Understanding Intercepts and Slope in Graphs and Equations

Intercepts: Where Lines Meet Axes

Intercepts are the points where a line crosses the coordinate axes. The x-intercept is where the line touches the x-axis $with y=0$, while the y-intercept is where it touches the y-axis $with x=0$.

Finding intercepts from a graph is straightforward - just look for where the line crosses each axis. For example, if a line crosses the x-axis at (5,0) and the y-axis at (0,5), these are your intercepts.

When working with a table of values, find the x-intercept by locating where y=0, and find the y-intercept where x=0. If these exact values aren't in your table, you might need to extend the pattern to find them.

Remember This! When a point is on the x-axis, its y-coordinate is always 0. When a point is on the y-axis, its x-coordinate is always 0.

Finding Intercepts and Understanding Slope

To find intercepts from an equation, use a simple two-step approach. For the x-intercept, substitute y=0 into the equation and solve for x. For the y-intercept, substitute x=0 and solve for y.

For example, with x + y = 5:

• For x-intercept: x + 0 = 5, so x = 5, giving (5,0)
• For y-intercept: 0 + y = 5, so y = 5, giving (0,5)

Slope measures how steep a line is - essentially how much the line rises or falls as it moves horizontally. The formula for slope is m = rise/run or m = $y₂ - y₁$/$x₂ - x₁$.

When finding slope from a graph, select two points on the line and calculate how much the line rises (vertical change) divided by how much it runs (horizontal change).

Quick Tip: A positive slope means the line rises as you move right, while a negative slope means the line falls as you move right.

Calculating Slope

Finding slope from a graph requires picking two points on the line and applying the rise/run formula. Count the vertical change (rise) and horizontal change (run), then divide. For instance, if going from one point to another requires moving up 3 units and left 3 units, the slope would be 3/-3 = -1.

When given two specific points like (4,1) and (1,4), use the slope formula directly: m = $y₂ - y₁$/$x₂ - x₁$ = $4 - 1$/$1 - 4$ = 3/-3 = -1

The resulting slope of -1 tells you that for every unit you move right, the line drops by 1 unit. This is valuable information that helps you understand the line's behavior.

Pro Tip: When calculating slope, it doesn't matter which point you label as (x₁,y₁) and which as (x₂,y₂), as long as you're consistent in the numerator and denominator.