# 1.3 Linear Functions.

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1.3 Linear Functions

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

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

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

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.

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 x (x2,y2) △y=y2-y1 (x2,y1) (x1,y1) △x=x2-x1

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

Slope For a line with slope m, m>0
y 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

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 x 1,Write an equation defining f
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

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