Let's break down and solve the given problems:

Problem 2.2: Find the equation of the line shown on the Cartesian plane.

From the graph, the line appears to have the following characteristics:

1. Y-intercept: The line crosses the y-axis at $y = 4$.
2. Slope: We can calculate the slope (m) by choosing two points on the line.
   • One point is $(0, 4)$ (the y-intercept).
   • Another point is $(-2, 6)$.

The slope formula is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute $(x_1, y_1) = (0, 4)$ and $(x_2, y_2) = (-2, 6)$: $m = \frac{6 - 4}{-2 - 0} = \frac{2}{-2} = -1$ Thus, the slope is $-1$.

Using the slope-intercept form of the equation of a line: $y = mx + b$ Where $m = -1$ and the y-intercept $b = 4$, the equation of the line is: $y = -x + 4$

Problem 2.3: For the function $y = 2x + 4$

The function provided is already in slope-intercept form $y = mx + b$, where:

• $m = 2$ (the slope),
• $b = 4$ (the y-intercept).

This function has a slope of 2 and crosses the y-axis at $y = 4$. It is not related to the graph from problem 2.2, but rather is a different line, likely for comparison.

Summary:

• 2.2: The equation of the line on the graph is $y = -x + 4$.
• 2.3: The function is already in the form $y = 2x + 4$.

Do you need further clarification or details?

Relative Questions:

1. How do we find the slope of a line when given two points?
2. What does the slope represent in the context of a line on a Cartesian plane?
3. How does the y-intercept affect the position of the line?
4. Can two lines with the same slope have different equations? Why or why not?
5. How can we tell if two lines are parallel just by looking at their equations?

Tip:

To find the equation of any line, always start by identifying the slope and the y-intercept from the graph or using points on the line.