Mastering Quadratic Functions: Understanding Properties, Intercepts, Symmetry, and the Graph

quadratic function

A quadratic function is a polynomial function of degree 2

A quadratic function is a polynomial function of degree 2. It has the general form:

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

where a, b, and c are constants. The highest power of x in a quadratic function is 2.

The graph of a quadratic function is a curve called a parabola. The shape of the parabola depends on the sign of the coefficient a:

1. If a > 0, the parabola opens upwards. In this case, the vertex of the parabola is the lowest point on the graph, and the graph increases as you move away from the vertex in both directions.

2. If a < 0, the parabola opens downwards. In this case, the vertex of the parabola is the highest point on the graph, and the graph decreases as you move away from the vertex in both directions. To find the x-intercepts of a quadratic function, you need to solve the equation f(x) = 0. This means you need to find the values of x for which the function equals zero. The x-intercepts are the points where the parabola intersects the x-axis. To find the y-intercept of a quadratic function, you can substitute x = 0 into the function and solve for y. The y-intercept is the point where the parabola intersects the y-axis. The axis of symmetry is a vertical line that passes through the vertex of the parabola. Its equation can be found using the formula x = -b/2a. The vertex of a quadratic function can also be found using the formula x = -b/2a. Once you find the x-coordinate of the vertex, you can substitute it back into the function to find the corresponding y-coordinate. Finally, the graph of a quadratic function is symmetric with respect to the axis of symmetry. This means that if (x, y) is a point on the graph, then (-x, y) is also a point on the graph.

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