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Bell Ringer 1) If (x) = 3x + 2, then what is the solution of f(2). Hint: substitute 2 in for x. 2) If f(x) = 2x2 – 3x + 4, then what is f(3), or what’s the solution when you substitute 3 in for x? Linear Functions Function Notationa linear function written in the form y = mx + b where y is written as a function f. x-coordinate This is read as ‘f of x’ f(x) = mx + b slope y-intercept f(x) is another name for y. It means “the value of f at x.” g(x) or h(x) can also be used to name functions Domain and Range = values of ‘x’ for which the function is defined. Domain = the values of f(x) where ‘x’ is in the domain of the function f. Range The graph of a function f is the set of all points (x, f(x)). Graphing a Function To graph a function: (1) make a table by substituting into the function. (2) plot the points from your table and connect the points with a line. (3) identify the domain and range, (if restricted) Graph a Function Graph the Function SOLUTION STEP 1 STEP 2 Make a table by choosing a few values for x and then finding values for y. -2 -1 STEP 3 Plot the points. Notice the points appear on a line. Connect the points drawing a line through them. f ( x) 2 x 3 x f(x) = 2x – 3 0 1 2 f(x) -7 -5 -3 -1 1 f ( x) 2 x 3 The domain and range are not restricted therefore, you do not have to identify. Graph a Function 1 – Graph the function f(x) = 2 x + 4 with domain x ≥ 0. Then identify the range of the function. STEP 1 x 0 2 4 6 8 Make a table. y 4 3 2 1 0 STEP 2 Plot the points. Connect the points with a ray because the domain is restricted. f ( x) 1 x4 2 STEP 3 Identify the range. From the graph, you can see that all points have a y-coordinate of 4 or less, so the range of the function is y ≤ 4. Family of Functions is a group of functions with similar characteristics. For example, functions that have the form f(x) = mx + b constitutes the family of linear functions. Parent Linear Function The most basic linear function in the family of all linear functions is called the PARENT LINEAR FUNCTION which is: f(x) = x f(x) = x x -5 -2 0 1 3 f(x) -5 -2 0 1 3 Real-Life Functions A cable company charges new customers $40 for installation and $60 per month for its service. The cost to the customer is given by the function f(x) = 60x +40 where x is the number of months of service. To attract new customers, the cable company reduces the installation fee to $5. A function for the cost with the reduced installation fee is g(x) = 60x + 5. Graph both functions. How is the graph of g related Real-Life Functions The graphs of both functions are shown. Both functions have a slope of 60, so they are parallel. The y-intercept of the graph of g is 35 less than the graph of f. So, the graph of g is a vertical translation of the graph of f. Write an Equation Given a Slope and a Point Write the Equation using Point-Slope Form 1 m and (1,6) 3 Step 1: Plug it in y y1 m( x x1 ) Point- Slope Form A Challenge Write the equation of a line in point-slope form that passes through (-3,6) and (1,-2) Hint: Find the change in y and the change in x. Change is determined by subtraction. Reminder that slope is rise or run. A Challenge Can you write the equation of a line in point-slope form that passes through (-3,6) and (1,-2) 6+2 =- 8 =- 2 m= -3 – 1 4 1 y – y1 = m(x – x1) y + 2 = - 2 (x – 1) Calculate the slope. Use m = - 2 and the point (1, -2). Point-Slope Form y = -2(x – 1) - 2 y = -2x + 2 - 2 y = -2x + 0 Slope-Intercept Form y = -2x Equations of Parallel Lines Write an equation for the line that contains (5, 1) and is parallel to 3 m= 5 3 y 1 x 5 5 3 y x4 5 Equations of Perpendicular Lines Find the equation of the line that contains (0, -2) and is perpendicular to y = 5x + 3 1 m= 5 1 y2 x 5