Final answer:
To write the equation of the quadratic function, we can use the given information about the axis of symmetry, range, and x-intercept. By substituting the values into the equation for a quadratic function in the form y = a(x - h)^2 + k and solving for the unknowns, we find the equation y = -2x^2 + 12x - 12.
Step-by-step explanation:
To write an equation that represents the quadratic function, we can use the given information.
First, we know that the axis of symmetry is at x = 1. This means that the vertex of the parabola is at (1, y), where y is the y-coordinate of the vertex.
Second, the range of the function is (-∞, 4]. This means that the y-values of the function are greater than or equal to negative infinity and less than or equal to 4.
Finally, we know that the x-intercept is at (5, 0). This means that the function crosses the x-axis at x = 5.
Based on this information, we can write the equation in the form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Since the axis of symmetry is x = 1, the vertex is (1, k). We also know that the function crosses the x-axis at x = 5, so we can use this point to find the value of a.
Substituting the value of the vertex (1, k) and the point (5, 0) into the equation, we can solve for a and find the equation that represents the quadratic function.
After rearranging and solving the equation, we obtain the equation y = -2x^2 + 12x - 12.