The graph of 
y=f(x)
 contains the point 
(6,4)
. What point must be on the graph of 
y=
f(11−5x)+8/10

The point \( (6,4) \) lies on the graph of \( y = f(x) \). Now, we are asked to determine what point must be on the graph of \( y = f(11 - 5x) + \frac{8}{10} \).

### Step 1: Analyze the transformation
The new function \( y = f(11 - 5x) + \frac{8}{10} \) is a transformation of \( f(x) \), which involves two parts:
1. **Horizontal Transformation**: \( 11 - 5x \)
   - This expression involves a reflection (due to the negative sign) and a horizontal scaling by a factor of 5, as well as a shift by 11 units.
2. **Vertical Transformation**: \( + \frac{8}{10} \)
   - This results in a vertical shift of \( +0.8 \) units.

### Step 2: Plug in the point \( x = 6 \)
The point \( (6, 4) \) is on the graph of \( y = f(x) \), meaning \( f(6) = 4 \). Now, let's find what happens to the corresponding \( x \)-value in the transformed graph.

For the new function \( f(11 - 5x) \), set \( x = 6 \) and substitute it into the expression for \( x \):

\[
x' = 11 - 5(6) = 11 - 30 = -19
\]

Thus, the corresponding \( x \)-value in the transformed graph is \( -19 \).

### Step 3: Apply the vertical shift
Now, evaluate the \( y \)-value at this new \( x \)-value. Since \( f(6) = 4 \), the corresponding \( y \)-value for the transformed function is:

\[
y = f(-19) + \frac{8}{10} = 4 + 0.8 = 4.8
\]

### Step 4: Conclusion
Thus, the point \( (-19, 4.8) \) must be on the graph of \( y = f(11 - 5x) + \frac{8}{10} \).

---

Would you like further clarification or details on any part of the process?

### Related Questions:
1. How does the reflection in \( f(11 - 5x) \) affect the graph?
2. How would the point change if the transformation was \( y = f(2x + 3) \)?
3. What happens to the graph when the vertical shift is negative?
4. How would the graph change if it was \( f(11 - 5x) - 0.8 \) instead?
5. How can we find the inverse of the function \( f(11 - 5x) \)?

**Tip:** For horizontal transformations, pay attention to signs and scaling factors to accurately determine the new \( x \)-coordinates.

The graph of y=f(x) contains the point (6,4). What point must be on the graph of y=f(11−5x)+8/10?

Explore how to find the point on the transformed graph of y=f(11−5x)+0.8 given that the original graph contains the point (6, 4). Learn about function transformations, including horizontal shifts, reflections, and vertical shifts, with a step-by-step solution. Ideal for Grades 10-12.

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