:
y = ___________________ f(x) = __________________
Day 3: Review Graphing Absolute Value Functions
(with and without a calculator)
x y x
x y
- Which absolute value function(s) open up? (there may be more than one answer!) A. y = -2| x – 5| B. y = | x + 1| – 7 C. y = -| x + 4| + 8 D. y = ¼| x – 9|
- Which absolute value function(s) are vertically stretched? (there may be more than one answer!) A. y = -2| x – 5| B. y = | x + 1| – 7 C. y = -| x + 4| + 8 D. y = ¼| x – 9|
- Which absolute value function(s) have an absolute minimum at the vertex? (there may be more than one answer!) A. y = -2| x – 5| B. y = | x + 1| – 7 C. y = -| x + 4| + 8 D. y = ¼| x – 9|
- Given f ( x ) = | x + 9|. The vertex of the function moves from (0, 0) nine units ______: A. left B. right C. up D. down
- Sketch What is the range? What is the end behavior? The vertex is… (circle all that apply) A) a relative minimum B) a relative maximum C) an absolute minimum D) an absolute maximum
Day 5: Absolute Value Equations Objective : To understand the definition of absolute value and to know how to use this definition in solving absolute value equations. Absolute Value means __________________________________________________ Absolute Value Equations What it MEANS: Graph an Absolute Value Equation on a Number Line
1. |x| = 4
As distance: |x - 0| = 4 “the set of points whose distance from 0 is 4” Another way – think of two FUNCTIONS. Where are they EQUAL? Graph y=|x| and y= The solution(s) is/are the x-coordinates of the points of intersection
3. |x - 4| = 3
As distance: the set of points whose distance from ____ is equal to ___ As functions - What two functions are we looking at here? Where are they EQUAL?
How we SOLVE ALGEBRAICALLY: To Solve an Absolute Value Equation
- Isolate the absolute value symbol on one side of the equal sign
- Break the equation into 2 derived equations – the positive case and the negative case
- Solve both equations
- Check your solutions (WARNING: There may be extraneous solutions!)
- |x+3| = 8 2. -3|x - 1| + 2 = – 4 Let’s verify our answers graphically (this is how you can use your calculator to check your hw) Y 1 = Y 2 = Y 1 = Y 2 =
- |2x + 12| = 4x 4. |4x + 5| = 2x + 4 x y x y
Algebra 2 WARM-UP (before absolute value Inequalities) Review from Algebra 1 ~ solving linear inequalities (We will use this today!) A) Inequality Symbols : < __________________ ,
___________________ , ____________________ Don’t forget switch the sign of the inequality when multiplying / dividing by a negative # Switch Don’t switch -3x < 9 3x < -12 Original Problem(s) x > -3 x < -4 Solution(s) B) Graphing Linear Inequalities: Closed circle Open Circle , < , > C) Solve the following linear inequalities , then graph each solution: EX 1] 3x + 12 < 9 EX 2] 4x – 3 6x + 15 EX 3] –4x + 16 > 4 EX 4] –3x – 6 3x + 6
D) Graphing Compound Inequalities EX] -1 < x < 2 EX] x -2 or x > 1 Graph the following inequalities.
- or Solve the compound inequality, and then graph your solution.
- or
Day 6: Absolute Value Inequalities
Solve and Graph the Absolute Value Inequality:
| x + 3 | ≥ 5 AND or OR? Verify graphically
Graph your solution:
3. | x – 2 | > 4 AND or OR?
Verify graphically
Does this look like AND or OR?
4. | 5x + 1 | ≥ 16 AND or
OR?
Verify graphically
Does this look like AND or OR?
. | 5x + 1 | – 4 ≤ 14 AND or OR?
Verify graphically
Try these:
- |3x + 1|+ 2 < 8
- 2|x - 3|- 2 ≥ 8
