Download Linear Relation and Function Notes

Algebra 2
Linear Relations and Functions
Name__________________________________________
Warm-up:
1) Given f ( x)  3x 2  2 x  1 and g ( x)  x  4 , find
a)
( f  g )( x)
b) ( f  g )( x)
2) Solve each equation. Check your solution.
a) 3m  7  13
b) 3x  7  15
2.2 Notes: Relations that have straight line graphs are called linear relations. An equation such as x  y  5 is called a
linear equation. A linear equation has no operations other than addition, subtraction, and multiplication of a variable by
a constant. The variables may not be multiplied together or appear in the denominator. A linear equation does not
contain variables with exponents other than 1. The graph of a linear equation is always a line.
Linear Equations
Nonlinear equations
4 x  5 y  16
2 x  6 y 2  25
x  10
y  x 2
2
y   x 1
3
x  xy  
y
1
x
2
y
5
8
1
x
A linear function is a function with ordered pairs that satisfy a linear equation. Any linear function can be written in the
form f ( x)  mx  b where m and b are real numbers.
Example 1: State whether each is a linear function. Write yes or no. Explain.
a.
f ( x)  8 
d. f ( x) 
3
x
4
5
x6
b. f ( x ) 
2
x
e. g ( x)  
c. g ( x, y )  3 xy  4
3
1
x
2
3
Example 2: Real World Example--The growth rate of a sample of Bermuda grass is given by the function
f ( x)  5.9 x  3.25 , where f(x) is the total height in inches x days after an initial measurement.
a. How tall is the sample after 3 days
Algebra 2
Linear Relations and Functions
Name__________________________________________
b. The term 3.25 in the function represents the height of the grass when it was initially measured. The sample is
how many times as tall after 3 days?
c. If the Bermuda grass is 50.45 inches tall, how many days has it been since it was last cut?
d. Is it reasonable to think that this rate of growth can be maintained for long periods of time? Explain.
Standard Form: Any linear equation can be written in standard form. Ax+By=C, where A, B, and C are integers with a
greatest common factor of 1, A  0 , and A and B are not both zero.
Example 3: Write 
3
x  8 y  15 in standard form. Identify A, B, and C.
10
You Try: 2 y  4 x  5
You Try: 3x  6 y  9  0
Since two points determine a line, one way to graph a linear function is to find the points at which the graph intersects
each axis and connect them with a line. The y coordinate of the point at which the graph crosses the y-axis is called the
y-intercept. Likewise, the x-coordinate of the point at which it crosses the x-axis is called the x-intercept.
Example 4: Find the x-intercept and the y-intercept of
the graph of 2 x  3 y  8  0 . Then graph the
equation.
You Try: Find the x-intercept and the y-intercept of the
graph of 2 x  5 y  10  0 . Then graph the equation.
Algebra 2
Linear Relations and Functions
Name__________________________________________
The slope of a line is the ratio of the change in the y-coordinates to the corresponding change in the x-coordinates. The
slope of a line is the same as its rate of change.
Suppose a line passes through points at (x1,y1) and (x2,y2).
Slope =
y 2  y1 y
=
x2  x1 x
Example 4: Find the slope of a line that passes through (-4,3) and (2,5).
You Try: Find the slope of the line that passes through each pair of points.
a) (1,-3) and (3,5)
b) (-8,11) and (24,-9)