Can Absolute Values of Quadratic Functions Determine Their Discriminants?

In summary, a quadratic function is a polynomial function with a degree of 2, represented by the form f(x) = ax^2 + bx + c. To prove that a function is quadratic, you can check its form, degree, and graph, or use the quadratic formula to solve for its roots. The standard form of a quadratic function is f(x) = ax^2 + bx + c, and to prove that two quadratic functions are equal, you can solve for the variables or compare their graphs. A quadratic function can have a negative coefficient of x^2, resulting in a downward parabola.
  • #1
Mathick
23
0
Let \(\displaystyle f(x)\) and \(\displaystyle g(x)\) be quadratic functions such as the inequality \(\displaystyle \left| f(x) \right| \ge \left| g(x) \right| \) is hold for all real \(\displaystyle x\) . Prove that \(\displaystyle \left| \Delta_f \right| \ge \left| \Delta_g \right|\). For quadratic function \(\displaystyle f(x)=ax^2+bx+c \), then \(\displaystyle \Delta=b^2-4ac. \)

I have no idea how I could start this task. Please, help!
 
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  • #2
Hi Mathick,

This was actually a challenge problem that anemone posted http://mathhelpboards.com/challenge-questions-puzzles-28/prove-b-4ac-8804-b-4ac-15129.html. A solution is provided in the last post.
 

Related to Can Absolute Values of Quadratic Functions Determine Their Discriminants?

1. What is a quadratic function?

A quadratic function is a polynomial function of the form f(x) = ax^2 + bx + c. It is a type of function that has a degree of 2, which means the highest power of the variable is 2. It is commonly represented by a parabola and has a U-shaped graph.

2. How do you prove that a function is quadratic?

To prove that a function is quadratic, you can check if it follows the form f(x) = ax^2 + bx + c. You can also check if the function has a degree of 2 and if its graph is a parabola. Additionally, you can use the quadratic formula to solve for the roots of the function. If the function has two distinct real roots, it is quadratic.

3. What is the standard form of a quadratic function?

The standard form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. It is the most common way of representing a quadratic function.

4. How do you prove that two quadratic functions are equal?

To prove that two quadratic functions are equal, you can set them equal to each other and solve for the variables. If the resulting equations have the same solutions, then the functions are equal. You can also graph the functions and see if they have the same shape and intercepts.

5. Can a quadratic function have a negative coefficient of x^2?

Yes, a quadratic function can have a negative coefficient of x^2. When the coefficient of x^2 is negative, the parabola will open downwards instead of upwards. This does not change the fact that it is still a quadratic function.

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