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2 years ago

Write the equation of the trigonometric graph.

Mathematics

1 answer:

prisoha [69]2 years ago

6 0

Answer:

$f(x)= \sin(6x)+3$

Step-by-step explanation:

The parent function of this graph is: y = sin(x)

The sine function is periodic, meaning it repeats forever.

Standard form of a sine function:

$f(x) = \sf A \sin (B(x + C)) + D$

A = amplitude (height from the mid-line to the peak)
2π/B = period (horizontal distance between consecutive peaks)
C = phase shift (horizontal shift - positive is to the left)
D = vertical shift

The parent function y = sin(x) has the following:

Amplitude (A) = 1
Period = 2π
Phase shift (C) = 0
Vertical shift (D) = 0
Mid-line: y = 0

From inspection of the given graph:

Amplitude (A) = 1
$\sf Period=\dfrac{\pi}{12}-\dfrac{-\pi}{4}=\dfrac{\pi}{3}$
Phase shift (C) = 0
Vertical shift (D) = +3 (as mid-line is y = 3)

$\sf If\:Period=\dfrac{\pi}{3} \implies \dfrac{2 \pi}{B}=\dfrac{\pi}{3}\implies B=6$

Substituting the values into the standard form:

$\implies f(x) =1 \sin (6(x + 0)) + 3$

$\implies f(x) = \sin (6x) + 3$

Therefore, the equation of the given trigonometric graph is:

$f(x)= \sin(6x)+3$

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Answer: " 2x (2x - 1) (x + 1) " .

______________________________________

Step-by-step explanation:

______________________________________

Given:

f(x) = 9x³ + 2x² − 5x + 4 ;

g(x) = 5x³ − 7x + 4 ;

______________________________________

What is: f(x) − g(x) ?

______________________________________

Plug in: " 9x³ + 2x² − 5x + 4 " for: " f(x) " ;

and: " (5x³ − 7x + 4) " ; for: "g(x)" ;

______________________________________

→ " f(x) − g(x) =

" 9x³ + 2x² − 5x + 4 − (5x³ − 7x + 4) " .

______________________________________

Rewrite this expression as:

→ " 9x³ + 2x² − 5x + 4 − 1(5x³ − 7x + 4) " .

→ {since: " 1 " ; multiplied by "any value" ; is equal to that same value.}.

______________________________________

Now, let us example the following portion of the expression:

______________________________________

" − 1(5x³ − 7x + 4) "

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Note the "distributive property" of multiplication:

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→ a(b + c) = ab + ac ;

______________________________________

Likewise:

→ a(b + c + d) = ab + ac + ad .

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As such:

______________________________________

→ " − 1(5x³ − 7x + 4) " ;

______________________________________

= (-1 * 5x³) + (-1 * 7x) + (-1 * 4) ;

= - 5x³ + (-7x) + (-4) ;

= - 5x³ − 7x − 4 ;

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Now, add the "beginning portion of the expression" ; that is:

" f(x) " ; to the expression ; which is:

→ 9x³ + 2x² − 5x + 4 ;

→ as follows:

_______________________________________

→ 9x³ + 2x² − 5x + 4 − 5x³ − 7x − 4 ;

→ {Note that the: " - " sign; that is;

the "negative sign", in the term: " -5x³ " ;

becomes a: " − " sign; that is; a "minus sign" .}.

______________________________________

Now, combine the "like terms" of this expression; as follows:

+ 9x³ − 5x³ = + 4x³ ;

− 5x − 7x = − 2x ;

+ 4 − 4 = 0 ;

______________________________________

and we have:

______________________________________

→ " 4x³ + 2x² − 2x ".

______________________________________

Now, to write this answer in "factored form" :

Note that among all 3 (three) terms in this expression, each term has a factor of "2" . The lowest coefficient among these 3 (three) terms is "2" ; so we can "factor out" a "2".

Also, each of the 3 (three) terms in this fraction is a coefficient to a variable. That variable takes the form of "x". The term in this expression with the variable, "x"; with the lowest degree has the variable: "x" (i.e. "x¹ = x" ) ; so we can "factor out a "2x" (rather than just the number, "2".).

So, by factoring out a "2x" ; take the first term [among the 3 (three) terms in the expression] —which is: "4x³ " .

2x * (?) = 4x³ ? ;'

↔ $\frac{4x^3}{2x} =$ ? ;

→ 4/2 = 2 ;

$\frac{x^{3}}{x} = \frac{x^3}{x^1} = x^{(3-1)} = x^{2}$ ;

As such: 2x * (2x²) = 4x³ ;

___________________________________________

Now, by factoring out a "2x" ; take the second term [among the 3 (three) terms in the expression] — which is: "2x² " .

2x * (?) = 2x² ? ;

↔ $\frac{2x^{2}}{2x} =$ ?

→ 2/2 = 1 ;

→ $\frac{x^{2}}{x} = \frac{x^2}{x^1}= x^{(2-1)} } = x^1 = x$ ;

As such: 2x * (x) = 2x²

__________________________________________

Now, by factoring out a "2x" ; take the third term [among the 3 (three) terms in the expression] — which is: " − 2x " .

2x * (?) = - 2x ;

↔ $\frac{-2x}{2x} =$ -1 ;

As such: 2x * (-1) = − 2x .

__________________________________________

So:

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Given the simplified expression:

→ " 4x³ + 2x² − 2x " ;

We can "factor out' a: " 2x " ; and write the this answer is: "factored form" ; as:

__________________________________________

"2x (2x² + x − 1 ) . "

Now, we can further factor the:

" (2x² + x − 1) " ; portion;

Note: "(2x² + x - 1)" =

2x² + 2x - 1x -1 = (2x -1) + x (2x - 1 ) =

(2x - 1) ( x + 1)

_______________________________________

Now, bring down the "2x" ; and write the Full "factored form" ; as follows:

_______________________________________

→ " 2x (2x - 1) (x + 1) " .

_______________________________________

Hope this helps!

Wishing you the best!

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