Basic graphs used in transformations

Basic Graphs

Linear graphs are straight-line graphs derived from equations of the form y = mx + c where ‘m’ is the gradient and ‘c’ is the y-intercept.
Quadratic graphs result from equations of the form y = ax² + bx + c, where ‘a’, ‘b’, and ‘c’ are constants. They generate a ‘U’ or ‘n’ shape called a parabola.
Cubic graphs come from equations of the form y = ax³ + bx² + cx + d and can create a variety of S and W shapes.
Circle graphs can be formed from an equation in the form of (x - h)² + (y - k)² = r². The centre of the circle is at (h,k) and ‘r’ is the radius.
Exponential graphs, derived from equations like y = a^x , are famous for their steep incline or decline.
Trigonometric graphs such as sine, cosine, and tangent come from the trigonometric functions and tend to look like waves.

Transformations of Graphs

Translations move a graph left, right, up, or down without altering its shape. This involves adding or subtracting a value from the ‘x’ or ‘y’ variable.
Reflections flip a graph over the x-axis or the y-axis. This involves changing the sign of the function, either -f(x) for an x-axis reflection or f(-x) for a y-axis reflection.
Stretches can be vertical or horizontal, essentially pulling the graph up or pushing it wider respectively. Multiply ‘y’ by a constant for a vertical stretch or ‘x’ by a constant for a horizontal stretch.
Rotations are not normally used in basic GCSE algebra.

Examples

If you’ve y = x², translating it by y = (x-2)² + 3 will move the graph 2 units to the right and three units upward.
To reflect the graph y = x² in the x-axis, use the equation y = -x².
The equation y = 2x² would stretch the graph y = x² vertically by a factor of 2.

Conclusion

Recognising these basic graphs and understanding the transformations that can be applied to them is vital in GCSE algebra.
These principles help in sketching functions, understanding their characteristics, and predicting the impact of changes to the original function.
It’s key to practice by sketching these transformations to get familiar with them. Regular and meticulous revision of transformations will secure these mathematical concepts in your memory.