What is the relationship between the vertical and horizontal shifts of a quadratic parent function and its corresponding g(x) function?
The relationship between the vertical and horizontal shifts of a quadratic parent function and its corresponding g(x) function can be understood by examining the general form of a quadratic function. The general form of a quadratic function is given by:
f(x) = a(x - h)^2 + k
In this equation, (h, k) represents the vertex of the parabola, and a determines the direction and steepness of the curve.
Now, consider the g(x) function, which is obtained from the parent function by applying vertical and horizontal shifts. The general form of g(x) can be expressed as:
g(x) = a(x - h')^2 + k'
In this equation, (h', k') represents the new vertex of the parabola, which reflects the shifts.
To understand the relationship between vertical and horizontal shifts, follow these steps:
1. Vertical Shift: The k' value in the g(x) equation represents the vertical shift of the parabola. If k' is positive, the parabola moves upward by k' units; if k' is negative, it moves downward by |k'| units. When k' = 0, there is no vertical shift.
2. Horizontal Shift: The h' value in the g(x) equation represents the horizontal shift of the parabola. If h' is positive, the parabola moves to the right by h' units; if h' is negative, it moves to the left by |h'| units. When h' = 0, there is no horizontal shift.
Therefore, the vertical shift in the g(x) function is represented by k', and the horizontal shift is represented by h'. By analyzing these two values, you can understand how the parabola shifts from the original parent function.