algebra 1 – first quarter
classwork #3: lesson 1.4 – solving absolute value equations
name: adrianna george
period: 5th
score:
date: 9/3/25
direction: explore absolute value functions, recognize their v-shape, and identify the vertex.

part a: warm-up (dok 1 – recall)
1. evaluate:
a. |-5| = ______
b. |7| = ______
c. |0| = ______
2. in your own words, what does absolute value mean? ______

part b: finding the vertex (dok 2/3 – reasoning)
3. find the vertex for each:
a. y = |x + 5| → vertex: ______
b. y = |x - 2| - 4 → vertex: ______
c. y = -|x - 1| + 3 → vertex: ______
4. explain your reasoning for one of your answers: ______

part c: rigor question – analyze & compare (dok 3)
5. compare the graphs of y = |x - 4| and y = -|x - 4|.
a. how are their vertices the same?
b. how are their shapes different?

part d: rigor question – real-world application (dok 4)
6. a skateboard ramp has a v-shape modeled by the function y = |x| + 2.
a. what is the vertex?
b. what is the new equation if the ramp is shifted to the right by 3 units and up by 1 unit? what is the new vertex?

Question

algebra 1 – first quarter
classwork #3: lesson 1.4 – solving absolute value equations
name: adrianna george
period: 5th
score:
date: 9/3/25
direction: explore absolute value functions, recognize their v-shape, and identify the vertex.

part a: warm-up (dok 1 – recall)
1. evaluate:
   a. |-5| = ______
   b. |7| = ______
   c. |0| = ______
2. in your own words, what does absolute value mean? ______

part b: finding the vertex (dok 2/3 – reasoning)
3. find the vertex for each:
   a. y = |x + 5| → vertex: ______
   b. y = |x - 2| - 4 → vertex: ______
   c. y = -|x - 1| + 3 → vertex: ______
4. explain your reasoning for one of your answers: ______

part c: rigor question – analyze & compare (dok 3)
5. compare the graphs of y = |x - 4| and y = -|x - 4|.
   a. how are their vertices the same?
   b. how are their shapes different?

part d: rigor question – real-world application (dok 4)
6. a skateboard ramp has a v-shape modeled by the function y = |x| + 2.
   a. what is the vertex?
   b. what is the new equation if the ramp is shifted to the right by 3 units and up by 1 unit? what is the new vertex?

Accepted Answer

### Part A: Warm - Up
#### 1. Evaluate
##### a. $|-5|$
# Explanation:
## Step1: Recall absolute value definition
The absolute value of a number $a$ is $|a|=\begin{cases}a, & a\geq0\-a, & a < 0\end{cases}$. For $- 5$, since $-5<0$, $|-5|=-(-5)$
## Step2: Calculate the result
$-(-5) = 5$
# Answer: $5$

##### b. $|7|$
# Explanation:
## Step1: Recall absolute value definition
For $7$, since $7\geq0$, $|7| = 7$
# Answer: $7$

##### c. $|0|$
# Explanation:
## Step1: Recall absolute value definition
For $0$, $|0|=0$ (by the definition of absolute value, the distance of $0$ from $0$ on the number line is $0$)
# Answer: $0$

#### 2. What does absolute value mean?
# Brief Explanations:
The absolute value of a number represents the distance of that number from $0$ on the number line. Distance is always non - negative, so the absolute value of a number is always greater than or equal to $0$. For example, $|-3|$ is the distance from $-3$ to $0$ on the number line, which is $3$, and $|4|$ is the distance from $4$ to $0$ on the number line, which is $4$.
# Answer: The absolute value of a number is its distance from 0 on the number line (and is non - negative).

### Part B: Finding the Vertex
The general form of an absolute value function is $y = |x - h|+k$, where $(h,k)$ is the vertex of the V - shaped graph.

#### 3. Find the vertex for each
##### a. $y = |x + 5|$
# Explanation:
## Step1: Rewrite in general form
We can rewrite $y=|x + 5|$ as $y=|x-(-5)|+0$
## Step2: Identify $h$ and $k$
Comparing with $y = |x - h|+k$, we have $h=-5$ and $k = 0$
# Answer: Vertex: $(-5,0)$

##### b. $y=|x - 2|-4$
# Explanation:
## Step1: Compare with general form
Comparing $y = |x - 2|-4$ with $y=|x - h|+k$, we have $h = 2$ and $k=-4$
# Answer: Vertex: $(2,-4)$

##### c. $y=-|x - 1|+3$
# Explanation:
## Step1: Compare with general form
Comparing $y=-|x - 1|+3$ with $y = |x - h|+k$ (the negative sign affects the direction of the graph, not the vertex location), we have $h = 1$ and $k = 3$
# Answer: Vertex: $(1,3)$

#### 4. Explain your reasoning for ONE of your answers (taking part 3a as an example)
# Brief Explanations:
The general form of an absolute value function is $y=|x - h|+k$, where the vertex is $(h,k)$. For the function $y = |x+5|$, we can rewrite it as $y=|x-(-5)|+0$. By comparing with the general form, we see that $h=-5$ and $k = 0$. So the vertex is the point $(-5,0)$ because in the absolute value function $y = |x - h|+k$, the vertex occurs at the point where the expression inside the absolute value ($x - h$) is equal to $0$, i.e., when $x=h$, and the $y$ - coordinate is $k$.
# Answer: For $y = |x + 5|$, rewrite as $y=|x-(-5)|+0$. Using $y = |x - h|+k$ (vertex $(h,k)$), $h=-5,k = 0$, so vertex is $(-5,0)$.

### Part C: Rigor Question - Analyze & Compare
#### 5. Compare the graphs of $y = |x - 4|$ and $y=-|x - 4|$
##### a. How are their vertices the same?
# Brief Explanations:
For the function $y = |x - 4|$, using the general form $y=|x - h|+k$, we have $h = 4$ and $k = 0$, so the vertex is $(4,0)$. For the function $y=-|x - 4|$, we can rewrite it as $y=-|x - 4|+0$. Comparing with $y = |x - h|+k$ (the negative sign affects the direction, not the vertex), we have $h = 4$ and $k = 0$. So both functions have their vertices at the point $(4,0)$ because the $x$ - value that makes the expression inside the absolute value zero is $x = 4$ for both functions, and the vertical shift $k$ is $0$ for both.
# Answer: Both vertices are at $(4,0)$

##### b. How are their shapes different?
# Brief Explanations:
The graph of $y = |x - 4|$ is a V - shaped graph that opens upwards (because the coefficient of the absolute value term is positive, $1$). The graph of $y=-|x - 4|$ is a V - shaped graph that opens downwards (because the coefficient of the absolute value term is negative, $- 1$). So the shape (the direction of opening) of the two graphs is different; one opens up and the other opens down.
# Answer: $y = |x - 4|$ opens up, $y=-|x - 4|$ opens down.

### Part D: Rigor Question - Real - World Application
#### 6. A skateboard ramp has a V - shape modeled by the function $y = |x|+2$
##### a. What is the vertex?
# Explanation:
## Step1: Compare with general form
The general form of an absolute value function is $y = |x - h|+k$. For $y = |x|+2$, we can rewrite it as $y=|x-0|+2$
## Step2: Identify $h$ and $k$
Comparing with $y = |x - h|+k$, we have $h = 0$ and $k = 2$
# Answer: Vertex: $(0,2)$

##### b. What is the new equation if the ramp is shifted to the right by 3 units and up by 1 unit? What is the new vertex?
# Explanation:
## Step1: Recall transformation rules
To shift a function $y = f(x)$ to the right by $a$ units, we replace $x$ with $x - a$ in the function. To shift a function up by $b$ units, we add $b$ to the function.
For the function $y = |x|+2$, shifting to the right by $3$ units: replace $x$ with $x - 3$, so we get $y=|x - 3|+2$. Then shifting up by $1$ unit: add $1$ to the function, so $y=|x - 3|+2 + 1=|x - 3|+3$
## Step2: Find the new vertex
Using the general form $y = |x - h|+k$ for $y=|x - 3|+3$, we have $h = 3$ and $k = 3$
# Answer: New equation: $y = |x - 3|+3$; New vertex: $(3,3)$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part A: Warm - Up

1. Evaluate

a. \(|-5|\)

Step1: Recall absolute value definition

Step2: Calculate the result

Step1: Recall absolute value definition

Step1: Recall absolute value definition

Answer:

b. \(|7|\)