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Calculating the slope is a crucial step when defining the equation of a line. The slope determines the direction and steepness of the line. To find the slope \( m \) that connects two points, we use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula represents the ratio of the vertical change \((y_2 - y_1)\) to the horizontal change \((x_2 - x_1)\) between two points \((x_1, y_1)\) and \((x_2, y_2)\).

• A positive slope means the line goes upwards as it moves from left to right.
• A negative slope implies the line goes downwards.
• A zero slope indicates a horizontal line, while an undefined slope suggests a vertical line.

In our exercise, we have the points \((-3,-4)\) and \((1,4)\). Applying the slope formula:
\[ m = \frac{4 - (-4)}{1 - (-3)} = \frac{8}{4} = 2 \]This slope tells us the line increases by 2 units vertically for every 1 unit it moves horizontally.

Once we have the slope, the point-slope form helps in quickly finding the equation of the line. The point-slope form of a line’s equation is:
\[ y - y_1 = m(x - x_1) \]Here,

• \( (x_1,y_1) \) is a point on the line,
• \( m \) is the slope as previously calculated.

For our example, using the point \((-3,-4)\) and the slope \(m = 2\), the point-slope form becomes:
\[ y - (-4) = 2(x - (-3)) \]Solve this to simplify it into the slope-intercept form:
\[ y + 4 = 2(x + 3) \] \[ y + 4 = 2x + 6 \] \[ y = 2x + 6 - 4 \] \[ y = 2x + 2 \] This final form is the more familiar slope-intercept form \(y = mx + b\), where \(b\) is the y-intercept. In this case, \(b = 2\).

Graphing a linear equation is the visual representation of the relationship between two variables, often x and y. To graph the equation of a line, you can follow these simple steps:
1. **Identify Points:** Begin with plotting the given points, for example, \((-3,-4)\) and \((1,4)\) on a graph.
2. **Draw the Line:** Using a ruler, draw a line through these points. Make sure the line extends across the graph, covering all quadrants through the points.
3. **Use the Y-Intercept and Slope:** Since the equation of our line is \(y = 2x + 2\), it intercepts the y-axis at 2. From this point, use the slope, which is \(2\). This means for each unit you go right on the x-axis, go up 2 units on the y.
Graphing helps understand how the line behaves. It shows its intercepts, slope, and allows you to easily compare it to other lines.