1. true or false: the absolute value standard form is ( y = a|x - h| + k ) (with \standard form\ handwritten).
2. in which direction does the graph of ( y = |x + 3| + c ) shift as ( c ) decreases?
a. right
b. left
c. up
d. down (d is circled)
3. which of the following absolute value equations has reflected across the ( x )-axis, a vertical stretch by a factor of 2 and shifted 4 units to the right?
a. ( f(x) = \frac{1}{2}|x + 4| )
b. ( f(x) = -\frac{1}{2}|x + 4| )
c. ( f(x) = 2|x - 4| )
d. ( f(x) = -2|x - 4| )
for 4-7, determine the transformations applied to the absolute parent function, ( y = |x| ):

| equation | transformations |
| --- | --- |
| 4. ( y = |x - 3| + 5 ) | |
| 5. ( y = -|x| - 7 ) | |
| 6. ( y = \frac{1}{2}|x - 8| ) | |
| 7. ( y = -3|x + 4| ) | |

for 8-10, describe the transformations that have occurred to the graph from its parent function ( y = |x| ), then write the corresponding equation.

8. graph (v-shape on grid, vertex at (0,2)) with ( y = |x| + 2 ) written.
9. graph (v-shape, vertex shifted right and down) with ( y = |x - 1| - 4 ) written.
10. graph (reflected, shifted left and up) with ( y = -|x + 2| + 3 ) written.

Question

1. true or false: the absolute value standard form is ( y = a|x - h| + k ) (with \standard form\ handwritten).
2. in which direction does the graph of ( y = |x + 3| + c ) shift as ( c ) decreases?
   a. right
   b. left
   c. up
   d. down (d is circled)
3. which of the following absolute value equations has reflected across the ( x )-axis, a vertical stretch by a factor of 2 and shifted 4 units to the right?
   a. ( f(x) = \frac{1}{2}|x + 4| )
   b. ( f(x) = -\frac{1}{2}|x + 4| )
   c. ( f(x) = 2|x - 4| )
   d. ( f(x) = -2|x - 4| )
for 4-7, determine the transformations applied to the absolute parent function, ( y = |x| ):

| equation | transformations |
| --- | --- |
| 4. ( y = |x - 3| + 5 ) |  |
| 5. ( y = -|x| - 7 ) |  |
| 6. ( y = \frac{1}{2}|x - 8| ) |  |
| 7. ( y = -3|x + 4| ) |  |

for 8-10, describe the transformations that have occurred to the graph from its parent function ( y = |x| ), then write the corresponding equation.

8. graph (v-shape on grid, vertex at (0,2)) with ( y = |x| + 2 ) written.
9. graph (v-shape, vertex shifted right and down) with ( y = |x - 1| - 4 ) written.
10. graph (reflected, shifted left and up) with ( y = -|x + 2| + 3 ) written.

Accepted Answer

### Question 1
# Explanation:
The standard form of an absolute value function is $ y = a|x - h| + k $, where $(h, k)$ is the vertex. So the statement is true.
# Answer:
True

### Question 2
# Explanation:
## Step1: Recall vertical shift rule
For a function $ y = f(x) + c $, when $ c $ decreases, if $ c $ is positive and we make it smaller (or negative and more negative), the graph shifts down.
## Step2: Analyze $ y = |x + 3| + c $
Here, $ c $ is the vertical shift parameter. As $ c $ decreases, the graph moves down.
# Answer:
d. Down

### Question 3
# Explanation:
## Step1: Recall transformations
- Reflection over $ x $-axis: $ a $ is negative.
- Vertical stretch by factor $ 2 $: $ |a| = 2 $.
- Shift 4 units right: $ h = 4 $ (so $ x - 4 $ in the absolute value).
## Step2: Analyze options
- Option a: $ \frac{1}{2}|x + 4| $ – no reflection, stretch factor $ \frac{1}{2} $, shift left 4. Incorrect.
- Option b: $ -\frac{1}{2}|x + 4| $ – reflection, stretch factor $ \frac{1}{2} $, shift left 4. Incorrect.
- Option c: $ 2|x - 4| $ – no reflection, stretch factor 2, shift right 4. Incorrect.
- Option d: $ -2|x - 4| $ – reflection (negative $ a $), stretch factor 2 ($ |a| = 2 $), shift right 4 ($ x - 4 $). Correct.
# Answer:
d. $ f(x) = -2|x - 4| $

### Question 4
# Explanation:
## Step1: Identify $ h $ and $ k $
For $ y = |x - 3| + 5 $, compare with $ y = |x - h| + k $. Here, $ h = 3 $, $ k = 5 $.
## Step2: Determine transformations
- Shift right 3 units (because $ h = 3 $, $ x - 3 $) and shift up 5 units (because $ k = 5 $).
# Answer:
Shift 3 units right, 5 units up

### Question 5
# Explanation:
## Step1: Identify $ a $ and $ k $
For $ y = -|x| - 7 $, $ a = -1 $, $ k = -7 $.
## Step2: Determine transformations
- Reflection over $ x $-axis (because $ a = -1 $) and shift down 7 units (because $ k = -7 $).
# Answer:
Reflect over $ x $-axis, shift down 7 units

### Question 6
# Explanation:
## Step1: Identify $ a $ and $ h $
For $ y = \frac{1}{2}|x - 8| $, $ a = \frac{1}{2} $, $ h = 8 $.
## Step2: Determine transformations
- Vertical compression by factor $ \frac{1}{2} $ (because $ |a|=\frac{1}{2}<1 $) and shift right 8 units (because $ h = 8 $, $ x - 8 $).
# Answer:
Vertical compression by $ \frac{1}{2} $, shift right 8 units

### Question 7
# Explanation:
## Step1: Identify $ a $ and $ h $
For $ y = -3|x + 4| $, rewrite as $ y = -3|x - (-4)| $, so $ a = -3 $, $ h = -4 $.
## Step2: Determine transformations
- Reflection over $ x $-axis (because $ a = -3<0 $), vertical stretch by factor 3 (because $ |a| = 3>1 $), shift left 4 units (because $ h=-4 $, $ x - (-4)=x + 4 $).
# Answer:
Reflect over $ x $-axis, vertical stretch by 3, shift left 4 units

### Question 8
# Explanation:
## Step1: Analyze the graph and equation
The equation is $ y = |x| + 2 $. Compare with parent $ y = |x| $.
## Step2: Determine transformation
Adding 2 to the parent function shifts the graph up 2 units.
# Answer:
Transformation: Shift up 2 units. Equation: $ y = |x| + 2 $

### Question 9
# Explanation:
## Step1: Analyze the graph
The vertex is at $ (1, -1) $, so the equation is $ y = |x - 1| - 1 $ (since vertex form is $ y = |x - h| + k $, $ h = 1 $, $ k = -1 $).
## Step2: Determine transformation
From $ y = |x| $, shifting right 1 unit (because $ x - 1 $) and down 1 unit (because $ -1 $).
# Answer:
Transformation: Shift right 1 unit, shift down 1 unit. Equation: $ y = |x - 1| - 1 $

### Question 10
# Explanation:
## Step1: Analyze the equation $ y = -|x + 2| + 3 $
Compare with parent $ y = |x| $.
## Step2: Determine transformations
- $ a=-1 $: Reflect over $ x $-axis.
- $ x + 2=x - (-2) $: Shift left 2 units.
- $ +3 $: Shift up 3 units.
# Answer:
Transformation: Reflect over $ x $-axis, shift left 2 units, shift up 3 units. Equation: $ y = -|x + 2| + 3 $

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QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Recall vertical shift rule

Step2: Analyze \( y = |x + 3| + c \)

Step1: Recall transformations

Step2: Analyze options

Answer:

Question 2