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Quadratic Functions Graphs of Quadratic Functions The graph of any quadratic function is called a parabola. Parabolas are shaped like cups, as shown in the graph below. If the coefficient of x2 is positive, the parabola opens upward; otherwise, the parabola opens downward. The vertex (or turning point) is the minimum or maximum point. The Standard Form of a Quadratic Function The quadratic function f (x) = a(x - h)2 + k, a0 is in standard form. The graph of f is a parabola whose vertex is the point (h, k). The parabola is symmetric to the line x = h. If a > 0, the parabola opens upward; if a < 0, the parabola opens downward. Graphing Parabolas With Equations in Standard Form To graph f (x) = a(x - h)2 + k: 1. Determine whether the parabola opens upward or downward. If a > 0, it opens upward. If a < 0, it opens downward. 2. Determine the vertex of the parabola. The vertex is (h, k). 3. Find any x-intercepts by replacing f (x) with 0. Solve the resulting quadratic equation for x. 4. Find the y-intercept by replacing x with zero. 5. Plot the intercepts and vertex. Connect these points with a smooth curve that is shaped like a cup. Text Example Graph the quadratic function f (x) = -2(x - 3)2 + 8. Solution We can graph this function by following the steps in the preceding box. We begin by identifying values for a, h, and k. Standard form f (x) = a(x - h)2 + k a = -2 h=3 k=8 Given equation f (x) = -2(x - 3)2 + 8 Step 1 Determine how the parabola opens. Note that a, the coefficient of x 2, is -2. Thus, a < 0; this negative value tells us that the parabola opens downward. Text Example cont. Step 2 Find the vertex. The vertex of the parabola is at (h, k). Because h = 3 and k = 8, the parabola has its vertex at (3, 8). Step 3 Find the x-intercepts. Replace f(x) with 0 in f(x) = -2(x - 3)2 + 8. 0 = -2(x - 3)2 + 8 Solve for x. Add 2(x - 3)2 to both sides of the equation. 2(x - 3) = 8 2 (x - 3)2 = 4 (x - 3) = 2 x - 3 = -2 x=1 Find x-intercepts, setting f (x) equal to zero. or or Divide both sides by 2. Apply the square root method. x-3=2 x=5 Express as two separate equations. Add 3 to both sides in each equation. The x-intercepts are 1 and 5. The parabola passes through (1, 0) and (5, 0). Text Example cont. Step 4 Find the y-intercept. Replace x with 0 in f(x) = -2(x - 3)2 + 8. f(0) = -2(0 - 3)2 + 8 = -2(-3)2 + 8 = -2(9) + 8 = -10 The y-intercept is –10. The parabola passes through (0, -10). Step 5 Graph the parabola. With a vertex at (3, 8), x-intercepts at 1 and 5, and a y-intercept at –10, the axis of symmetry is the vertical line whose equation is x = 3. The Vertex of a Parabola Whose Equation Is f (x) = ax 2 + bx + c Consider the parabola defined by the quadratic function f (x) = ax 2 + bx + c. The parabola's vertex is at -b -b , f 2a 2a Example Graph the quadratic function f (x) = -x + 6x -. 2 Solution: Step 1 Determine how the parabola opens. Note that a, the coefficient of x 2, is -1. Thus, a < 0; this negative value tells us that the parabola opens downward. Step 2 Find the vertex. We know the x-coordinate of the vertex is – b/2a. We identify a, b, and c to substitute the values into the equation for the x-coordinate: x = -b/(2a) = -6/2(-1) = 3. The x-coordinate of the vertex is 3. We substitute 3 for x in the equation of the function to find2 the y-coordinate: y = f (3) = -3 + 6(3) - 2 = -9 + 18 - 2 = 7 the parabola has its vertex at (3,7). Example Graph the quadratic function f (x) = -x2 + 6x -. Step 3 Find the x-intercepts. Replace f (x) with 0 in f (x) = -x2 + 6x - 2. 0 = -x2 + 6x - 2 a = -1,b = 6,c = -2 -b b 2 - 4ac x= 2a 2 -6 6 - 4(-1)(-2) = 2(-1) -6 36 - 8 -2 -6 28 -6 2 7 = = -2 -2 = 3 7 = Example Graph the quadratic function f (x) = -x2 + 6x -. Step 4 - 2. Find the y-intercept. Replace x with 0 in f (x) = -x2 + 6x f (0) = -02 + 6 • 0 - 2 = - The y-intercept is –2. The parabola passes 10through (0, -2). 8 Step 5 Graph the parabola. 6 4 2 -10 -8 -6 -4 -2 2 -2 -4 -6 -8 -10 4 6 8 10 Minimum and Maximum: Quadratic Functions • Consider f(x) = ax2 + bx +c. 1. If a > 0, then f has a minimum that occurs at x = -b/(2a). This minimum value is f(-b/(2a)). 2. If a < 0, the f has a maximum that occurs at x = -b/(2a). This maximum value is f(-b/(2a)). Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions 1. 2. 3. 4. 5. Read the problem carefully and decide which quantity is to be maximized or minimized. Use the conditions of the problem to express the quantity as a function in one variable. Rewrite the function in the form f(x) = ax2 + bx +c. Calculate -b/(2a). If a > 0, then f has a minimum that occurs at x = -b/(2a). This minimum value is f(-b/(2a)). If a < 0, the f has a maximum that occurs at x = b/(2a). This maximum value is f(-b/(2a)). Answer the question posed in the problem.