Understanding Quadratic Graphs

A quadratic graph is a type of polynomial graph produced by a quadratic function. Quadratic functions have the general form y=ax²+bx+c, where ‘a’, ‘b’ and ‘c’ are constants.
The highest power in a quadratic equation is 2 (since ‘x’ is squared). This gives the graph its unique ‘U’-shape, known as a parabola.
If the value of ‘a’ is positive, the parabola opens upwards. If ‘a’ is negative, the parabola opens downwards.

Intercepts and Roots of Quadratic Graphs

The x-intercept(s) or root(s) of a quadratic graph are the points where the graph crosses the x-axis. These can be found by setting ‘y’ to 0 in the equation and solving for ‘x’.
Quadratic graphs have a y-intercept, which is found by setting ‘x’ to 0 in the quadratic equation, resulting in ‘y’ = ‘c’.
A quadratic graph can have zero, one or two x-intercepts. These represent the solutions or “roots” to the quadratic equation.

The peak or the lowest point of a quadratic graph is known as the vertex. The vertex is a turning point of the graph. The vertex can be found using the formula -b/2a to find the ‘x’ coordinate, and then substitifying this value into the equation to find the corresponding ‘y’ coordinate.
Every quadratic graph has an axis of symmetry, a vertical line that cuts the parabola into two symmetric halves. The equation for this line is x=h where ‘h’ is the x-coordinate of the vertex.

When sketching a quadratic graph, it’s helpful to identify the key features first: the y-intercept, x-intercepts (if any), the vertex and the axis of symmetry. Plotting these points accurately will result in a more precise sketch.
It’s important to remember that the graph is a smooth curve, and not composed of straight lines.
Always indicate the key points (vertex, intercepts) on the graph.

Quadratic graphs can undergo various transformations including translation, reflection, and stretching or shrinking.
A quadratic graph will shift up or down when a constant is added or subtracted to the function (vertical translation).
When a constant is added or subtracted to the ‘x’ value in the quadratic equation, this represents a horizontal translation.
The graph can be reflected in the x-axis when ‘y’ is multiplied by -1. This also happens in the quadratic equation if ‘a’ is a negative value.
A vertical stretch or shrink happens when the ‘a’ value in the quadratic equation is more or less than 1.

For successful work with quadratic graphs, always note the key points and features of the graph like the vertex, axis of symmetry, intercepts etc.
Practice sketching and interpreting quadratic graphs. Pay attention to different forms of quadratic equations and how they affect the graph’s appearance.
Show all your workings when plotting or analysing a quadratic graph, this is especially important when finding the x-intercepts and the vertex. Troubleshoot by checking your calulations if your graph doesn’t look right. Don’t forget, the graph should always be a smooth curved shape.