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Message: <blockquote data-quote="Lovtus" data-source="post: 28725261" data-attributes="member: 532745">1.2) Quadratic function.A quadratic function is a type of function in which the highest degree of the variable is two. The general form of a quadratic function is:f(x) = ax^2 + bx + cwhere a, b, and c are constants. The graph of a quadratic function is a parabola, which can either be concave up or concave down, depending on the sign of the leading coefficient a.The vertex of the parabola is the point where the function reaches its minimum or maximum value, and is given by the formula:x = -b / 2ay = f(-b / 2a)The axis of symmetry is a vertical line that passes through the vertex, and is given by the equation:x = -b / 2aQuadratic functions can be used to model a variety of phenomena, such as the trajectory of a projectile, the shape of a bridge arch, or the growth of a population.Here are some sample problems involving quadratic functions:Problem 1: Find the vertex, axis of symmetry, and y-intercept of the quadratic function f(x) = x^2 - 4x + 5.Solution: To find the vertex, we first need to find the x-coordinate, which is given by x = -b / 2a = -(-4) / 2(1) = 2. To find the y-coordinate, we substitute x = 2 into the function: f(2) = 2^2 - 4(2) + 5 = 1. Therefore, the vertex is (2, 1). The axis of symmetry is x = 2. To find the y-intercept, we set x = 0: f(0) = 0^2 - 4(0) + 5 = 5. Therefore, the y-intercept is (0, 5).Problem 2: A ball is thrown upward from the ground with an initial velocity of 32 feet per second. The height h (in feet) of the ball at time t (in seconds) is given by the quadratic function h(t) = -16t^2 + 32t. Find the maximum height of the ball and the time it takes to reach that height.Solution: The maximum height of the ball is reached at the vertex of the parabola. To find the vertex, we use the formula x = -b / 2a = -32 / 2(-16) = 1. The time it takes to reach the maximum height is 1 second. To find the maximum height, we substitute t = 1 into the function: h(1) = -16(1)^2 + 32(1) = 16. Therefore, the maximum height of the ball is 16 feet.</blockquote>

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