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

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)

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
\[

$$\begin{aligned} &y = -3x^2 \implies \text{opens downwards, vertically stretched (narrower) } \implies \text{Graph (c)}\\ &y = -\frac{1}{3}x^2 \implies \text{opens downwards, vertically compressed (wider) } \implies \text{Graph (d)} \end{aligned}$$

\]
</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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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
\[

$$\begin{aligned} &y = -3x^2 \implies \text{opens downwards, vertically stretched (narrower) } \implies \text{Graph (c)}\\ &y = -\frac{1}{3}x^2 \implies \text{opens downwards, vertically compressed (wider) } \implies \text{Graph (d)} \end{aligned}$$

\]
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<answer>

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

</answer>

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