2Quiz: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.
3Linear FunctionDefinition: A function f defined by f(x)=ax+b, where a and b are real numbers, is called a linear function.f(x)=ax+ba: coefficient of xb: constant termx is input, f(x) is output
4Linear Function Algebraically Geometrically f(x)=ax+b If a point(m,n) is on the line f(x)=ax+ b, then n=a*m+byx
5Zero of a FunctionDefinition: 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.
6x-,y-Intercept f(x)=ax+b y-Intercept: the output of f(0) y x-Intercept:zero of the function
7Graphing a linear function by x-,y-intercepts Exercise: graph the line given by 2x+5y=15yxSolution:step1: finding x-,y-intercepts by solving linear equations.x-intercept: 7.5y-intercept: 3108642step2: plotting corresponding points on the xy-plane.-2-4-6-8-10step3: connecting the two points by a straight line and extend the line over two ends.
8SlopeDefinition: 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.
10SlopeExercise: 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=6yx108642-2-4-6-8-10Line 1Line 2
11Slope For a line with slope m, m>0 ym>0The graph rises from left to rightm<0The graph falls from left to rightm=0The line is horizontalm is undefinedThe line is verticalxyxyxyx
12Slope-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.
13Exercise 1, Graph the line containing (2,4) with a slope of m=-3/4 2, Graph the line represented by y=2x-53, 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?
14exercise y x 1,Write an equation defining f 108642-2-4-6-8-101,Write an equation defining f2, Find the slope, y-intercept, x-intercept3, Find any zeros of f