Question

Which of the following could be the equation of the function below? on a coordinate plane, a curve crosses the y-axis at y = negative 3. it has a maximum at y = 1 and a minimum at y = negative 3. it goes through 1 cycle at pi. y = negative 2 cosine (4 (x pi)) minus 1 y = 2 cosine (x pi) 1 y = negative 2 cosine (2 (x pi)) minus 1 y = 2 cosine (4 (x pi)) 2

Answer 1

The equation of the function which is graphed on the right is specified by: Option C: y = -2cos(2(x + π)) -1

How to find the function which was used to make graph?

There are many tools we can use to find the information of the relation which was used to form the graph.

A graph contains data of which input maps to which output.

Analysis of this leads to the relations which were used to make it.

For example, if the graph of a function is rising upwards after a certain value of x, then the function must be having increasingly output for inputs greater than that value of x.

If we know that the function crosses x axis at some point, then for some polynomial functions, we have those as roots of the polynomial.

What are some properties of a cosine function?

Suppose that we've got:

$f(x) = a \cos (bx - c) + d$

Then, this function has:

• Period = $\dfrac{2\pi}{b}$
• Phase shift = $\dfrac{c}{b}$
• Maximum value: a + d
• Minimum value: -a + d

For this case, we're given that:

• The function crosses y-axis at  y = -3
• The maximum value of the function is y = 1
• The minimum value of the funciton is y = -3
• It goes through 1 cycle at pi.

All the options are of cosine function, so let the function be:

$f(x) = a \cos (bx - c) + d$

Then, as per the given data about max and min. values, we get:

d+a=1

d-a = -3

Adding both equations,

2d = -2 => d = -1

Thus, -1+a = 1, and therefore a = 2

Since it goes through 1 cycle at pi, so its period = π

Thus, we get:

$Period = \pi = 2\pi/b\\\\or\\\\b = 2$

Now since there is only third option which has d= -1, and b =2 so assuming at least one option was true, we get the needed function as described by the option C.

Thus, the equation of the function which is graphed on the right is specified by: Option C: y = -2cos(2(x + π)) -1

Learn more about amplitude and period of a cosine function here:

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