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

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) Using the *Quadratic Transformations* knowledge point $$ \begin{aligned} &f(x) = x^2,\quad g(x) = 3(x-4)^2 - 2\ &\text{Horizontal shift: } x \to x - 4 \implies \text{right by } 4 \text{ units}\ &\text{Vertical shift: } \text{subtract } 2 \implies \text{down by } 2 \text{ units} \end{aligned} $$ ## Determine direction and width of g(x) Using the *Vertical Stretching* knowledge point $$ \begin{aligned} &\text{Leading coefficient } a = 3 > 0 \implies \text{opens upwards}\ &|a| = 3 > 1 \implies \text{narrower (vertically stretched) than } f(x) \end{aligned} $$ ## Match positive leading coefficients Using the *Vertical Stretching* and *Vertical Shrinking* knowledge points $$ \begin{aligned} &y = 3x^2 \implies \text{opens upwards, vertically stretched (narrower) } \implies \text{Graph (b)}\ &y = \frac{1}{3}x^2 \implies \text{opens upwards, vertically compressed (wider) } \implies \text{Graph (a)} \end{aligned} $$ ## Match negative leading coefficients Using the *Vertical Stretching* and *Vertical Shrinking* knowledge points $$ \begin{aligned} &y = -3x^2 \implies \text{opens downwards, vertically stretched (narrower) } \implies \text{Graph (c)}\ &y = -\frac{1}{3}x^2 \implies \text{opens downwards, vertically compressed (wider) } \implies \text{Graph (d)} \end{aligned} $$ | 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" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Analyze transformations of g(x)

Determine direction and width of g(x)

Match positive leading coefficients

Match negative leading coefficients

Answer: