91Ó°ÊÓ

To solve problems involving linear functions, understanding the slope formula is crucial. The slope of a line, often represented by the letter m, shows how steep the line is and the direction it goes. For a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
This formula tells us how much the y-values (vertical changes) increase or decrease for a one-unit increase in x (horizontal change).
In our example, the points (0, -6) and (1, -3) are used to find the slope. Plugging the values into the formula, we get \[ m = \frac{-3 - (-6)}{1 - 0} = \frac{3}{1} = 3. \]
So, the slope of the line is 3. This means for every 1 unit increase in x, y increases by 3 units.

Once we have the slope, we can form the equation of the line using the slope-intercept form, which is: \[ y = mx + b \]. Here, m is the slope and b is the y-intercept, where the line crosses the y-axis.
In our exercise, we know the slope m is 3, and the line goes through the point (0, -6). Since this point is on the y-axis, -6 is our y-intercept b.
Therefore, substituting these values into the formula, we get the equation: \[ y = 3x - 6 \].
This equation now represents our linear function and can be used to find y-values for any given x-value.

Substituting values into the equation of a line helps us find specific points. Once we have the line's equation, we can substitute any x-value to find its corresponding y-value.
For instance, in the given problem, we need to find y when x is 4.5. Using the equation \[ y = 3x - 6 \], we substitute 4.5 for x: \[ y = 3(4.5) - 6 \]
This simplifies to: \[ y = 13.5 - 6 = 7.5 \].
Therefore, the corresponding y-value when x is 4.5 is 7.5. Substitution is a powerful method to locate specific points on a line quickly.

In precalculus, linear functions are fundamental concepts that pave the way to more advanced topics in calculus and other areas of mathematics.
Understanding how to analyze linear functions involves key skills such as calculating slopes, forming equations, and substituting values.
Grasping these concepts is essential, as they reappear in various forms throughout higher mathematical studies. Linear functions represent relationships where changes in one variable lead to consistent changes in another, reflecting both real-world scenarios and more theoretical applications.
By mastering these precalculus concepts, you build a strong foundation for further mathematical education.