Finding equations of sine and cosine curves

To form an equation of a (simple) sine or cosine curve from a graph or from given information, we need to identify the key features of the cyclic function.  We can begin by writing the general form of the equations. Namely:

$f\left(x\right)=a\sin\left(bx-c\right)+d$f(x)=asin(bx−c)+d or $f\left(x\right)=a\cos\left(bx-c\right)+d$f(x)=acos(bx−c)+d

From this point, we need to determine the values of the constants $a$a, $b$b, $c$c and $d$d using the information at hand. Recall from previous chapters that:

• The vertical translation is the value of $d$d in the general equations. It is the central value of the function and it allows us to construct an imaginary line that the function appears to oscillate around.
• The amplitude is the maximum deviation of the function from the central line. This is the value of $a$a in the general form of the equation. 
• The period is the length of one complete cycle. The value of the period is equal to $\frac{360^\circ}{b}$360°b​. If we know the period, we can deduce the value of $b$b.
• The phase shift is the horizontal translation of the graph compared with an unmodified sine or cosine graph. The phase shift is equal to $\frac{c}{b}$cb​. If the shift is to the right, then $c$c is negative.
• If the sine curve starts at the central line but decreases, or the cosine curve starts at the minimum and increases, this can be interpreted as a reflection.  If there is a reflection, then the value of $a$a has to be negative.  (A reflection is equivalent to a phase shift of a suitable size and either approach will lead to a valid formula.)

 

Worked Example 1

Find the equation of the cosine curve that has undergone the following transformations:

a vertical translation of $4$4 units upwards
a vertical dilation by a multiple of $2$2
a horizontal translation resulting in a phase shift of $22\frac{1}{2}^\circ$2212​° to the left
a horizontal dilation by a multiple of $\frac{1}{2}$12​

The general form of the cosine equation is $f\left(x\right)=a\cos\left(bx-c\right)+d$f(x)=acos(bx−c)+d, for which we need values of $a$a, $b$b, $c$c and $d$d.  

We are told that the vertical translation is $4$4 units up. So, $d=4$d=4
We are told that there is a vertical dilation by the multiple $2$2. This means the amplitude is $a=2$a=2.

Given that there is a horizontal dilation by the multiple $\frac{1}{2}$12​, the period is adjusted from being $360^\circ$360° to half of that amount, which is $180^\circ$180°. Since $\frac{360^\circ}{b}=180^\circ$360°b​=180°, we have $b=2$b=2.

There is a phase shift of $22\frac{1}{2}^\circ$2212​° to the left. This is $\frac{c}{b}$cb​. (Remember that shifting left is positive.) So, $\frac{c}{2}=22\frac{1}{2}^\circ$c2​=2212​°  and therefore $c=45^\circ$c=45°.

There is no reflection. So, the value of $a$a will be positive.

Putting all this together into the general form we determine the equation:

$f\left(x\right)=2\cos\left(2x+45^\circ\right)+4$f(x)=2cos(2x+45°)+4. 

 

Worked Example 2

Find the equation of the curve in the graph below. 

We mark in the visible details.

The maximum is at $y=0.1$y=0.1 and the minimum is at $y=-1.1$y=−1.1. The average of these locates the centre line, $y=-0.5$y=−0.5. The amplitude is $0.1-(-0.5)=0.6$0.1−(−0.5)=0.6.

Crests of the waveform occur at $-180^\circ$−180° and $540^\circ$540° so that the period is $540-(-180)=720^\circ$540−(−180)=720°. This is twice the standard period of $360^\circ$360°. Therefore, the period constant is $\frac{1}{2}$12​.

The curve looks like a cosine curve shifted $180^\circ$180° to the left. It could also be interpreted as a sine curve reflected in the $x$x-axis and this would make the amplitude constant negative with no phase shift.

Considering this as a sine function with no phase shift and inserting the constant values into $y(x)=a\sin\left(bx-c\right)+d$y(x)=asin(bx−c)+d, we have

$y(x)=-0.6\sin\frac{x}{2}-0.5$y(x)=−0.6sinx2​−0.5

If we think of it as a sine function without a reflection but with a shift of $360^\circ$360° to the right, we would write the formula as

$y(x)=0.6\sin\left(\frac{x}{2}-180^\circ\right)-0.5$y(x)=0.6sin(x2​−180°)−0.5

Or, if we think of it as a cosine curve shifted to the left by $180^\circ$180°, we could write the formula as

$y(x)=0.6\cos\left(\frac{x}{2}+90^\circ\right)-0.5$y(x)=0.6cos(x2​+90°)−0.5

You should verify that these forms are equivalent.

 

Practice Questions

Question 1

Question 2

Question 3

 

 

Adding in the phase shift

What are the different ways that we can transform the graph of the sine or cosine function? Recall that a transformation can be described algebraically using the variables $a$a, $b$b, $c$c, and $d$d, where

$\sin\left(x\right)\text{ }\rightarrow\text{ }a\sin\left(bx-c\right)+d$sin(x) → asin(bx−c)+d

We read this as "$\sin\left(x\right)$sin(x) transforms into $a\sin\left(bx-c\right)+d$asin(bx−c)+d". In this lesson we will focus on the role of the variable $c$c in horizontal translations or phase shifts and how it works in conjunction with other transformations.

Worked example 3

The graph of $y=\cos x$y=cosx has been shifted $30^\circ$30° to the right. State the equation of the new graph.

Think: A shift to the right is a horizontal translation, so we know the new graph will have an equation of the form $y=\cos\left(x-c\right)$y=cos(x−c). We need to determine the magnitude and sign of the variable $c$c.

Do: The graph of $y=\cos x$y=cosx crosses the $y$y-axis at $\left(0,1\right)$(0,1). Since the whole graph is shifted by $30^\circ$30° to the right, this point will move to $\left(30^\circ,1\right)$(30°,1). We can substitute these coordinates into the form of the new equation to obtain the value of $c$c.

$y$y	$=$=	$\cos\left(x-c\right)$cos(x−c)
$1$1	$=$=	$\cos\left(30^\circ-c\right)$cos(30°−c)	(Substituting the point $\left(30^\circ,1\right)$(30°,1))
$0$0	$=$=	$30^\circ-c$30°−c	(Using the fact that $\cos\theta=1$cosθ=1 when $\theta=0$θ=0)
$c$c	$=$=	$30^\circ$30°	(Adding $c$c to both sides of the equation)

In the final line we see that $c=30^\circ$c=30°, so that the equation of the new graph is $y=\cos\left(x-30^\circ\right)$y=cos(x−30°).

Reflect: If the horizontal translation had been to the left, we would have found that $c=-30^\circ$c=−30°, so that the new graph would have the equation $y=\cos\left(x-\left(-30^{\circ}\right)\right)=\cos\left(x+30^{\circ}\right)$y=cos(x−(−30∘))=cos(x+30∘).

Worked example 4

Consider the graph of the form $f\left(x\right)=\sin\left(b\left(x+c\right)\right)$f(x)=sin(b(x+c)) shown below.

State the equation of the graph. Give your answer in the form where $c$c is the least positive value.

Think: Compared to the graph of $y=\sin x$y=sinx, the graph of $f\left(x\right)$f(x) has been "squashed" horizontally due to the period change caused by $b$b, and translated horizontally due to the phase shift caused by $c$c.

Do: To determine the period change, recall that $y=\sin x$y=sinx has a period of $360^\circ$360°, so that one whole cycle covers $360^\circ$360° along the $x$x-axis. This means that one whole cycle of $\sin bx$sinbx will cover $\frac{360^\circ}{b}$360°b​ along the $x$x-axis. From the graph we can see that the period of $f\left(x\right)$f(x) is $120^\circ$120°, so $b=\frac{360}{120}=3$b=360120​=3.

Next, we see that the point $\left(0^\circ,0\right)$(0°,0) on the graph of $y=\sin x$y=sinx has been translated to the point $\left(-45^\circ,0\right)$(−45°,0) on the graph of $f\left(x\right)$f(x). This corresponds to a phase shift of $\left(-45\right)^\circ$(−45)°, which means the value of the variable $c$c is $-45^\circ$−45°.

Now we can combine the working above and state the equation of the graph as $f\left(x\right)=\sin\left(3\left(x+45^\circ\right)\right)$f(x)=sin(3(x+45°)).

Reflect: How would the value of $b$b and $c$c change if we were told that the equation of the graph had the form $\sin\left(bx+c\right)$sin(bx+c) instead of $\sin\left(b\left(x+c\right)\right)$sin(b(x+c))?

Practice questions

Question 4

Question 5

Question 6