Transformations can change the intercepts of a function. A vertical translation will shift the y-intercepts up or down, while a horizontal translation will move the x-intercepts left or right. Reflections over the x-axis or y-axis can change the quadrant in which the intercepts lie, potentially moving them to the opposite side of the axis. Dilations can either bring the intercepts closer to or farther from the origin, depending on whether the dilation is a compression or a stretch. It's essential to consider these effects when analysing the transformed function's graph.

Transformations can change the appearance of trigonometric functions but not their inherent periodic nature. For instance, translating or reflecting a sine or cosine function will move or flip its graph, but the function will still repeat its values in regular intervals. However, horizontal dilations can change the length of one period. For example, if the sine function is horizontally dilated by a factor of 2, its period will be halved. It's crucial to recognise the effect of transformations on the periodicity to understand the function's behaviour over different intervals.

Yes, translations and reflections don't change the shape of a function's graph. While they can move the graph to a different location on the coordinate plane or flip it over an axis, the inherent shape remains the same. For example, translating a parabola upwards will shift its position, but it will still look like a parabola. Similarly, reflecting a cubic function over the y-axis will produce a mirror image, but the curve's nature remains unchanged. Only dilations and some advanced transformations can alter the original shape of a function's graph.

Transformations like translations and vertical dilations won't change the overall increasing or decreasing nature of a function. However, reflections can. Reflecting a function over the x-axis will change its direction. For instance, an increasing function will become decreasing after such a reflection. Similarly, reflecting over the y-axis can reverse the direction in which the function increases or decreases. It's vital to understand these effects, especially when analysing real-world scenarios modelled by functions, as the behaviour of the function can provide insights into the situation it represents.

When multiple transformations are applied to a function, they are performed in a specific order. For instance, if a function is first reflected over the x-axis and then translated upwards, the reflection is done first, followed by the translation. It's essential to understand the sequence of transformations to accurately determine the final position and shape of the function. However, some transformations, like translations, are commutative, meaning their order doesn't affect the outcome. But, for non-commutative transformations like dilation followed by a reflection, the order matters and can produce different results.