Question

Determine the domain, range, vertex, and the intervals of increase and decrease of y = 0.6(t + 5)² - 8 without the use of a graph. Express your interval notation and vertex as an ordered pair. Explain how you determined the domain, range, and the intervals of increase and decrease in words.

A) Domain: All Real Numbers, Range: [-8, [infinity]), Vertex: (-5, -8), Increasing: (-[infinity], -5), Decreasing: (-5, [infinity])
B) Domain: All Real Numbers, Range: (-[infinity], -8], Vertex: (-5, -8), Increasing: (-5, [infinity]), Decreasing: (-[infinity], -5)
C) Domain: All Real Numbers, Range: [-8, [infinity]), Vertex: (5, -8), Increasing: (-5, [infinity]), Decreasing: (-[infinity], -5)
D) Domain: All Real Numbers, Range: (-[infinity], -8], Vertex: (5, -8), Increasing: (-[infinity], -5), Decreasing: (-5, [infinity])

Answer 1

The domain of y = 0.6(t + 5)² - 8 is all real numbers, and the range is [-8, infinity) with the vertex at (-5, -8). The function is increasing on the interval (-5, infinity) and decreasing on the interval (-infinity, -5).

Step-by-step explanation:

When analyzing the function y = 0.6(t + 5)² - 8, we can determine its domain, range, vertex, and the intervals of increase and decrease without graphing. The domain of a quadratic function is always all real numbers since there are no restrictions on the values that t can take.

The range of the function can be found by observing the direction of the parabola and its vertex. Since the coefficient of the squared term (0.6) is positive, the graph opens upwards, making the vertex the lowest point on the graph. Therefore, the range will be all real numbers greater than or equal to the y-coordinate of the vertex.

The vertex can be found by identifying the value of t that makes the expression within the square (t+5) equal to zero; thus the vertex is (-5, -8), where -5 is the x-coordinate, and -8 is the y-coordinate (the minimum value of y).

For the intervals of increase and decrease, since the parabola opens upwards, it is decreasing from negative infinity until it reaches the vertex, and then it starts increasing from the vertex to infinity. Therefore, the function is increasing on the interval (-5, infinity) and decreasing on the interval (-infinity, -5).