Answer:
D. The function is increasing when x > 0
Explanation:
The quadratic parent function is represented by the equation y = x². This function creates a U-shaped curve called a parabola when graphed. Now, let's go through the options and explain why each is incorrect until we arrive at the correct one.
A. The function is always increasing:
This statement is incorrect. The quadratic function y = x² is not always increasing. It increases for positive values of x but decreases for negative values of x. For example, when x is negative, like -2 or -3, the corresponding y-values are positive, like 4 and 9, respectively. So, the function is not always increasing.
A. The function is never increasing:
This statement is incorrect. The quadratic function y = x² is not always decreasing. It increases for positive values of x but decreases for negative values of x. Same explanation as option A.
C. The function is increasing when x < 0:
This statement is incorrect as well. The function is actually decreasing when x > 0.
D. The function is increasing when x > 0:
This statement is correct. The function y = x² is increasing when x > 0. As you move to the right along the positive x-axis, the corresponding y-values increase. For example, when x = 1, y = 1; when x = 2, y = 4; when x = 3, y = 9, and so on. This pattern continues as you move to larger positive x-values.
Hence, D is correct.
Additional Information:
Parent Function: The parent function of a family of functions is the most basic form or simplest function in that family.
Increasing Function: A function is considered increasing on an interval if, as the input values (x) increase, the corresponding output values (y) also increase. In other words, the function is moving "upwards" as you move from left to right along the x-axis.
Decreasing Function: A function is considered decreasing on an interval if, as the input values (x) increase, the corresponding output values (y) decrease. In other words, the function is moving "downwards" as you move from left to right along the x-axis.
Constant Function: A function is considered constant on an interval if the output values (y) remain the same regardless of the input values (x). In other words, the function does not change its value as you move from left to right along the x-axis.