A quadratic function, is a polynomial function of the form `f(x) = ax^2 + bx + c, x!= 0.`
Therefore, if your function has an `"x^2"` term, then it is a quadratic function.
if the problem is in the `y = ax^2 + bx + c ` form x≠ 0
Characteristics of Quadratic Functions
1. Standard form is y = ax2 + bx + c, where a≠ 0.
2. The graph is a parabola, a u-shaped figure.
3. The parabola will open upward or downward.
4. A parabola that opens upward contains a vertex that is a minimum point.
A parabola that opens downward contains a vertex that is a maximum point.
Click , to view Parabola that opens upward and Parabola that opens downward.
5. The domain of a quadratic function is all real numbers.
6. To determine the range of a quadratic function, ask yourself
- Is the vertex a minimum or
- What is the y-value of the vertex?
If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value.
If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value.
7. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form x = n, where n is a real number. Click More Images to view Parabola that opens upward. Its axis of symmetry is the vertical line x =0.
8. The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots, solutions, and solution sets. Each quadratic function will have two, one, or no x-intercepts.
Consider a polynomial function
The degree of P(x) is 2. Thus P(x) is quadratic polynomial. Also it can be said that P(x) is quadratic function. To determine if function is quadratic, first check if function is polynomial, second ceheck its degree if degree is 2 then quadratic otherwise not quadratic.