 Periodic functions > Trigonometric graphs and equations
123456Trigonometric graphs and equations

## Theory

Take a look at the applet

The graph of the function $g\left(x\right)=a\cdot sin\left(b\left(x+c\right)\right)+d$ is a sinusoid. That is a graph that results when you transform the graph of a standard sine function $f\left(x\right)=sin\left(x\right)$ . The graph of $g$ has the following properties:

• the amplitude (maximum deviation from the equilibrium) is $a$ ;

• the period is $\frac{2\pi }{b}$ so $b=\frac{2\pi }{\text{period}}$ ;

• the horizontal translation is $-c$ ;

• the equilibrium is the line $y=d$ .

Take a look at the following applet:

The graph of the function $g\left(x\right)=a\cdot cos\left(b\left(x+c\right)\right)+d$ is a sinusoid. That is a graph that results when you transform the graph of a standard cosine function $f\left(x\right)=cos\left(x\right)$ . The graph of $g$ has the following properties :

• the amplitude ( maximum deviation from the equilibrium ) is $a$ ;

• the period is $\frac{2\pi }{b}$ so $b=\frac{2\pi }{\text{period}}$ ;

• the horizontal translation is $-c$ ;

• the equilibrium is the line $y=d$ .