Linear Functions and Equations

A linear function is a function whose graph is a straight line. The line can"t be vertical, due to the fact that then we wouldn"t have a function, but any type of other type of right line is fine. Now, are you prepared to make words "slope" a component of your life? Okay, here we go...

You are watching: An equation whose graph is a straight line

The following graphs show linear functions.

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Let"s find a couple of point out whose coordinates are nice and easy to job-related with and see what the rise and also run are between those two points. Use the undergarment visual if you"d like. It won"t assist you through this problem, yet no one"s avoiding you.

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The steep is:

Sample Problem

Find the steep of the line displayed below.

If we shot to use the formula to a upright line, we"ll it is in in trouble. Since the "run"" between any two clues on a vertical line is 0, and we can"t division by 0, the steep of a vertical line is undefined. So, the steep of the line x = 1 is undefined.

Makes sense, because it would take some powerful thighs come run directly up a upright mountain. If girlfriend attempted to perform so, you"d shortly be undefined together well.

Sample Problem

Find the steep of the line that passes v the clues (1, 3) and also (2, 7).

We can discover the slope of a heat if given any type of two point out on the line.We know component of the line will look like this:

To gain from the suggest (1, 3) to the allude (2, 7), we need to relocate right 1 and up 4:

That means the steep of the heat is. Yodelay-hee-hoo!

Sample Problem

Find the slope of the line that goes with (-3, 1) and (2, -2).

Part of the line looks like this:

The street we take trip to gain from one value of x to the various other is 3 + 2 = 5, since very first we have to travel indigenous x = -3 come x = 0 and then native x = 0 to x = 2. We have a layover at the y-axis, wherein we have the right to grab a quick bite that vastly overpriced fast food while us wait because that our connecting line.

To obtain from one value of y to the other, an initial we travel from y = 1 come y = 0 and also then native y = 0 come y = -2, for a total rise that -3. Once again, us couldn"t gain a direct flight. Ah, well. It"ll provide us much more time to review this book we"ve been working on.

Thus the steep of this heat is.

Be careful: It"s usual to make mistakes calculating the rise and also run once there are negative coordinates involved. To stop mistakes, us recommend drawing a picture to assist with the calculations. If art isn"t your thing, find a mountain or publication a flight so you deserve to live out one of our previous examples. An ext expensive and also time-consuming to gain the point throughout that way, but it"ll certainly drive the idea home.

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Well, currently we can read off the steep of a line from a graph or from any type of two clues on the line. We"re feeling an excellent about ourselves. How about graphing a heat if given a single point and a slope?

Sample Problem

Graph the line that goes through (0, 0) and also has a slope of 2.

Let"s begin by illustration the allude we"re given:

We"re said the line has a slope of 2, which method as x moves end 1, y goes up 2: