1. 1.3 Linear Functions

2. 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.

3. 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
b: constant term
x is input, f(x) is output

4. Linear Function Algebraically Geometrically f(x)=ax+b
If a point(m,n) is on the line f(x)=ax+ b, then n=a*m+b y x

5. 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.

6. x-,y-Intercept f(x)=ax+b y-Intercept: the output of f(0) y
x-Intercept:
zero of the function

7. Graphing a linear function by x-,y-intercepts
Exercise: graph the line given by 2x+5y=15 y x
Solution:
step1: finding x-,y-intercepts by solving linear equations.
x-intercept: 7.5
y-intercept: 3 10 8 6 4 2
step2: plotting corresponding points on the xy-plane. -2 -4 -6 -8
-10
step3: connecting the two points by a straight line and extend the line over two ends.

8. 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.

9. Slope
Geometrically: y x
(x2,y2)
△y=y2-y1
(x2,y1)
(x1,y1)
△x=x2-x1

10. Slope
Exercise: Find the slope of the line through Line 1:(4,2) and (3,4) Answer: m=-2 Line2:(-1,3)and(-2,-3) Answer: m=6 y x 10 8 6 4 2 -2 -4 -6 -8
-10
Line 1
Line 2

11. Slope For a line with slope m, m>0y
m>0
The graph rises from left to right
m<0
The graph falls from left to right
m=0
The line is horizontal
m is undefined
The line is vertical x y x y x y x

12. 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.

13. 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?

14. exercise y x 1,Write an equation defining f10 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

15. Homework PG.30: 10-40(M10); 50-85(m5) KEY: 30,60,85
Reading: 1.4 equation of lines