practice a

in exercises 1-3, graph the function. compare the graph to the graph of $f(x) = x^2$.
1. $g(x) = x^2 + 4$
2. $h(x) = x^2 + 7$
3. $h(x) = x^2 - 2$

in exercises 4-6, graph the function. compare the graph to the graph of $f(x) = x^2$.
4. $g(x) = -x^2 + 1$
5. $h(x) = -x^2 - 3$
6. $j(x) = 3x^2 - 2$

in exercises 7 and 8, describe the transformation from the graph of $f$ to the graph of $g$. then graph $f$ and $g$ in the same coordinate plane. write an equation that represents $g$ in terms of $x$.
8. $f(x) = \frac{1}{3}x^2 - 1$

Question

practice a

in exercises 1-3, graph the function. compare the graph to the graph of $f(x) = x^2$.
1. $g(x) = x^2 + 4$
2. $h(x) = x^2 + 7$
3. $h(x) = x^2 - 2$

in exercises 4-6, graph the function. compare the graph to the graph of $f(x) = x^2$.
4. $g(x) = -x^2 + 1$
5. $h(x) = -x^2 - 3$
6. $j(x) = 3x^2 - 2$

in exercises 7 and 8, describe the transformation from the graph of $f$ to the graph of $g$. then graph $f$ and $g$ in the same coordinate plane. write an equation that represents $g$ in terms of $x$.
8. $f(x) = \frac{1}{3}x^2 - 1$

Accepted Answer

{ "quality": "clear", "question_count": 8, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Quadratic Transformations", "Graphing Quadratic Functions" ], "current_concepts": [ "Quadratic Transformations", "Graphing Quadratic Functions" ] } ## Analyze transformations for Exercises 1–3 $$ \begin{aligned} &\text{1. } g(x) = x^2 + 4 \implies \text{Vertical translation up by 4 units.}\ &\text{2. } h(x) = x^2 + 7 \implies \text{Vertical translation up by 7 units.}\ &\text{3. } k(x) = x^2 - 2 \implies \text{Vertical translation down by 2 units.} \end{aligned} $$ ## Analyze transformations for Exercises 4–6 $$ \begin{aligned} &\text{4. } g(x) = -x^2 + 1 \implies \text{Reflection in the } x\text{-axis and vertical translation up by 1 unit.}\ &\text{5. } h(x) = -x^2 - 3 \implies \text{Reflection in the } x\text{-axis and vertical translation down by 3 units.}\ &\text{6. } j(x) = 3x^2 - 2 \implies \text{Vertical stretch by a factor of 3 and vertical translation down by 2 units.} \end{aligned} $$ ## Analyze transformations for Exercises 7 and 8 $$ \begin{aligned} &\text{7. [Not fully visible in image]}\ &\text{8. } f(x) = \frac{4}{3}x^2 - 1 \implies \text{Vertical stretch by a factor of } \frac{4}{3} \text{ and vertical translation down by 1 unit.} \end{aligned} $$ ## Question 1 The graph of $g(x) = x^2 + 4$ is a vertical translation 4 units up of the graph of $f(x) = x^2$. ## Question 2 The graph of $h(x) = x^2 + 7$ is a vertical translation 7 units up of the graph of $f(x) = x^2$. ## Question 3 The graph of $k(x) = x^2 - 2$ is a vertical translation 2 units down of the graph of $f(x) = x^2$. ## Question 4 The graph of $g(x) = -x^2 + 1$ is a reflection in the $x$-axis and a vertical translation 1 unit up of the graph of $f(x) = x^2$. ## Question 5 The graph of $h(x) = -x^2 - 3$ is a reflection in the $x$-axis and a vertical translation 3 units down of the graph of $f(x) = x^2$. ## Question 6 The graph of $j(x) = 3x^2 - 2$ is a vertical stretch by a factor of 3 and a vertical translation 2 units down of the graph of $f(x) = x^2$. ## Question 8 The graph of $f(x) = \frac{4}{3}x^2 - 1$ is a vertical stretch by a factor of $\frac{4}{3}$ and a vertical translation 1 unit down of the graph of $f(x) = x^2$. { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze transformations for Exercises 1–3

Analyze transformations for Exercises 4–6

Analyze transformations for Exercises 7 and 8

Answer:

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 8