Linear Functions

By: Elizabeth A. and Valarie P.

Definition:

A function is a linear function if F(X)=ax+b, for real numbers A and B.

When a linear function is written in the form Ax+By=C, it is said to be in standard form.

The graph of a linear function is a straight line. To graph a linear function, find at least two of its ordered pairs, plot them, and draw a line through them.

Examples:

• Graph 5x-2y=10

• Solution

Examples (cont.)

• Let x= 0.5(0)-2y= 10• -2y= 10• Y= -5• (0,-5)

• Let x= 0will give us the y-intercept• Solve for y

Examples (cont.)

• Let y= 0.5x-2(0)= 10• 5x= 10• X= 2• (2,0)

• Let y= 0 will give us the x-intercept

Examples (cont.)

• Let x= 4.5(4)-2y= 10• 20-2y= 10• -2y= -10• Y= 5• (4,5)

• 3rd point is a check• These points would be plotted and a line

drawn through them to complete the graph.

Pictures

This is a picture of the Chapel Bridge in Lucern, Switzerland. The roof of this footbridge, originally constructed in 1333, can be modeled by a linear function. The origin is set at the base of the stone tower. Given this somewhat arbitrary origin, the roof of the bridge could be modeled by the function y = 0.25x + 2.

Review

Definition:• A linear function is a polynomial of degree one. The

graph of a linear function is a straight line. The rate of change of a linear function is called the slope of the function. In a linear function, the average and instantaneous rate of change are always the same.

Formula for a Linear Function:• General Form: • Slope-Intercept Form: • Point-Slope Form: • Two-Point Form:

Review (cont.)• You need only two points to graph a linear function.

These points may be chosen as the x and y intercepts of the graph for example.

• Determine the x intercept, set f(x) = 0 and solve for x. • 2x + 4 = 0

• x = -2 • Determine the y intercept, set x = 0 to find f(0). • f(0) = 4

• The graph of the above function is a line passing through the points (-2 , 0) and (0 , 4) as shown below.

The end…