in exercises 1-6, graph the function. compare the graph to the graph of $f(x) = x^2$.

1. $g(x) = 4x^2$
2. $h(x) = 1.5x^2$
3. $j(x) = \frac{1}{3}x^2$
4. $g(x) = -3x^2$
5. $k(x) = -\frac{5}{2}x^2$
6. $n(x) = -0.5x^2$

in exercises 7-9, use a graphing calculator to graph the function. compare the graph to the graph of $y = -5x^2$.

7. $y = 5x^2$
8. $y = -0.5x^2$
9. $y = -0.05x^2$

10. the arch support of a bridge can be modeled by $y = -0.00125x^2$, where $x$ and $y$ are measured in feet.
a. the width of the arch is 800 feet. describe the domain of the function. explain.
b. graph the function using the domain in part (a). find the height of the arch.

11. is the $y$-intercept of the graph of $y = ax^2$ always 0? explain.

in exercises 12-15, determine whether the statement is always, sometimes, or never true. explain your reasoning.

12. the graph of $f(x) = ax^2$ is narrower than the graph of $g(x) = dx^2$ when $d = -a$.

13. the graph of $f(x) = ax^2$ opens in the same direction as the graph of $g(x) = dx^2$ when $d = |a|$.

14. the graph of $f(x) = (ax)^2$ opens in the same direction as the graph of $g(x) = dx^2$ when $a^2 = d$.

15. the graph of $f(x) = (ax)^2$ is narrower than the graph of $g(x) = dx^2$ when $0

Question

in exercises 1-6, graph the function. compare the graph to the graph of $f(x) = x^2$.

1. $g(x) = 4x^2$
2. $h(x) = 1.5x^2$
3. $j(x) = \frac{1}{3}x^2$
4. $g(x) = -3x^2$
5. $k(x) = -\frac{5}{2}x^2$
6. $n(x) = -0.5x^2$

in exercises 7-9, use a graphing calculator to graph the function. compare the graph to the graph of $y = -5x^2$.

7. $y = 5x^2$
8. $y = -0.5x^2$
9. $y = -0.05x^2$

10. the arch support of a bridge can be modeled by $y = -0.00125x^2$, where $x$ and $y$ are measured in feet.
a. the width of the arch is 800 feet. describe the domain of the function. explain.
b. graph the function using the domain in part (a). find the height of the arch.

11. is the $y$-intercept of the graph of $y = ax^2$ always 0? explain.

in exercises 12-15, determine whether the statement is always, sometimes, or never true. explain your reasoning.

12. the graph of $f(x) = ax^2$ is narrower than the graph of $g(x) = dx^2$ when $d = -a$.

13. the graph of $f(x) = ax^2$ opens in the same direction as the graph of $g(x) = dx^2$ when $d = |a|$.

14. the graph of $f(x) = (ax)^2$ opens in the same direction as the graph of $g(x) = dx^2$ when $a^2 = d$.

15. the graph of $f(x) = (ax)^2$ is narrower than the graph of $g(x) = dx^2$ when $0 < a^2 < d$.

Accepted Answer

{ "quality": "clear", "question_count": 15, "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": [ "Quadratic Domain and Range", "Quadratic Modeling", "Quadratic Parameter Analysis" ], "current_concepts": [ "Quadratic Transformations", "Vertical Stretching", "Vertical Shrinking", "Quadratic Domain and Range", "Quadratic Modeling", "Quadratic Parameter Analysis" ] } ## Analyze Exercises 1–6 Using the **Quadratic Transformations**, **Vertical Stretching**, and **Vertical Shrinking** knowledge points $$ \begin{aligned} &\text{For } y = ax^2:\ &1.\ g(x) = 4x^2: |a| = 4 > 1 \implies \text{vertical stretch by a factor of 4, opens up.}\ &2.\ h(x) = 1.5x^2: |a| = 1.5 > 1 \implies \text{vertical stretch by a factor of 1.5, opens up.}\ &3.\ j(x) = \frac{1}{3}x^2: |a| = \frac{1}{3} < 1 \implies \text{vertical shrink by a factor of } \frac{1}{3}\text{, opens up.}\ &4.\ g(x) = -3x^2: |a| = 3 > 1, a < 0 \implies \text{vertical stretch by a factor of 3 and reflection in the x-axis.}\ &5.\ k(x) = \frac{5}{2}x^2: |a| = 2.5 > 1 \implies \text{vertical stretch by a factor of 2.5, opens up.}\ &6.\ n(x) = -0.5x^2: |a| = 0.5 < 1, a < 0 \implies \text{vertical shrink by a factor of 0.5 and reflection in the x-axis.} \end{aligned} $$ ## Analyze Exercises 7–9 Using the **Quadratic Transformations** knowledge point $$ \begin{aligned} &\text{Compare to } y = -5x^2:\ &7.\ y = 5x^2: |5| = |-5| \implies \text{reflection in the x-axis.}\ &8.\ y = -0.5x^2: |-0.5| < |-5| \implies \text{vertical shrink (wider) and opens in the same direction.}\ &9.\ y = -0.05x^2: |-0.05| < |-5| \implies \text{vertical shrink (much wider) and opens in the same direction.} \end{aligned} $$ ## Solve Exercise 10 Using the **Quadratic Modeling** knowledge point $$ \begin{aligned} &\text{Equation: } y = -0.00125x^2\ &\text{a. Width is } 800\text{ ft. Assuming symmetry about the y-axis: } -400 \le x \le 400.\ &\text{b. Height is the maximum value of } y \text{ minus the value at the boundaries:}\ &y(0) = 0\ &y(400) = -0.00125(400)^2 = -200\ &\text{Height } = 0 - (-200) = 200\text{ feet.} \end{aligned} $$ ## Solve Exercise 11 Using the **Quadratic Parameter Analysis** knowledge point $$ \begin{aligned} &\text{Let } x = 0:\ &y = a(0)^2 = 0\ &\text{The y-intercept is always } 0\text{ because substituting } x=0 \text{ always yields } y=0. \end{aligned} $$ ## Analyze Exercises 12–15 Using the **Quadratic Parameter Analysis** knowledge point $$ \begin{aligned} &12.\ |a| = |-a| = |d| \implies \text{They have the same width. Statement is Never True.}\ &13.\ d = |a| \ge 0 \implies g(x) \text{ opens up. } f(x) \text{ opens in the same direction only if } a > 0\text{. Statement is Sometimes True.}\ &14.\ f(x) = a^2x^2 \text{ and } g(x) = dx^2. \text{ Since } a^2 = d \ge 0\text{, both coefficients are positive. Statement is Always True.}\ &15.\ f(x) = a^2x^2 \text{ is narrower than } g(x) = dx^2 \text{ when } a^2 > d\text{. Given } a^2 < d\text{, } f(x) \text{ is wider. Statement is Never True.} \end{aligned} $$ | No. | Answer | |---|---| | 1 | Vertical stretch by a factor of 4; opens up. | | 2 | Vertical stretch by a factor of 1.5; opens up. | | 3 | Vertical shrink by a factor of $\frac{1}{3}$; opens up. | | 4 | Vertical stretch by a factor of 3 and reflection in the x-axis; opens down. | | 5 | Vertical stretch by a factor of 2.5; opens up. | | 6 | Vertical shrink by a factor of 0.5 and reflection in the x-axis; opens down. | | 7 | Reflection in the x-axis; same width. | | 8 | Vertical shrink (wider); opens in the same direction. | | 9 | Vertical shrink (wider); opens in the same direction. | | 10a | Domain: $-400 \le x \le 400$. The arch is symmetric about the y-axis with a total width of 800 feet. | | 10b | Height: 200 feet. | | 11 | Always; substituting $x = 0$ into $y = ax^2$ always yields $y = 0$. | | 12 | Never; since $|a| = |-a|$, the graphs have the same width. | | 13 | Sometimes; true only when $a > 0$. | | 14 | Always; since $a^2 = d$, both coefficients are positive and equal, so both open up. | | 15 | Never; since $a^2 < d$, the graph of $f(x)$ is wider, not narrower. | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze Exercises 1–6

Analyze Exercises 7–9

Solve Exercise 10

Solve Exercise 11

Analyze Exercises 12–15

Answer:

No.	Answer
2	Vertical stretch by a factor of 1.5; opens up.
3	Vertical shrink by a factor of \(\frac{1}{3}\); opens up.
4	Vertical stretch by a factor of 3 and reflection in the x-axis; opens down.
5	Vertical stretch by a factor of 2.5; opens up.
6	Vertical shrink by a factor of 0.5 and reflection in the x-axis; opens down.
7	Reflection in the x-axis; same width.
8	Vertical shrink (wider); opens in the same direction.
9	Vertical shrink (wider); opens in the same direction.
10a	Domain: \(-400 \le x \le 400\). The arch is symmetric about the y-axis with a total width of 800 feet.
10b	Height: 200 feet.
11	Always; substituting \(x = 0\) into \(y = ax^2\) always yields \(y = 0\).
12	Never; since \(	a	=	-a	\), the graphs have the same width.
13	Sometimes; true only when \(a > 0\).
14	Always; since \(a^2 = d\), both coefficients are positive and equal, so both open up.
15	Never; since \(a^2 < d\), the graph of \(f(x)\) is wider, not narrower.