is the minimum value of f The parabola opens down if the vertex is a maximum

# Is the minimum value of f the parabola opens down if

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is the minimum value of f . The parabola opens down if ; the vertex is a maximum point and f - b 2 a ! is the maximum value of f . 56
The y -intercept is the value of f at x = 0. That is, the y -intercept is The x -intercepts, if any, are found by solving the quadratic equation ax 2 + bx + c = 0. Recall the Quadratic Formula . x = If the discriminant b 2 - 4 ac > 0, the graph has If the discriminant b 2 - 4 ac = 0, the graph has If the discriminant b 2 - 4 ac = 0, the graph will touch the x -axis at its If the discriminant b 2 - 4 ac < 0, the graph has When the quadratic function is in the form: f ( x ) = a ( x - h ) 2 + k , we can use what we learned about transformations in Sec 3.5 to identify the vertex. h is the shift and k is the shift. The vertex is . In this form, it is still true that if a > 0 the parabola opens and if a < 0 the parabola opens . Recall, again, Section 2.2 . The y -intercept is the value of f at x = 0. That is, the y -intercept = f (0). You must calculate it. The x -intercepts, if any, are found by letting y = 0. That is, the x -intercepts, if any, are found by solving the quadratic equation a ( x - h ) 2 + k = 0. Recall Section 1.2 . This equation is best solved by the Square-root Property . Do not . *************** Summary Standard Form ( h, k ) Form f ( x ) = ax 2 + bx + c f ( x ) = a ( x - h ) 2 + k vertex - b 2 a , f - b 2 a !! ( h, k ) a a > 0 opens up same a < 0 opens down same y -int let x = 0, solve for y same y -int= c calculate the y -intercept x -int(s) let y = 0, solve for x same (factor or quad. formula) (use square-root property) 57
Match each function with one of the graphs below. f ( x ) = - x 2 - 1 f ( x ) = ( x + 1) 2 - 1 f ( x ) = x 2 - 2 x f ( x ) = x 2 + 2 x + 1 f ( x ) = x 2 + 2 x + 2 x y 1 x y 1 -1 1 -1 x y 1 x y 1 x y 1 58
Graph the function f ( x ) = ( x - 3) 2 - 10 by starting with the graph of y = x 2 and using transformations (shifting, compressing, stretching, and/or reflecting.) x y Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y -intercept, and any x -intercepts. Determine the domain and range. Determine where the function is increasing and where it is decreasing. f ( x ) = - x 2 +4 x f ( x ) = ( x +2) 2 - 4 f ( x ) = x 2 +6 x +9 x y x y x y 59
f ( x ) = - 3 x 2 + 3 x - 2 f ( x ) = 2 x 2 + 5 x + 3 x y x y Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. f ( x ) = 4 x 2 - 8 x + 3 f ( x ) = - 3 x 2 + 12 x + 1 f ( x ) = ( x - 3) 2 + 5 f ( x ) = - 2( x + 3) 2 - 7 Determine, without graphing, the interval on which the given function is increasing and on which it is decreasing. f ( x ) = 4 x 2 - 8 x + 3 f ( x ) = - 3 x 2 + 12 x + 1 f ( x ) = ( x - 3) 2 + 5 f ( x ) = - 2( x + 3) 2 - 7 60
The John Deere company has found that the revenue, in dollars, from sales of riding mowers is a function of the unit price p , in dollars, that it charges. If the revenue R is R ( p ) = - 1 2 p 2 + 1900 p what unit price p should be charged to maximize revenue? What is the maximum revenue? The marginal cost C (in dollars) of manufacturing x cell phones (in thousands) is given by C ( x ) = 5 x 2 - 200 x + 4000 . How many cell phones should be manufactured to minimize the marginal cost? What is the minimal marginal cost?