Question

A parabola opens upward on a coordinate plane. The vertex of the parabola is at the point left-parenthesis 1 comma negative 4 right-parenthesis. The parabola passes through the x axis at x equals negative 1 and x equals 3. It passes through the y axis at y equals negative 3.
Question
Use the graph to answer the question.
The figure shows the graph of the function y equals left-parenthesis x minus 1 right-parenthesis squared minus 4. Identify the x-values for which the function is increasing.
(1 point)
Responses

left-parenthesis x vertical line x greater than negative 4 right-parenthesis
Image with alt text: left-parenthesis x vertical line x greater than negative 4 right-parenthesis

left-parenthesis x vertical line x less than negative 1 or x greater than 3 right-parenthesis
Image with alt text: left-parenthesis x vertical line x less than negative 1 or x greater than 3 right-parenthesis

left-parenthesis x vertical line x greater than 1 right-parenthesis
Image with alt text: left-parenthesis x vertical line x greater than 1 right-parenthesis

left-parenthesis x vertical line x greater-than-or-equal to 0 right-parenthesis
Image with alt text: left-parenthesis x vertical line x greater-than-or-equal to 0 right-parenthesis
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Answer 1

To determine the x-values for which the function \( y = (x - 1)^2 - 4 \) is increasing, we can analyze the vertex and the shape of the parabola.

1. **Vertex:** The vertex of the parabola is at the point \( (1, -4) \). This means that the function has its minimum value at \( x = 1 \).

2. **Direction of the Parabola:** Since the parabola opens upward (as indicated), it will be decreasing on the interval \( (-\infty, 1] \) and increasing on the interval \( [1, +\infty) \).

Thus, the function starts increasing at \( x = 1 \) and continues to increase for all values greater than 1.

Therefore, the x-values for which the function is increasing are:
**(x | x > 1)**.

This corresponds to the last option in your responses.