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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