Use the graph of the function f to find approximations of the given values.

The x y-coordinate plane is given. A curve with 2 parts is graphed.
The first part is linear, enters the window in the third quadrant, goes up and right, passes through the point (−9, −6), passes through the point (−6, −3), crosses the x-axis at x = −3, crosses the y-axis at y = 3, and ends at the open point (3, 6).
The second part is a curve, begins at the closed point (3, −9), goes up and right becoming more steep, passes through the point (6, 0), goes up and right becoming less steep, passes through the point (9, 9), changes direction at the approximate point (10.1, 9.9), goes down and right becoming more steep, and exits the window in the first quadrant.
(a)
f(−3)
(b)
f(3)
(c)
f(9)
(d)
f(9) − f(3)
9 − 3

Let's solve the given problems using the graph of the function \( f(x) \).

### (a) \( f(-3) \)
From the graph description, \( f(x) \) crosses the x-axis at \( x = -3 \). This means that the value of \( f(-3) \) is 0, because the x-axis represents the points where the function equals zero.

Thus, \( f(-3) = 0 \).

### (b) \( f(3) \)
At \( x = 3 \), the graph transitions from a linear section to a curved section. The linear part ends with an open point at \( (3, 6) \), but the second part begins at a closed point at \( (3, -9) \). Since the closed point represents the actual value of the function at \( x = 3 \), we have:

Thus, \( f(3) = -9 \).

### (c) \( f(9) \)
From the graph description, \( f(x) \) passes through the point \( (9, 9) \), so the value of \( f(9) \) is simply the y-coordinate at that point.

Thus, \( f(9) = 9 \).

### (d) \( f(9) - f(3) \)
We already found that \( f(9) = 9 \) and \( f(3) = -9 \), so:

\[
f(9) - f(3) = 9 - (-9) = 9 + 9 = 18
\]

### (e) \( \frac{f(9) - f(3)}{9 - 3} \)
We calculated \( f(9) - f(3) = 18 \), and the denominator is \( 9 - 3 = 6 \), so:

\[
\frac{f(9) - f(3)}{9 - 3} = \frac{18}{6} = 3
\]

### Final Answers:
(a) \( f(-3) = 0 \)  
(b) \( f(3) = -9 \)  
(c) \( f(9) = 9 \)  
(d) \( f(9) - f(3) = 18 \)  
(e) \( \frac{f(9) - f(3)}{9 - 3} = 3 \)

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Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the slope of the linear part of the graph between points \( (-9, -6) \) and \( (-3, 0) \)?
2. What would be the equation of the linear part of the function?
3. How would you describe the concavity of the curved portion of the graph?
4. What are the coordinates of the minimum and maximum points on the graph?
5. What is the average rate of change of the function over the interval \( [3, 9] \)?

**Tip**: Understanding function behavior from graphs often involves identifying key points (intercepts, peaks, troughs) and noting any changes in slope or curvature.

Use the graph of the function f to find approximations of the given values. The x y-coordinate plane is given. A curve with 2 parts is graphed. The first part is linear, enters the window in the third quadrant, goes up and right, passes through the point (−9, −6), passes through the point (−6, −3), crosses the x-axis at x = −3, crosses the y-axis at y = 3, and ends at the open point (3, 6). The second part is a curve, begins at the closed point (3, −9), goes up and right becoming more steep, passes through the point (6, 0), goes up and right becoming less steep, passes through the point (9, 9), changes direction at the approximate point (10.1, 9.9), goes down and right becoming more steep, and exits the window in the first quadrant. (a) f(−3) (b) f(3) (c) f(9) (d) f(9) − f(3) 9 − 3

Learn how to interpret the graph of a piecewise function to find specific function values and calculate the average rate of change. This problem covers key mathematical concepts such as function analysis, graph interpretation, and the application of the slope formula. Suitable for students in Grades 9-12.

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