integrated math 2
chapter 3 test
name:
date:
period:
directions: show all work for each problem in the space shown and then mark your answer on the answer sheet provided

given $f(x) = x^2$ and $g(x) = 3(x-4)^2 - 2$
graph both functions and then choose true or false for each statement given.

1) $g(x)$ is $f(x)$ translated down 4 units and right 4 unit.
true / false

2) $g(x)$ is $f(x)$ translated right 4 units and down 2 unit.
true / false

3) $g(x)$ opens upwards.
true / false

4) $g(x)$ is wider than $f(x)$.
true / false

match the following equations on the left with their correct graph on the right.

5) $y = 3x^2$
6) $y = -3x^2$
7) $y = \frac{1}{3}x^2$
8) $y = -\frac{1}{3}x^2$

Question

integrated math 2
chapter 3 test
name:
date:
period:
directions: show all work for each problem in the space shown and then mark your answer on the answer sheet provided

given $f(x) = x^2$ and $g(x) = 3(x-4)^2 - 2$
graph both functions and then choose true or false for each statement given.

1) $g(x)$ is $f(x)$ translated down 4 units and right 4 unit.
true / false

2) $g(x)$ is $f(x)$ translated right 4 units and down 2 unit.
true / false

3) $g(x)$ opens upwards.
true / false

4) $g(x)$ is wider than $f(x)$.
true / false

match the following equations on the left with their correct graph on the right.

5) $y = 3x^2$
6) $y = -3x^2$
7) $y = \frac{1}{3}x^2$
8) $y = -\frac{1}{3}x^2$

Accepted Answer

{ "quality": "clear", "question_count": 8, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Quadratic Transformations", "Vertical Stretching", "Vertical Shrinking" ], "new_concepts": [], "current_concepts": [ "Quadratic Transformations", "Vertical Stretching", "Vertical Shrinking", "Parabola Vertex Form" ] } ## Analyze transformations of $g(x) = 3(x-4)^2 - 2$ Using the *Quadratic Transformations* knowledge point $$ \begin{aligned} &f(x) = x^2 \implies g(x) = a(x-h)^2 + k\ &a = 3, \quad h = 4, \quad k = -2\ &\text{Translation: right } 4 \text{ units, down } 2 \text{ units} \end{aligned} $$ ## Determine vertical scaling and orientation of $g(x)$ Using the *Vertical Stretching* knowledge point $$ \begin{aligned} &a = 3 > 0 \implies \text{opens upwards}\ &|a| = 3 > 1 \implies \text{vertically stretched (narrower than } f(x)\text{)} \end{aligned} $$ ## Evaluate statements 1 to 4 Using the *Quadratic Transformations* and *Vertical Stretching* knowledge points $$ \begin{aligned} &\text{Statement 1: } g(x) \text{ is translated down 4 and right 4. } \implies \text{False}\ &\text{Statement 2: } g(x) \text{ is translated right 4 and down 2. } \implies \text{True}\ &\text{Statement 3: } g(x) \text{ opens upwards. } \implies \text{True}\ &\text{Statement 4: } g(x) \text{ is wider than } f(x). \implies \text{False} \end{aligned} $$ ## Match equations 5 and 6 to graphs Using the *Vertical Stretching* knowledge point $$ \begin{aligned} &y = 3x^2: a = 3 > 0 \text{ (opens up, narrow)} \implies \text{Graph b}\ &y = -3x^2: a = -3 < 0 \text{ (opens down, narrow)} \implies \text{Graph c} \end{aligned} $$ ## Match equations 7 and 8 to graphs Using the *Vertical Shrinking* knowledge point $$ \begin{aligned} &y = \frac{1}{3}x^2: a = \frac{1}{3} > 0 \text{ (opens up, wide)} \implies \text{Graph a}\ &y = -\frac{1}{3}x^2: a = -\frac{1}{3} < 0 \text{ (opens down, wide)} \implies \text{Graph d} \end{aligned} $$ | No. | Answer | |---|---| | 1 | False | | 2 | True | | 3 | True | | 4 | False | | 5 | b | | 6 | c | | 7 | a | | 8 | d | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze transformations of \(g(x) = 3(x-4)^2 - 2\)

Determine vertical scaling and orientation of \(g(x)\)

Evaluate statements 1 to 4

Match equations 5 and 6 to graphs

Match equations 7 and 8 to graphs

Answer: