In this applet we will explore how to change an equation to shift its graph up or down or left or to the right or left. This can help us spotting familiar graphs in new positions. This will also help us graph unfamiliar equations more quickly.
To shift the graph of a function y=f(x) straight upwards, we add a constant to the right hand side e.g. by adding 2 to the right hand side of the formula y = x2 we get y = x2+2 and the graph is shifted up by 2 units.
To shift the graph of equation y = f(x) straight down, we subtract we add a negative constant to the right hand side of the formula y = f(x)
To shift the graph of y = f(x) left or right , we add positive or negative constant to x respectively e.g. if by adding 2 to x in y = x2 to get y = (x+2) 2 , shifts the graph 2 units to the left.
In general we can say
- y = f(x) + b , shifts the graph up by b units if b > 1 or shifts it down by |b| units if b <0
- y = f(x-a) , shifts the graph right a units if a > 0 or shifts it left |a| units if a<0.
Right Click any where on the graph and select ‘Trace On’ , then drag sliders ‘a’ or ‘b’.
Every function can be classified as an even function, an odd function, or neither. Even functions have the characteristic that f (x) = f (-x). They are symmetrical with respect to the y-axis. A line segment joining the points f (x) and f (-x) will be perfectly horizontal , shown by dotted blue line in the applet below.
Odd functions have the characteristic that f (x) = - f (-x). They are symmetrical with respect to the origin. A line segment joining the points f (x) and f (-x) always contains the origin , shown by dotted red line in the applet below.
Some of the most common even functions are y = k , where k is a constant, y= x2 , and y = cos(x) . Some of the most common odd functions are y = x3 and y = sin(x) . Some functions that are neither even nor odd include y = x - 4 , y = cos(x) + 1.
Formal tests for symmetry:
1. y – axis : replace x with –x , produces an equivalent equation
2. x - axis : replace y with – y , produces an equivalent equation
3. origin : replace x with –x and y with –y , produces equivalent equation.
In the following applet use check box to select between odd or even functions. Also you can enter any function in the input box to see whether it is odd or even by checking the symmetry about y-axis or origin.
In mathematics, a function is a relation between a set of inputs and a set of potential outputs with the property that each input is related to exactly one output. An example of such a relation is defined by the rule f(x) = x2, which relates an input x to its square, which are both real numbers. The output of the function f corresponding to an input x is denoted by f(x) (read "f of x"). If the input is –3, then the output is 9, and we may write f(–3) = 9.
Thus we have two quantities (called "variables") and we observe that there is a relationship between them. If we find that for every value of the first variable there is only one value of the second variable, then we say: "The second variable is a function of the first variable."
The first variable is the independent variable (usually written as x), and the second variable is the dependent variable (usually written as y).
The domain of a function is the set of all possible x-values which will make the function work and will output real y-values.
The range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function.
The following applet shows the graph of sine function. Its domain ( which can be tracked by point B) and range(which can be tracked by point C) are respectively (-∞ , ∞) and [-1,1].
