The quadratic function f is graphed in the xy-plane shown. The function will be transformed to form a second function, g. For which transformation of f will the equation f(x) = g(x) have two solutions?

A. g(x) = f(x) + 2

B. g(x) = f(x) - 2
C. g(x) = -f(x) + 2
D. g(x) = -f(x) - 2

Answer: B. g(x) = f(x) - 2

To determine which transformation of the quadratic function f will result in two solutions for f(x) = g(x), we need to consider the properties of quadratic functions.

A quadratic function is defined by the equation f(x) = ax^2 + bx + c, where a, b, and c are constants.

The number of solutions to the equation f(x) = g(x) is determined by the discriminant, which is given by the formula Δ = b^2 - 4ac.

If Δ > 0, the quadratic equation has two distinct real solutions.
If Δ = 0, the quadratic equation has one real solution.
If Δ < 0, the quadratic equation has no real solutions.

Now, let's consider the possible transformations of f that could affect the discriminant:

1. Vertical Translation: Adding or subtracting a constant to the equation does not change the discriminant, as the term involving x^2 remains the same.

2. Horizontal Translation: Shifting the quadratic function horizontally by a constant amount does not affect the discriminant.

3. Vertical Scaling: Multiplying the entire equation by a positive constant does not change the discriminant.

4. Horizontal Scaling: Multiplying the x-term by a constant affects the discriminant. The discriminant is multiplied by the square of the scaling factor, hence changing the number of solutions.

Therefore, the transformation that will result in two solutions for f(x) = g(x) is a horizontal scaling. By multiplying the x-term by a constant (greater than 1), we change the discriminant, potentially increasing the number of solutions.

To create a quadratic equation with two solutions, we need to transform the original quadratic function f in such a way that it opens upwards (like a "U" shape) and intersects the x-axis at two distinct points.

There are two common transformations that can achieve this: vertical stretching and horizontal translation. Let's explore each one separately:

1. Vertical Stretching:
- If we stretch the graph of f vertically, it becomes narrower and steeper. This transformation can be achieved by multiplying the function f by a constant greater than 1 (e.g., f(x) = k * x^2, where k > 1).
- By applying vertical stretching, the parabola gets closer to the x-axis, and if it is stretched enough, it can intersect the x-axis at two distinct points.

2. Horizontal Translation:
- If we translate the graph of f horizontally, it moves left or right. This transformation can be achieved by adding or subtracting a constant to the function f (e.g., f(x) = (x - h)^2, where h is a non-zero constant).
- By applying horizontal translation, the parabola can move sideways, changing the x-intercepts and potentially creating two distinct points of intersection with the x-axis.

Therefore, the two transformations that can result in the equation f(x) = g(x) having two solutions are vertical stretching and horizontal translation.