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3 years ago

The graph of a sinusoidal function has a minimum point at (0,3)(0,3) and then intersects its midline at (5π,5)

Mathematics

1 answer:

bekas [8.4K]3 years ago

8 0

Answer: F(x) = 2*sin(x/10 +(3/2)*pi) + 5

Step-by-step explanation:

The information that we have is that:

We have a minimum at (0, 3)

the midline is at (5*pi, 5)

This is a sinusoidal function, so we can write one generic one as:

F(x) = A*sin(c*x + p) + B.

where A and B are constants, c is the frequency and p is a phase

First, the minimum of the sine function is when sin(x) = -1, and this happens at (3/2)*pi

We know that this minimum is at x = 0.

sin(c*0 + p) = -1

Then p = 3/2*pi.

So our function is:

F(x) = A*sin(c*x +(3/2)*pi) + B.

Now, we know that F(0) = 3, so:

3 = A*sin(c*0 +(3/2)*pi) + B = -A + B.

now we can use the other hint, the midpoint of the sine function is when sin(x) = 0, and this happens at x = 0 and x = pi, particularlly as we here have a phase of 3/2*pi, we should find x = 2*pi.

then:

c*5*pi + (3/2)*pi = 2*pi

c*5 + 3/2 = 2

c*5 = 2 - 3/2 = 1/2

C = 1/2*5 = 1/10

So our function is

F(x) = A*sin(x/10 +(3/2)*pi) + B

and we know that when x = 5*pi, F(5*pi) = 5, so:

5 = F(x) = A*sin(5*pi/10 +(3/2)*pi) + B

5 = B

and we aready knew that:

- A + B = 3

-A + 5 = 3

A = 5 - 3 = 2

So our equation is:

F(x) = 2*sin(x/10 +(3/2)*pi) + 5

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Step-by-step explanation:

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helpful links:

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3 years ago

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Step-by-step explanation:

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3 years ago

Solve x2+4x+6=0 by graphing

klemol [59]

Answer: x = -2±√2 i or no real roots. See the explanation below.

Step-by-step explanation:

If you draw the graph, you will notice that there are no common points or you can say that there are no x-intercepts.

This means that the equation doesn't have any real roots.

If there are 2 common points (2 x-intercepts) then there are 2 real roots (answers.)

If there is only one common point (vertex is on x-axis) then there is only 1 real root.

If there aren't any common points (no x-intercepts) then there are no real roots.

So first, I'm gonna convert the equation into function and convert the standard form to vertex form.

$x^2+4x+6=0\\y=x^2+4x+6\\y=(x^2+4x+4)-4+6\\y=(x+2)^2+2$

We notice that the vertex is at (-2,2), the graph doesn't have x-intercepts (Graph below)

So the answer is clearly complex.

If you solve this by solving the equation then it'll be.

$x^2+4x+6=0\\x= \frac{-b+-\sqrt{b^2-4ac}}{2a} \\x=\frac{-4+-\sqrt{16-24} }{2}\\x=\frac{-4+-\sqrt{-8} }{2}\\\\x=\frac{-4+-2\sqrt{2}i }{2}\\\\x={-2+-1\sqrt{2}i }\\\\$

The answer is complex.

However, you can find the value of x easily by graphing. For example.

$x^2+6x+5=0$

Factor then you get

$(x+5)(x+1)=0\\x=-5,-1$

Or you can convert it to vertex form then graph.

$y=x^2+6x+5$

$y=(x^2+6x+9)-9+5\\y=(x+3)^2-4$

The vertex is at (-3,-4) then graph parabola (x-intercepts determine the roots of quadratic equation. For the graph, parabola intercepts x at (-5,0) and (-1,0) that means both -5 and -1 both are the value of x.

Note:

From the standard form, $y=ax^2+bx+c$ The graph is called Parabola, and it's Quadratic Function.

a is how the graph is, for example. If a>0, the graph is supine parabola and if a<0 then the graph is inverted parabola.

|a| is determined if the graph is wide or narrow, the more the value of a is, the more narrow it will be. The less the value of a is, the more wide it will be.

b is what determined the change of graph (likely vertex). Without b, then it'd be $ax^2+k$ which only move y-axis and not x-axis.

c is what determined the y-intercept and y-axis. If there is no b or the value of b is 0 then the vertex is at (0,c). If there is the value of b then c is only determined the y-intercept.

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3 years ago

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Step-by-step explanation:

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2 years ago

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Solving for matrices

muminat

Answer:

Step-by-step explanation:

The augmented matrix for the system of three equaitons is

$\left(\begin{array}{ccccc} 3&-4&-5&|&-27\\5&2&-2&|&11\\5&-4&4&|&-7\end{array}\right)$

Multiply the first row by 5, the second row by -3 and add these two rows:

$\left(\begin{array}{ccccc} 3&-4&-5&|&-27\\0&-26&-19&|&-168\\5&-4&4&|&-7\end{array}\right)$

Subtract the third row from the second:

$\left(\begin{array}{ccccc} 3&-4&-5&|&-27\\0&-26&-19&|&-168\\0&6&-6&|&18\end{array}\right)$

Divide the third row by 6:

$\left(\begin{array}{ccccc} 3&-4&-5&|&-27\\0&-26&-19&|&-168\\0&1&-1&|&3\end{array}\right)$

Now multiply the third equation by 26 and add it to the second row:

$\left(\begin{array}{ccccc} 3&-4&-5&|&-27\\0&-26&-19&|&-168\\0&0&-45&|&-90\end{array}\right)$

You get the system of three equations:

$\left\{\begin{array}{r}3x-4y-5z=-27\\-26y-19z=-168\\-45z=-90\end{array}\right.$

From the third equation

$z=\dfrac{90}{45}=2.$

Substitute z=2 into the second equation:

$-26y-19\cdot 2=-168\\ \\-26y-38=-168\\ \\-26y=-168+38=-130\\ \\y=\dfrac{130}{26}=5.$

Now substitute z=2 and y=5 into the first equation:

$3x-4\cdot 5-5\cdot 2=-27\\ \\3x-20-10=-27\\ \\3x-30=-27\\ \\3x=-27+30=3\\ \\x=1.$

The solution is (1,5,2)

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3 years ago