Final answer:
The range of the quadratic function y = x² - 14x + 49 is the set of all real numbers y such that y is greater than or equal to 0, as determined by the vertex of the parabola, which is (7, 0).
Step-by-step explanation:
The student is asking about the range of the quadratic function y = x² - 14x + 49. To find the range, we first need to determine the vertex of the parabola, as the vertex will give us the maximum or minimum value of the quadratic function, depending on whether the parabola opens upwards or downwards. Since the coefficient of the x² term is positive, the parabola opens upwards, and therefore has a minimum value at its vertex.
To find the vertex, we use the formula x = -b/(2a), where a and b are coefficients from the quadratic equation ax² + bx + c. Here a=1 and b=-14. Substituting these values we get the x-coordinate of the vertex as x = 14/2 = 7. Plugging this back into the equation, we get y = 7² - 14(7) + 49 = 49 - 98 + 49 = 0. Therefore, the vertex is at point (7, 0).
Since the quadratic function has a minimum value at the vertex and it opens upwards, the range of the quadratic function is y ≥ 0. Any value of y less than 0 is not part of the range because the parabola does not extend below the x-axis in this case. Therefore, the correct range for this function is y.