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The slope is a fundamental concept when dealing with linear functions. It tells us how steep a line is. In mathematics, it's often represented by the letter \( m \).
The slope is defined as the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run). Basically, it answers the question, "If I move horizontally by a certain amount, how much will I move vertically?"

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A zero slope represents a horizontal line.
• An undefined slope means the line is vertical.

For the function \( f(x) = \frac{1}{3}x \), the slope is \( \frac{1}{3} \). This means for every 3 units you move to the right, you move 1 unit up. The sign of the slope, in this case, is positive, indicating the line will slant upwards as you go from left to right.

The y-intercept is where the graph of a function crosses the y-axis, which essentially gives us the starting point of the line when drawn on the coordinate plane.
In the equation format \( y = mx + b \), \( b \) represents the y-intercept.

• It's the value of y when x is zero.
• Graphically, it's where the line meets the y-axis.

In the function \( f(x) = \frac{1}{3}x \), the y-intercept \( b \) is zero. This means the line passes through the origin point (0,0). With a y-intercept of zero, the graph begins right at the starting point of both axes, providing a simple foundation from which to start plotting the line using the slope.

The coordinate plane is a two-dimensional surface where we can graph points, lines, and curves. It's composed of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). They meet at what's called the origin, a point represented as (0,0).
The coordinate plane helps us visualize mathematical functions.

• X-coordinates (horizontal values) tell how far to move right or left from the origin.
• Y-coordinates (vertical values) tell how far to move up or down.
• This setup allows every point on the plane to be described by an ordered pair \((x, y)\).

When we plotted the function \( f(x) = \frac{1}{3}x \), we started by plotting the y-intercept at the origin (0,0) on the coordinate plane. Next, we used the slope to find another point at (3,1) by moving 3 units to the right and 1 unit up from the origin. These points then help us draw the line, giving a visual representation of the function \( f(x) = \frac{1}{3}x \) on the plane.