function f(x) = -2x² + 12x - 10, showing the vertex, axis of symmetry, x-intercepts, y-intercept, and the maximum value. Demonstrate how the parabola opens downward and identify the domain and range.

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Let's analyze the quadratic function f of x equals negative 2 x squared plus 12 x minus 10. First, we note that the leading coefficient a equals negative 2, which means the parabola opens downward. We can find the vertex by using the formula negative b over 2a for the x-coordinate. This gives us x equals 3. Substituting this value back into the function, we get a y-coordinate of 8. So the vertex is at the point (3, 8). Since this is a parabola, the axis of symmetry passes through the vertex, giving us a vertical line at x equals 3. Now, let's find the intercepts of our function. To find the x-intercepts, we set f of x equal to zero and solve the equation negative 2 x squared plus 12 x minus 10 equals 0. Using the quadratic formula or factoring, we get x equals 1 or x equals 5. So our x-intercepts are the points (1, 0) and (5, 0). For the y-intercept, we substitute x equals 0 into our function, giving us f of 0 equals negative 10. So the y-intercept is the point (0, negative 10). Since our parabola opens downward, the vertex represents the maximum point of the function. Therefore, the maximum value of the function is 8, which occurs at x equals 3. Let's now identify the domain and range of our quadratic function. The domain of a quadratic function is all real numbers, which we write as the interval from negative infinity to positive infinity. This is because we can substitute any real number for x in our function and get a valid output. For the range, we need to consider the direction of opening. Since our parabola opens downward, the function has a maximum value of 8 but no minimum value. Therefore, the range is all real numbers less than or equal to 8, which we write as the interval from negative infinity to 8, with 8 included. This means the function can take any value from negative infinity up to and including 8, but no values greater than 8. Let's understand how our quadratic function relates to the basic parabola y equals x squared. We can rewrite our function in vertex form as f of x equals negative 2 times the quantity x minus 3 squared, plus 8. This form helps us see the transformations applied to the basic parabola. First, the basic parabola y equals x squared is stretched vertically by a factor of 2. Then, it's reflected across the x-axis due to the negative coefficient. Next, it's shifted 3 units to the right, as indicated by the x minus 3 term. Finally, it's shifted 8 units upward. These transformations give us our final parabola, which opens downward and has its vertex at the point (3, 8). Let's summarize what we've learned about the quadratic function f of x equals negative 2 x squared plus 12 x minus 10. This function represents a parabola that opens downward because the leading coefficient negative 2 is less than zero. The vertex of the parabola is at the point (3, 8), which represents the maximum value of the function. The parabola crosses the x-axis at two points: (1, 0) and (5, 0), which are the x-intercepts. It crosses the y-axis at the point (0, negative 10), which is the y-intercept. The domain of the function is all real numbers, written as the interval from negative infinity to positive infinity. The range is all real numbers less than or equal to 8, written as the interval from negative infinity to 8, inclusive. These key features provide a complete description of the behavior of our quadratic function.