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Let's go through each part of the question based on the function \( Q(t) = 12 - 11t \).

### (a) Lies six units above the graph of \( Q(t) \):
To create a function that lies six units above \( Q(t) \), we simply add 6 to \( Q(t) \):
\[
y(t) = Q(t) + 6 = (12 - 11t) + 6 = 18 - 11t
\]

### (b) Lies two units below the graph of \( Q(t) \):
To create a function that lies two units below \( Q(t) \), we subtract 2 from \( Q(t) \):
\[
y(t) = Q(t) - 2 = (12 - 11t) - 2 = 10 - 11t
\]

### (c) Has the same vertical intercept:
The vertical intercept is the value of the function when \( t = 0 \). For \( Q(t) = 12 - 11t \), the intercept is 12. Any function with the same vertical intercept will have a constant of 12. So, the answer is any function of the form:
\[
y(t) = 12 + mt \quad \text{(where \( m \) is any slope)}
\]

### (d) Has the same slope:
The slope of \( Q(t) \) is the coefficient of \( t \), which is -11. A function with the same slope must have this same \( t \)-coefficient. The general form would be:
\[
y(t) = mt + b \quad \text{where} \, m = -11
\]
So, the answer is:
\[
y(t) = -11t + b \quad \text{(any constant \( b \))}
\]

### (e) Has the same steepness, but the slope is positive:
The "steepness" refers to the absolute value of the slope. For \( Q(t) \), the steepness is \( | -11 | = 11 \), but we want the slope to be positive. The function will have the same steepness but a positive slope:
\[
y(t) = 11t + b \quad \text{(any constant \( b \))}
\]

Would you like more details on any of these? Here are five related questions to further explore the concepts:

1. How do we determine the slope and intercept of a linear function from its equation?
2. What transformations (like translations) can be applied to linear functions?
3. How does changing the slope affect the graph of a function?
4. Can two different functions have the same slope but different intercepts?
5. How do vertical shifts in a graph relate to addition or subtraction of constants?

**Tip:** When shifting a graph vertically, you simply add or subtract a constant to the entire function.

Given the function Q(t) = 12 - 11t, construct a related function whose graph: (a) Lies six units above the graph of Q(t). (b) Lies two units below the graph of Q(t). (c) Has the same vertical intercept. (d) Has the same slope. (e) Has the same steepness, but the slope is positive.

Explore how to modify the linear function Q(t) = 12 - 11t by performing various transformations, such as vertical shifts and altering the slope. This problem covers key concepts in algebra related to graph transformations, slopes, and intercepts. Suitable for high school students in Grades 9-12.

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