Quadratic Functions: A Brief Overview Definition A quadratic function is a function of the form f ( x) ax 2 bx c, where a, b, and c are real numbers and a 0. Observe that the highest power of x is 2 for a quadratic function. Graphs of Quadratic Functions Consider the quadratic function f ( x) ax 2 bx c. Then the graph of this function • is called a parabola • is symmetric with respect to a line, called its axis of symmetry • opens upward if a > 0 • opens downward if a < 0 • has a lowest or highest point, called its vertex a<0 a >0 Extreme Point of a quadratic function The amazing thing is we can easily find the x and y coordinates of the vertex of a parabola. If f ( x) ax 2 bx c, b then the x-coordinate of the vertex is x . 2a Once we know the x-coordinate of the vertex, we substitute it in the function to find the y-coordinate of the vertex. Example #1 Shown below is the graph of a quadratic function. Find: 10 a) the coordinates of each 9 intercept of the graph of 8 f(x). 7 b) the coordinates of the 5 vertex of the graph of f(x). 4 3 c) What is the smallest 2 possible value of this 1 function? -1 0 1 2 3 4 5 -1 -2 -3 -4 Example #2 Let f ( x) 3 x 2 12 x 5. Without graphing the function, a) say whether the graph opens upward or downward b) find the x and y coordinates of the vertex . Please verify these results by graphing the function on your graphing calculator. Example #3 Consider again f ( x) 3 x 2 12 x 5. Determine a) the y intercept of the graph of this function. b) the x intercepts of the graph of this function. Note that: • we can use the Zero feature to estimate the x-intercepts. • we can use the Quadratic Formula to find the exact coordinates of the x-intercepts. Quadratic Functions: Applications Quadratic functions in real life Certain real-life problems can be modeled by linear functions. However, there are others for which quadratic functions are more appropriate. Using graphing calculators Given a problem. We will use our knowledge of key features of quadratic functions to find a correct window. Then we can use the Zero, the Maximum, the Minimum, as well as the Intersection features of our graphing calculators to address important questions. Example #4 The hourly cost of C (in dollars) of manufacturing x cell phones at a plant can be modeled by C ( x) 5 x 2 200 x 4000. a) Determine the hourly fixed cost. b) How many cells phones should be manufactured to minimize the hourly cost? c) What is the minimum hourly cost? Example #5 It is known that the population of Roman Catholic nuns in the US has been changing. Suppose that for the year that is t years after 1970, the number of nuns (in thousands) can be modeled by P(t ) 0.15t 2 2t 123.75. a) According to this model, how many nuns were there in 1990? b) According to the model, when would the number of nuns in the US be just 60,000? c) According to this model, when did the population of Roman Catholic nuns in the US reach a peak? d) What was the largest population of Roman Catholic nuns in the US, rounded to the nearest 1000? Example #6 The price p and the quantity sold x of a certain product obey the demand equation 1 p( x) x 100 0 x 600 6 a) Express the revenue as a function of x. b) What is the revenue when 200 units are sold? c) What is the maximum revenue? d) What price should be charged to maximize revenue?
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