Quadratic graphs

A quadratic graph is a curve called a parabola and its equation is in the form of y = ax² + bx + c, where a, b, and c are constants.

The value of ‘a’ determines the direction and width of the parabola. If ‘a’ is positive, the graph opens upwards, and if ‘a’ is negative, the graph opens downwards. Larger

values make the graph narrower and smaller

values make the graph wider.

The value ‘c’ in the equation is the y-intercept, which is the point where the graph crosses the y-axis.
‘b’ influences the position of the axis of symmetry, a vertical line passing through the vertex of the parabola.
The solutions or roots of the quadratic equation are the x-coordinates where the graph touches or crosses the x-axis. These points are also called ‘zeros’ or ‘x-intercepts’.
You can find the x-intercepts by setting y = 0 and solving the resulting equation.
The axis of symmetry is given by the formula x = -b/2a.
The turning point or vertex of the parabola can be found using the axis of symmetry. Substitute the x value from the axis of symmetry into the quadratic equation to find the y-coordinate of the vertex.
Quadratics can have two real roots (the graph crosses the x-axis at two distinct points), one real root (the graph touches the x-axis at a single point), or no real roots (the graph doesn’t intersect with the x-axis). The term “discriminant” (b² - 4ac) can determine the number of roots.
Quadratic graphs have a line of symmetry half way between the two roots.
Completing the square is a method to rewrite the quadratic equation in the form (x-h)²=k, which shows you the vertex (h, k) directly.
Quadratic graphs are important in understanding many natural phenomena and are widely used in physics, engineering, and business economics.