https://file.aiquickdraw.com/mathbot/file/17264583035003nr9l350.jpg

Let's go over each part of the problem step by step:

### Given:
- The graph of \( f(x) \) contains the point \( (1, 1) \), meaning \( f(1) = 1 \).

Now let's address each question:

---

### (a) What point must be on the graph of \( f(x) + 1 \)?

When we add 1 to \( f(x) \), the entire graph of \( f(x) \) shifts **up by 1 unit**. This means:
- \( f(x) + 1 \) will have a new point where the \( y \)-coordinate increases by 1.

For the point \( (1, 1) \) on \( f(x) \), it becomes:
- \( f(1) + 1 = 1 + 1 = 2 \)

So the new point will be \( (1, 2) \).

---

### (b) What point must be on the graph of \( f(x + 1) \)?

When we replace \( x \) with \( x + 1 \), the graph of \( f(x) \) shifts **left by 1 unit**. This means:
- The point \( (1, 1) \) shifts to the left by 1 unit on the \( x \)-axis.

For the point \( (1, 1) \) on \( f(x) \), it becomes:
- \( (1 - 1, 1) = (0, 1) \)

So the new point will be \( (0, 1) \).

---

### (c) What point must be on the graph of \( f(x - 3) - 2 \)?

When we replace \( x \) with \( x - 3 \), the graph shifts **right by 3 units**, and subtracting 2 from \( f(x) \) shifts it **down by 2 units**.

For the point \( (1, 1) \) on \( f(x) \), it becomes:
- First, shift right by 3 units: \( (1 + 3, 1) = (4, 1) \)
- Then, shift down by 2 units: \( (4, 1 - 2) = (4, -1) \)

So the new point will be \( (4, -1) \).

---

### Summary of answers:
- (a) \( (1, 2) \)
- (b) \( (0, 1) \)
- (c) \( (4, -1) \)

Would you like more details on how these transformations work? Here are some related questions to deepen your understanding:

1. How do vertical transformations affect the graph of a function?
2. What happens to the graph of \( f(x) \) when \( x \) is replaced with \( x - a \)?
3. What is the effect of adding a constant to \( f(x) \)?
4. Can you explain the difference between horizontal and vertical shifts?
5. What does it mean geometrically when a function is shifted?

**Tip:** When adjusting functions for transformations, focus on one transformation at a time, such as shifts, stretches, or reflections.

The graph of f(x) contains the point (1,1). What point must be on the graph of (a) f(x) + 1, (b) f(x + 1), and (c) f(x - 3) - 2?

Explore function transformations with vertical and horizontal shifts. Given that the graph of f(x) contains the point (1,1), find the points on the graphs of f(x) + 1, f(x + 1), and f(x - 3) - 2. Learn about vertical and horizontal shifts in this problem suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation