Question

use a table of values to graph the function \\(f(x) = |x + 2| - 3\\). select the correct graph below.

Explanation:

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"concepts_used": [
"Function Transformations",
"Horizontal Translation",
"Vertical Translation"
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"new_concepts": [
"Absolute Value Graphing"
],
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"Horizontal Translation",
"Vertical Translation",
"Absolute Value Graphing"
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</pre_analysis><reasoning>

Identify the parent function and transformations

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &\text{Parent function: } y = |x|\\ &\text{Given function: } f(x) = |x + 2| - 1 \end{aligned}$$

\]

Determine the horizontal translation

Using the Horizontal Translation knowledge point
\[

$$\begin{aligned} &x + 2 = 0 \implies x = -2\\ &\text{Shift parent function left by } 2 \text{ units.} \end{aligned}$$

\]

Determine the vertical translation

Using the Vertical Translation knowledge point
\[

$$\begin{aligned} &\text{Constant term: } -1\\ &\text{Shift parent function down by } 1 \text{ unit.} \end{aligned}$$

\]

Locate the vertex and key points

We calculate the vertex and key coordinates to identify the correct graph:

• Vertex: The vertex of the parent function \( (0,0) \) shifts left \( 2 \) and down \( 1 \), giving the vertex at \( (-2, -1) \).
• y-intercept: Let \( x = 0 \):

\[
f(0) = |0 + 2| - 1 = 2 - 1 = 1 \implies (0, 1)
\]

• x-intercepts: Let \( f(x) = 0 \):

\[

x + 2

- 1 = 0 \implies

x + 2

= 1 \implies x + 2 = \pm 1 \implies x = -1 \text{ or } x = -3 \implies (-1, 0), (-3, 0)

\]

Compare with the given options

We analyze the vertex and intercepts of each graph:

• Graph A: The vertex is at \( (-2, 1) \). This is incorrect.
• Graph B: The vertex is at \( (2, 0) \). This is incorrect.
• Graph C: The vertex is at \( (1, 1) \). This is incorrect.
• Graph D: The vertex is at \( (-2, -1) \), the y-intercept is at \( (0, 1) \), and the x-intercepts are at \( (-3, 0) \) and \( (-1, 0) \). This matches our calculated points perfectly.

</reasoning><answer>
<mcq-option>(A) V-shaped graph with vertex at (-2, 1)</mcq-option>
<mcq-option>(B) V-shaped graph with vertex at (2, 0)</mcq-option>
<mcq-option>(C) V-shaped graph with vertex at (1, 1)</mcq-option>
<mcq-correct>(D) V-shaped graph with vertex at (-2, -1)</mcq-correct>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-2, -1]],
"properties": {
"name": "Vertex (-2, -1)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 1]],
"properties": {
"name": "(0, 1)",
"size": 4,
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "Math.abs(x + 2) - 1", "latex": "|x + 2| - 1"}, -10, 10],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3
}
}
]
}
</plot>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Absolute Value Graphing"
]
}
</post_analysis>

Answer:

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</pre_analysis><reasoning>

Identify the parent function and transformations

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &\text{Parent function: } y = |x|\\ &\text{Given function: } f(x) = |x + 2| - 1 \end{aligned}$$

\]

Determine the horizontal translation

Using the Horizontal Translation knowledge point
\[

$$\begin{aligned} &x + 2 = 0 \implies x = -2\\ &\text{Shift parent function left by } 2 \text{ units.} \end{aligned}$$

\]

Determine the vertical translation

Using the Vertical Translation knowledge point
\[

$$\begin{aligned} &\text{Constant term: } -1\\ &\text{Shift parent function down by } 1 \text{ unit.} \end{aligned}$$

\]

Locate the vertex and key points

We calculate the vertex and key coordinates to identify the correct graph:

• Vertex: The vertex of the parent function \( (0,0) \) shifts left \( 2 \) and down \( 1 \), giving the vertex at \( (-2, -1) \).
• y-intercept: Let \( x = 0 \):

\[
f(0) = |0 + 2| - 1 = 2 - 1 = 1 \implies (0, 1)
\]

• x-intercepts: Let \( f(x) = 0 \):

\[

x + 2

- 1 = 0 \implies

x + 2

= 1 \implies x + 2 = \pm 1 \implies x = -1 \text{ or } x = -3 \implies (-1, 0), (-3, 0)

\]

Compare with the given options

We analyze the vertex and intercepts of each graph:

• Graph A: The vertex is at \( (-2, 1) \). This is incorrect.
• Graph B: The vertex is at \( (2, 0) \). This is incorrect.
• Graph C: The vertex is at \( (1, 1) \). This is incorrect.
• Graph D: The vertex is at \( (-2, -1) \), the y-intercept is at \( (0, 1) \), and the x-intercepts are at \( (-3, 0) \) and \( (-1, 0) \). This matches our calculated points perfectly.

</reasoning><answer>
<mcq-option>(A) V-shaped graph with vertex at (-2, 1)</mcq-option>
<mcq-option>(B) V-shaped graph with vertex at (2, 0)</mcq-option>
<mcq-option>(C) V-shaped graph with vertex at (1, 1)</mcq-option>
<mcq-correct>(D) V-shaped graph with vertex at (-2, -1)</mcq-correct>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-2, -1]],
"properties": {
"name": "Vertex (-2, -1)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 1]],
"properties": {
"name": "(0, 1)",
"size": 4,
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "Math.abs(x + 2) - 1", "latex": "|x + 2| - 1"}, -10, 10],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3
}
}
]
}
</plot>

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"Mathematics",
"Geometry",
"Absolute Value Graphing"
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