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

1. \\(g(x)\\) is \\(f(x)\\) translated right 4 units and down 2 unit.

true / false

1. \\(g(x)\\) opens upwards.

true / false

1. \\(g(x)\\) is wider than \\(f(x)\\).

true / false

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

1. \\(y = 3x^2\\)
2. \\(y = -3x^2\\)
3. \\(y = \frac{1}{3}x^2\\)
4. \\(y = -\frac{1}{3}x^2\\)

Explanation:

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<reasoning>

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
\[

$$\begin{aligned} &y = \frac{1}{3}x^2: a = \frac{1}{3} > 0 \text{ (opens up, wide)} \implies \text{Graph a}\\ &y = -\frac{1}{3}x^2: a = -\frac{1}{3} < 0 \text{ (opens down, wide)} \implies \text{Graph d} \end{aligned}$$

\]
</reasoning>

<answer>

No.	Answer
2	True
3	True
4	False
5	b
6	c
7	a
8	d

</answer>

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Answer:

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<reasoning>

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
\[

$$\begin{aligned} &y = \frac{1}{3}x^2: a = \frac{1}{3} > 0 \text{ (opens up, wide)} \implies \text{Graph a}\\ &y = -\frac{1}{3}x^2: a = -\frac{1}{3} < 0 \text{ (opens down, wide)} \implies \text{Graph d} \end{aligned}$$

\]
</reasoning>

<answer>

No.	Answer
2	True
3	True
4	False
5	b
6	c
7	a
8	d

</answer>

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