https://file.aiquickdraw.com/mathbot/file/1726203276953lx9jqyil.jpg

The question asks to complete parts (a), (b), and (c) based on the graph showing the percentage of citizens trusting the government over the years.

### Analysis:

From the graph, the two points are:
- \( (4, 24) \)
- \( (16, 15) \)

#### Part (a):
To find the **point-slope form** of the equation using these two points.

1. **Slope (m)**:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{15 - 24}{16 - 4} = \frac{-9}{12} = -0.75
   \]

2. The point-slope form is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \( m = -0.75 \) and using the point \( (4, 24) \):
   \[
   y - 24 = -0.75(x - 4)
   \]

#### Part (b):
To convert to **slope-intercept form**:

1. Expand the equation from part (a):
   \[
   y - 24 = -0.75(x - 4)
   \]
   Distribute:
   \[
   y - 24 = -0.75x + 3
   \]
   Add 24 to both sides:
   \[
   y = -0.75x + 27
   \]
   The equation in slope-intercept form is \( f(x) = -0.75x + 27 \).

#### Part (c):
To predict the percentage of citizens trusting the government in **2025**:

1. 2025 is 22 years after 2003, so \( x = 22 \).

2. Using the slope-intercept equation \( f(x) = -0.75x + 27 \):
   \[
   f(22) = -0.75(22) + 27 = -16.5 + 27 = 10.5
   \]
   Rounded to the nearest integer: **11%**.

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Would you like more details on any step or have any further questions?

Here are 5 related questions:
1. How is the point-slope form derived from two points?
2. What does the slope represent in this context?
3. How would the equation change if the slope were positive?
4. What does the y-intercept (27) represent in this context?
5. How can we determine when trust in the government reaches 0% using the equation?

**Tip:** Understanding the connection between slope and the rate of change is key to interpreting linear functions in real-life contexts!

The graph shows the percentage of citizens of a certain country who trust in their government. Complete parts (a) through (c) below using the given points: (4, 24) and (16, 15).

This problem involves determining the linear equation based on two points on a graph representing the percentage of citizens who trust in the government. It explores the point-slope form and slope-intercept form to predict future values.

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