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When we talk about the 'slope' of a line, we're referring to how steep the line is. The slope indicates the rate at which the line rises or falls as we move from left to right along the x-axis. Mathematically, the slope is often represented by the letter 'm'. In the slope-intercept form of a linear function, written as \(f(x) = mx + b\), the coefficient of x (which we call 'm') is the slope.
To understand the slope better, we can rewrite it in simpler terms:

• If the slope is positive, the line rises as we move to the right.
• If the slope is negative, the line falls as we move to the right.
• If the slope is zero, the line is horizontal.

For example, in the function \(f(x)=\frac{4}{3}x+15\), the slope is \(\frac{4}{3}\). This means that for every 3 units we move to the right on the x-axis, the line rises by 4 units. The steeper the slope, the higher the value of m, and vice versa.

The 'y-intercept' is the point where the line crosses the y-axis. In mathematical terms, it's the value of y when x equals zero. In the slope-intercept form \(f(x) = mx + b\), the 'b' represents the y-intercept.
To find the y-intercept, you simply look at the value of 'b' in the equation. It's the constant term that doesn't involve x.
For instance, in the given function \(f(x) = \frac{4}{3} x + 15\), the y-intercept is 15. This means that when x is 0, the value of the function (or y) is 15. So, the line crosses the y-axis at the point (0, 15).
Remember, the y-intercept gives us a starting point for drawing the line on a graph. From this point, we can use the slope to determine the direction and steepness of the line.

Linear functions are a type of function that produce a straight line when graphed. They can be written in various forms, but the most common one is the slope-intercept form \(f(x) = mx + b\).
Here are some properties of linear functions:

• They maintain a constant rate of change.
• They produce a straight line on a graph.
• They have no exponents on their variables (which means x is not raised to any power other than 1).

Linear functions are very useful in various fields like physics, economics, and engineering because they accurately describe relationships with a constant rate of change.
For example, the function \(f(x) = \frac{4}{3}x + 15\) is a linear function. It tells us that for every unit increase in x, the value of f(x) increases by a factor of \(\frac{4}{3}\) while starting at 15 when x is zero. The slope and y-intercept make it easy to graph and interpret this relationship.