Now that we have looked at the different types of graphs that you need to sketch, we need to look at how we translate graphs. Remember that each graph is a function of x which can be written as y = f(x). For this example f(x)= x2, which is the graph below:
When f(x) is translated by +k, to be f(x) +k, the equation becomes y=f(x) +k
y=f(x+k) This will shift the graph ‘k’ amounts along the x axis. When we +k the graph moves k amount to the left. f(x +2) moves the graph two places to the left.
y=f(x-2) moved the graph two places to the right:
y=-f(x) reflects the graph around the x axis: The reflection in the y axis is f(-x) To show this I am going to use the graph f(x)=x3+ 3 (as y=x2 will look the same as it’s a symmetrical through the y axis).
When it comes to reflections you also need to know how to sketch inverse operations.
Key Fact: Inverse operations are always reflected in the line y=x. For example, the inverse of a logarithm is an exponential function.
As you can see any function’s inverse is symmetrical through the mirror line y=x. f(x) is always a reflection in y=x of f-1(x)
y=k(fx) is an enlargement of scale factor k in the y direction. Here we show an enlargement of a sine graph.
y=sinx and y=3sinx. You can see the stretch is in the y axis.
The waves double because in this example we’ve gone from y=cosx to y=cos(2x), which squishes the graph up 2 more times!