An In-Depth Guide to Quadratic Functions and Equations: Standard Form, Vertex, Axis of Symmetry, X-Intercepts, and Y-Intercept

Quadratic Function

A quadratic function is a function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants

A quadratic function is a function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. This form is called the standard form of a quadratic function.

The graph of a quadratic function is a parabola, which can open upwards or downwards depending on whether the coefficient a is positive or negative. The vertex of the parabola is the highest or lowest point on the graph, depending on the direction it opens.

To find the vertex of a quadratic function in standard form, we can use the formula x = -b/2a to find the x-coordinate of the vertex. Once we have the x-coordinate, we can substitute it into the quadratic function to find the corresponding y-coordinate.

Additionally, the axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two equal halves. The equation of the axis of symmetry can be found by using the x-coordinate of the vertex.

To find the x-intercepts (also known as roots or zeros) of a quadratic function, we set f(x) = 0 and solve the resulting equation. This can be done by factoring, completing the square, or using the quadratic formula.

The y-intercept is the point where the graph of the quadratic function intersects the y-axis. To find the y-intercept, we substitute x = 0 into the quadratic function and solve for y.

In summary, to work with quadratic functions, it is important to understand their standard form, how to find the vertex, axis of symmetry, x-intercepts, and y-intercept. Solving quadratic equations can involve various methods such as factoring, completing the square, or using the quadratic formula. These concepts are essential for analyzing quadratic functions and their graphs.

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