Back to Edexcel Transformations of Curves (H) Home
2.9 E) Reflections in the x axis
2.9 E) Reflections in the x axis
A reflection in the x-axis occurs when we have a negative function. For example, suppose that we have a function f(x), we can reflect the function in the x-axis by multiplying it by -1, which means that the function is -f(x).
Let’s have a look at an example and say that the function f(x) is:
Let’s have a look at an example and say that the function f(x) is:
We are able to plot the graph of f(x) by subbing in a range of x values and squaring them. The table below is for x values between -3 and 3.
We can sketch f(x) by plotting the coordinates above and then joining them together with a smooth curve. A sketch of f(x) is given below.
We are now going to sketch -f(x) and we will be using a table to find the coordinates for the points on the curve. The first 3 rows in the table will be the same as the table above (the green one). The next row will be where we multiply f(x) by -1 to obtain -f(x). The table is shown below.
The function f(x) and -f(x) have been sketched on the graph below; f(x) is in green and -f(x) is in blue.
The green curve is the function f(x) and the blue curve is the function -f(x)
From the two different curves, we can see that -f(x) is a reflection of f(x) in the x-axis.
Working Out Point
We are able to obtain the coordinates of points when we have a negative function by multiplying the y-coordinates by -1 (alternatively, you can view this as just changing the sign).
For example, we have a function g(x) and a sketch of this function is given below.
We are able to obtain the coordinates of points when we have a negative function by multiplying the y-coordinates by -1 (alternatively, you can view this as just changing the sign).
For example, we have a function g(x) and a sketch of this function is given below.
There are two points of interest on the above function; point A and point B. Point A has the coordinates (-3, 4) and point B has the coordinates (2, 5).
Question: Sketch the graph of -g(x) and find the coordinates A and B.
The translated graph is going to be reflected in the x-axis and we obtain the coordinates of the point A and B by multiplying the y coordinates by -1.
The coordinates of A are (-3, 4). The y coordinate of A is 4 and we multiply this by -1 (or change the sign). This tells us that the coordinates for A on the translated curve are (-3, -4).
Question: Sketch the graph of -g(x) and find the coordinates A and B.
The translated graph is going to be reflected in the x-axis and we obtain the coordinates of the point A and B by multiplying the y coordinates by -1.
The coordinates of A are (-3, 4). The y coordinate of A is 4 and we multiply this by -1 (or change the sign). This tells us that the coordinates for A on the translated curve are (-3, -4).
We do the same for B, which has the coordinates (2, -5). The y coordinate for B is -5 and we need to multiply this by -1 (or change the sign). This tells us that the coordinates for B on the translated curve are (2, 5)
The function g(x) and -g(x) are sketched on the graph below; g(x) is the blue curve and -g(x) is the orange curve.
The Rule for Working Out a Point
The rule for working out the coordinates of a point after it has been reflected in the x axis [e.g. -f(x)] is to multiply the y value by -1 (you change the sign of the y value). The x value remains the same.
The rule for working out the coordinates of a point after it has been reflected in the x axis [e.g. -f(x)] is to multiply the y value by -1 (you change the sign of the y value). The x value remains the same.
