https://r2.erweima.ai/mathbot/image/1725592277132rqynm7y0.jpg

The function provided in the image is:

\[
y = \frac{5}{2}x + 4
\]

This is a linear function (also called a first-degree function), where:
- The slope (m) is \( \frac{5}{2} \), meaning the line rises by 5 units for every 2 units it moves horizontally.
- The y-intercept (b) is 4, meaning the line crosses the y-axis at \( (0, 4) \).

### Characteristics:
1. **Slope (m)**: The slope of the line is \( \frac{5}{2} \). This indicates the steepness of the line and the direction of increase.
2. **Y-intercept (b)**: The y-intercept is 4, meaning the line crosses the y-axis at \( y = 4 \).
3. **X-intercept**: To find the x-intercept, set \( y = 0 \) and solve for \( x \):
   
   \[
   0 = \frac{5}{2}x + 4
   \]
   \[
   \frac{5}{2}x = -4
   \]
   \[
   x = -\frac{8}{5}
   \]

   So, the x-intercept is \( \left(-\frac{8}{5}, 0\right) \).

4. **Domain**: The domain of this function is all real numbers, \( x \in (-\infty, \infty) \), since the function is defined for all values of \( x \).
5. **Range**: The range is also all real numbers, \( y \in (-\infty, \infty) \), as the line continues infinitely in both directions.

### Tabulation:

| \( x \) | \( y \)                  |
|--------|-------------------------|
| -2     | \( \frac{5}{2}(-2) + 4 = -5 + 4 = -1 \) |
| 0      | \( \frac{5}{2}(0) + 4 = 4 \)           |
| 2      | \( \frac{5}{2}(2) + 4 = 5 + 4 = 9 \)   |
| 4      | \( \frac{5}{2}(4) + 4 = 10 + 4 = 14 \) |

Would you like more details or have any questions?

Here are five questions that extend this topic:
1. How do we interpret the slope in the context of the line's steepness?
2. What is the significance of the y-intercept in a linear function?
3. How can we verify the solution for the x-intercept algebraically?
4. What changes occur to the graph if the slope is negative?
5. How does modifying the y-intercept affect the graph's position?

**Tip**: To graph any linear function quickly, remember that the slope gives the rise/run ratio, and the y-intercept shows where the line crosses the y-axis.

Resuelva las siguientes funciones de primer grado y escriba sus características y con la tabulación: y = 5/2x + 4

Learn how to solve the linear function y = 5/2x + 4 by identifying its slope, y-intercept, x-intercept, and domain and range. This lesson covers key concepts such as linear function properties, slope-intercept form, and tabulation.

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