Question

Which of the following statements accurately describes the graph, domain, and range of the quadratic equation y = x² + x + 9 after sketching it using a graphing calculator?

a. Domain: all real numbers, Range: y≤8.75
b. Domain: x≥0, Range: y≥0
c. Domain: all real numbers, Range: y≥8.75
d. Domain: all real numbers, Range: y≥−8.75

Answer 1

The graph of the quadratic equation y = x² + x + 9 is a parabola that opens upward. The range of the function is y ≥ 35/4, which is approximately 8.75. The domain of the function is all real numbers.

Step-by-step explanation:

The graph of the quadratic equation y = x² + x + 9 is a parabola that opens upward. The vertex of the parabola represents the minimum point of the quadratic function, which occurs at the x-coordinate -b/2a.

Since the coefficient of x² is positive (a=1), the parabola opens upward and the vertex represents the minimum value.

Therefore, the range of the function is y ≥ minimum value, which is the y-coordinate of the vertex.

In this case, the minimum value is the y-coordinate at the vertex, which can be found using the formula:

y = (4ac - b²) / 4a

Substituting the values from the equation y = x² + x + 9, we get:

y = (4(1)(9) - 1²) / 4(1)

y = 35/4

Therefore, the range of the function is y ≥ 35/4, which is approximately 8.75. The domain of the function is all real numbers since there are no restrictions on the input (x-values) of the quadratic function.