\( 1 \leftarrow \) Consider the function \( f(x)=-2 x^{2}+16 x-9 \). a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. a. The function has a value.
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The function \( f(x) = -2x^2 + 16x - 9 \) is a quadratic function, and because the coefficient of \( x^2 \) is negative (-2), it opens downwards. This means it has a maximum value. To find the maximum value, we use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = 16 \). Plugging in these values gives us: \[ x = -\frac{16}{2 \times -2} = 4. \] Now, substituting \( x = 4 \) back into the function: \[ f(4) = -2(4)^2 + 16(4) - 9 = -32 + 64 - 9 = 23. \] Thus, the maximum value is 23, which occurs at \( x = 4 \). The function's domain is all real numbers, \( (-\infty, \infty) \), and since it opens downwards, its range is from the maximum value down to negative infinity, \( (-\infty, 23] \). Exploring quadratic functions further, you will find that their vertex gives you crucial information about their minima or maxima. This is a great way to visualize how these functions behave without a graph! If you dive into real-world applications, you'll see quadratic functions pop up in many fields like engineering and physics! They help in modeling trajectories, maximizing profits, or minimizing costs. Whether you're designing a parabolic bridge or optimizing a business, these functions are like the Swiss army knives of mathematics—versatile and essential!
