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Linear Functions

A linear function is an expression that gives you a straight line. The name is slightly revealing. Here, you will learn how to draw and recognize linear functions.

Theory

Linear Function

A linear function can be written on the form

f(x) = ax + b,

where a is the slope, and b is the intersection with the y-axis.

Finding the Slope and the Constant Term

You can find the slope of a line if you have the coordinates of two points on the line. Call the points (x1,y1) and (x2,y2). You use the following formulas for the slope a and the constant term b:

Rule

The Slope of a Linear Function

The straight line that goes through the points (x1,y1) and (x2,y2) has the slope

a = y2 y1 x2 x1,

and the constant term

b = y1 ax1.

A straight line with slope a=-1 intersecting the y-axis in (0,1)

Rule

Important Attributes of the Linear Function

  • The slope a tells you how much the graph is increasing/decreasing when x increases by 1.

  • If a > 0, the graph rises towards the right, and if a < 0, the graph is sinking towards the right.

  • The graph intersects the y-axis in the point b.

  • The graph is a straight line with coordinates (x,y) = (x,f(x)).

Example 1

Find the slope of the straight line that goes through the points (5, 2) and (3, 6), and find where it intersects with the y-axis

You choose (x1,y1) to be (3, 6) and (x2,y2) = (5, 2). The calculations would work even if you switched the points. You get

a = y2 y1 x2 x1 = 2 6 5 3 = 4 2 = 2.

You now know that the line decreases by 2 when you move one place to the right. Let’s see what the point of intersection with the y-axis is: b = y1 ax1 = 6 (2) × 3 = 6 + 6 = 12.

Thus, the point of intersection with the y-axis is (0, 12).

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