Slope of a Line

Look at these two hills. If you had to walk up one of them, which one would feel harder? 

2. Slope in Slope–Intercept Form

The equation of a line in slope-intercept form is:    y = mx + b, where m = slope  and b = the point where the line crosses the y-axis, which happens when x=0.)

Slope is the engine of this formula. It tells us exactly what the line is doing:

• Is it rising?

If the slope is +3, that means for every 1 step to the right, the line goes up three units - it’s rising.

• Falling?

If the slope is -2, the line drops two units for every 1 unit to the right. The graph is falling. 

• Staying constant?

If the slope is 0, the line is perfectly flat.  It doesn’t rise or fall like walking on a flat road.

• Rising slowly?

A small positive slope, like +½, means the line rises a little each time. It rises very gently.

• Rising quickly?

A steep slope like +12 means the line shoots upward fast like climbing a steep hill. 

The slope tells the whole story.

4. Real-World Example.  

Imagine you start a new job and the company pays you:   $15 per hour, and a $20 sign-on bonus

Your earnings can be written as a linear equation:  y = 15x + 20

Let’s interpret this:

Slope (15) = You earn $15 for every hour you work. This is your rate of change which means how quickly your pay increases.

Intercept (20) = Your starting amount.  This is the money you have even before working any hours.

When we graph this: Start at 20 on the y-axis. From there, follow the slope: up 15, right 1. Connect the points.

 You’ll see a line that increases steadily - that’s your income rising hour by hour. 

To summarize:

• Slope measures how fast something changes.

• In the form y= mx + b, the slope tells us the story of the line.

• And slope shows up everywhere - from hills to income to the way data changes over time.

Once you understand slope, graphing and real-world problem solving become much easier.

Calculus was invented by extending the idea of slope from straight lines to curves. Without slope, there would be no way to measure instantaneous change, which is the very foundation of calculus.