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Question 1: The linear function $y = 5x$ is translated by $T = \begin{pmatrix} 5 \\ 1 \end{pmatrix}$. What is the resulting equation?

To translate a function, we apply the translation vector $T$ to the original coordinates.

The translation moves every point by $(x, y) \rightarrow (x + 5, y + 1)$. Substituting this into the original equation:

Original equation: $y = 5x$ After translation: $y - 1 = 5(x - 5)$

Simplifying: $y - 1 = 5x - 25$ $y = 5x - 24$

So, the resulting equation is $y = 5x - 24$. The correct answer should be $y = 5x - 24$.

Question 2: The quadratic function $y = 18 - 2x^2$ is rotated about the origin $O(0, 0)$. If the resulting function is $g(x)$, the value of $g(6)$ is...?

A rotation about the origin by $180^\circ$ (which is most likely implied here) changes the sign of both $x$ and $y$.

The original function is $y = 18 - 2x^2$.

After a 180° rotation, the function becomes $-y = 18 - 2(-x)^2$, which simplifies to:

$y = -(18 - 2x^2) = -18 + 2x^2$ So, $g(x) = 2x^2 - 18$.

Now, evaluate $g(6)$:

$g(6) = 2(6)^2 - 18 = 2(36) - 18 = 72 - 18 = 54$

The correct value of $g(6)$ is 54.

Question 3: The exponential function $y = 4^x$ is transformed by a factor of 3 and the center is shifted to $(1, 1)$. What is the resulting equation?

The transformation scales the function by a factor of 3, so the function becomes $y = 3 \cdot 4^x$.

The shift of the center to $(1, 1)$ modifies the function. Normally, a shift would affect the argument $x$ and $y$-value as follows:

New function: $y - 1 = 3 \cdot 4^{(x - 1)}$.

Simplifying: $y = 3 \cdot 4^{x - 1} + 1$

Thus, the new function is $y = 3 \cdot 4^{x - 1} + 1$.

Would you like further details or explanations on any of these problems?

Here are 5 additional practice questions based on this topic:

1. What is the general rule for translating a function by a vector $T = \begin{pmatrix} a \\ b \end{pmatrix}$?
2. How do you apply a 180° rotation to a general quadratic function?
3. What happens to the graph of an exponential function when it's vertically scaled by a factor?
4. How do you shift the graph of a function horizontally and vertically?
5. What is the effect of rotating a linear function about the origin?

Tip: When translating functions, pay close attention to how each transformation (scaling, shifting, or rotating) alters the function's equation.