Find the range of the quadratic function and the maximum or minimum value of the function. Identify the intervals over which the function is increasing or decreasing. \( y=-2 x^{2}+4 x+3 \)

We are given the quadratic function:   y = -2x² + 4x + 3 Step 1. Find the vertex. For a quadratic function in the form y = ax² + bx + c, the x-coordinate of the vertex is given by:   x = -b/(2a). Here, a = -2 and b = 4, so:   x = -4 / (2 × -2) = -4 / -4 = 1. Now, substitute x = 1 into the function to find the y-coordinate:   y = -2(1)² + 4(1) + 3 = -2 + 4 + 3 = 5. So, the vertex is at (1, 5). Since a = -2 is negative, the parabola opens downward, meaning the vertex is a maximum point. Step 2. Determine the range. Because the vertex is the maximum point, the function's maximum value is 5 and y can take any value less than or equal to 5. Thus, the range is:   (−∞, 5]. Step 3. Identify intervals of increase and decrease. For a downward-opening parabola, the function increases as it approaches the vertex from the left and decreases after the vertex. More specifically: - The function is increasing on the interval (−∞, 1] because as x moves toward 1 from the left, y increases. - The function is decreasing on the interval [1, ∞) because as x moves away from 1 toward the right, y decreases. Summary: • Maximum value: 5 at x = 1. • Range: (−∞, 5]. • Increasing on: (−∞, 1]. • Decreasing on: [1, ∞).