Starting with the graph of 

𝑓

(

𝑥

)

=

3

𝑥

, write the equation of the graph that results when:

(a) 

𝑓

(

𝑥

)

 is shifted 5 units upward. 

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=
   

(b) 

𝑓

(

𝑥

)

 is shifted 8 units to the left. 

𝑦

=
   

(c) 

𝑓

(

𝑥

)

 is reflected about the x-axis and the y-axis. 

𝑦

=
 

Let's analyze the given transformations step by step. The original function is:

\[
f(x) = 3x
\]

### (a) Shift 5 units upward:

To shift a function **upward** by \( c \) units, you add \( c \) to the function. In this case, the shift is 5 units upward, so the new function is:

\[
y = 3x + 5
\]

### (b) Shift 8 units to the left:

To shift a function **to the left** by \( c \) units, you replace \( x \) with \( x + c \) in the function. Since the shift is 8 units to the left, you substitute \( x \) with \( x + 8 \), so the new function is:

\[
y = 3(x + 8)
\]

Expanding this:

\[
y = 3x + 24
\]

### (c) Reflection about the x-axis and y-axis:

- **Reflection about the x-axis**: Multiply the function by \(-1\).
- **Reflection about the y-axis**: Replace \( x \) with \( -x \).

Applying both transformations to the original function \( f(x) = 3x \):

1. Reflecting about the **x-axis** gives \( -3x \).
2. Reflecting about the **y-axis** gives \( -3(-x) = 3x \).

Combining both reflections, we get:

\[
y = -3(-x) = 3x
\]

However, reflecting about both axes will result in the original function since it gets reversed twice.

Therefore, after reflecting about both axes, the equation remains:

\[
y = -3x
\]

---

### Summary of results:

- **(a) Shift 5 units upward**: \( y = 3x + 5 \)
- **(b) Shift 8 units to the left**: \( y = 3x + 24 \)
- **(c) Reflected about the x-axis and y-axis**: \( y = -3x \)

Would you like more details or clarification?

---

Here are 5 related questions to explore further:

1. How would the equation change if the function were shifted 5 units downward instead of upward?
2. What would happen if the function were shifted 8 units to the right instead of to the left?
3. How do you reflect a function about the y-axis alone?
4. How does the slope change under different transformations?
5. What is the effect of combining shifts and reflections?

**Tip**: When applying transformations to a graph, always think of shifts as changes to the inputs (horizontal) or outputs (vertical) and reflections as flipping the graph across an axis.

Starting with the graph of f(x) = 3x, write the equation of the graph that results when:
(a) f(x) is shifted 5 units upward. y = 
(b) f(x) is shifted 8 units to the left. y = 
(c) f(x) is reflected about the x-axis and the y-axis. y =

Explore the transformations of the linear function f(x) = 3x, including a 5-unit upward shift, an 8-unit leftward shift, and reflections about the x-axis and y-axis. Learn the step-by-step process for modifying the equation, suitable for students in Grades 8-10.

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