1.3 Linear Functions

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Quiz:
Write a linear function general form(or a linear
function has the form of ):
f(x)=_______
You can choose any letters to symbolize coefficients or
constants.
Linear Function
 Definition: A function f defined by f(x)=ax+b, where a
and b are real numbers, is called a linear function.
f(x)=ax+b
a: coefficient
of x
x is input,
f(x) is
output
b: constant
term
Linear Function
Algebraically
Geometrically
y
f(x)=ax+b
x
If a point(m,n) is on the line
f(x)=ax+ b, then n=a*m+b
Zero of a Function
Definition: Let f be a function. Then any number c for
which f(c)=0 is called a zero of the function f.
ex: f(x)=2x+4
solve for the linear equation f(x)=0, we get x=-2, then
-2 is a zero of f(x) defined above.
x-,y-Intercept
y-Intercept:
the output of f(0)
f(x)=ax+b
y
x-Intercept:
zero of the function
x
Graphing a linear function by x-,yintercepts
Exercise: graph the line given by 2x+5y=15
y
Solution:
step1: finding x-,y-intercepts by
solving linear equations.
10
8
6
x-intercept: 7.5
y-intercept: 3
step2: plotting corresponding
points on the xy-plane.
step3: connecting the two points
by a straight line and extend the
line over two ends.
4
2
-10
-8
-6
-4
-2
-2
-4
-6
-8
-10
2
4
6
8
10
x
Slope
Definition: Geometrically, slope is a numerical measure
of the steepness of a line and can be interpreted as the
ratio of rise to run (or rate of change).
Form: m= △y/ △x=(y1-y2)/(x1-x2), where x1-x2 ≠0
Notice: the order of points doesn’t matter, but when
using the formula to compute m, start with the x,y
values of the same point.
Slope
Geometrically:
y
(x2,y2)
△y=y2-y1
x
(x1,y1)
△x=x2-x1
(x2,y1)
Slope
Exercise: Find the slope of the line through
y
Line 1:(4,2) and (3,4)
10
8
Answer: m=-2
6
4
Line2:(-1,3)and(-2,-3)
Answer: m=6
2
-10
-8
-6
-4
-2
-2
-4
-6
-8
Line 2
-10
2
4
6
8
10
x
Line 1
Slope
 For a line with slope m,
m>0
The graph rises from left to
right
y
x
m<0
The graph falls from left to
right
y
m=0
The line is horizontal
y
x
x
m is undefined
The line is vertical
y
x
Slope-intercept form
 f(x)=mx+b or y = mx+b
m: slope
b: y-intercept
Compare the general form of a linear function to the
slope-intercept form.
Exercise
 1, Graph the line containing (2,4) with a slope of
m=-3/4
 2, Graph the line represented by y=2x-5
 3, Graph the line represented by x-3y=6.
 4, Graph the line through (5,-1)with undefined slope.
What is the equation of this line?
exercise
y
10
8
6
4
2
-10
-8
-6
-4
-2
-2
-4
-6
-8
-10
1,Write an equation defining f
2, Find the slope, y-intercept, x-intercept
3, Find any zeros of f
2
4
6
8
10
x
Homework
 PG.30: 10-40(M10); 50-85(m5)
 KEY: 30,60,85
 Reading: 1.4 equation of lines

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