Question

Is it possible to create a quadratic function with a minimum that has two distinct solutions that only lies within two quadrants? Explain. If so, give an example.

Answer 1

No

Explanation:

Quadratic functions with minimums, by default, have a domain of all real numbers (x∈R), this means any quadratic will lie in ≥ 2 quadrants (the positive left and right quadrant);

The only remaining question is it if lies in only these 2, a third as well or all 4 quadrants;

If the quadratic has no distinct solutions, this means it does not cross the x-axis, which means it is only above the x-axis and it will only therefore lie in the upper left and right quadrant;

If the quadratic has one solution, it would touch the x-axis but would not cross and enter either of the two quadrants below, therefore such a quadratic would only lie in the upper 2 quadrants;

If the quadratic has 2 distinct solutions, the graph must cross the x-axis falling into one or both of the quadrants below the x-axis;

Ultimately, the quadratic could have x-intercepts both to left of the y-axis, or both to the right or one to the left and one to the right;

In the case of both to one side, the portion of the function that lies under the x-axis will be in one of the lower quadrants, meaning the function will lie in 3 quadrants in total;

In the case of an x-intercept to the left and one to the right, then the portion of the function will lie in both negative quadrants under the y-axis, in which case the functions will occupy all 4 quadrants

So, the answer is no;

A quadratic function that has 2 distinct solutions will lie in either 3 or 4 quadrants, such a function cannot lie in only 2 quadratics.

Answer 2

• No

Explanation:

A quadratic function has a graph with one vertex.

This is a maximum or minimum value of the function.

Since the vertex is one point, it can only have a single unique pair of coordinates.

Example:

• f(x) = ax² + bx + c, with a > 0 and any values of b or c