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.