Mastering Quadratic Functions: Understanding Equations, Graphs, and Important Concepts

Quadratic Function

A quadratic function is a type of polynomial function of degree 2

A quadratic function is a type of polynomial function of degree 2. It can be written in the form:

f(x) = ax^2 + bx + c

Where a, b, and c are constants. The graph of a quadratic function is a curve called a parabola. The vertex of the parabola is the lowest or highest point on the graph, depending on the sign of the leading coefficient a.

To find the vertex of a quadratic function, you can use the formula:

x = -b / 2a

This formula gives you the x-coordinate of the vertex. To find the corresponding y-coordinate, you can substitute this x-value back into the equation and solve for y.

The axis of symmetry of the parabola is a vertical line that passes through the vertex. Its equation is x = -b / 2a.

The y-intercept of a quadratic function is the point where the graph crosses the y-axis. To find it, you can set x = 0 in the equation and solve for y.

Another important aspect of quadratic functions is the discriminant, which is given by the expression b^2 – 4ac. The discriminant tells you the nature of the solutions (or roots) of the quadratic equation.

1. If the discriminant is greater than zero (b^2 – 4ac > 0), then the quadratic equation has two distinct real roots.
2. If the discriminant is equal to zero (b^2 – 4ac = 0), then the quadratic equation has one real root (called a double root).
3. If the discriminant is less than zero (b^2 – 4ac < 0), then the quadratic equation has no real roots. It is also possible to express a quadratic function in factored form: f(x) = a(x - r1)(x - r2) Where r1 and r2 are the roots of the quadratic equation. This form can help to easily identify the x-intercepts of the parabola. When graphing a quadratic function, it can be helpful to determine the vertex, axis of symmetry, y-intercept, and x-intercepts. Additionally, you can find other points on the graph by plugging in different x-values into the equation. I hope this explanation helps you understand the concept of a quadratic function better. If you have any specific questions or need further clarification, feel free to ask.

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