Understanding Quadratic Graphs

Quadratic graphs represent quadratic equations, which are equations of the second degree.
A quadratic equation can be written in the form y = ax^2 + bx + c, where a, b and c are constants, and x is the variable. The value of a is never zero in quadratic equations.
Quadratic graphs will always be a curved line called a parabola, either opening upwards (a “smiley face”, if a > 0) or downwards (a “sad face”, if a < 0).

Sketching Quadratic Graphs

To plot a quadratic graph, first set up a grid with x and y axes, then substitute values into the equation to gain coordinate points (x, y).
Single points where the graph intersects with the x-axis (y = 0) or the y-axis (x = 0) are known as “roots” or “zeros”.
The line of symmetry is the value of the x-coordinate at the vertex (the maximum or minimum point of the curve). For any quadratic equation in the form y = ax^2 + bx + c, it is given by x = -b/2a.

A quadratic equation is solved by finding the roots (where y = 0), which are the x-coordinates where the graph crosses the x-axis.
Note that a quadratic equation may have two, one or no roots depending on the position of the graph.
If the equation is equal to zero (y = 0), then the roots of the equation are the solutions to the equation. These are the points where the graph intersects the x-axis.

Transformations can shift, stretch or reflect the graph. These can be in the x or y direction.
Vertical shifts (up or down) happen when a number is added or subtracted at the end of the equation.
Horizontal shifts (to the left or right) occur when a number is added or subtracted from ‘x’ in the bracketed part of the equation.
Reflections occur when the sign of ‘a’ is changed.
Stretches or shrinks happen when the absolute value of ‘a’ is greater or less than 1, respectively.