Write three different sentences that could describe the relationship between the quantities $27, $81

How to find the vertex calculus 2What is the vertex, focus and directrix of x^2 = 6y

son4ous [18]

Solution:

Given:

$x^2=6y$

Part A:

The vertex of an up-down facing parabola of the form;

$\begin{gathered} y=ax^2+bx+c \\ is \\ x_v=-\frac{b}{2a} \end{gathered}$

Rewriting the equation given;

$\begin{gathered} 6y=x^2 \\ y=\frac{1}{6}x^2 \\ \\ \text{Hence,} \\ a=\frac{1}{6} \\ b=0 \\ c=0 \\ \\ \text{Hence,} \\ x_v=-\frac{b}{2a} \\ x_v=-\frac{0}{2(\frac{1}{6})} \\ x_v=0 \\ \\ _{} \\ \text{Substituting the value of x into y,} \\ y=\frac{1}{6}x^2 \\ y_v=\frac{1}{6}(0^2) \\ y_v=0 \\ \\ \text{Hence, the vertex is;} \\ (x_v,y_v)=(h,k)=(0,0) \end{gathered}$

Therefore, the vertex is (0,0)

Part B:

A parabola is the locus of points such that the distance to a point (the focus) equals the distance to a line (directrix)

Using the standard equation of a parabola;

$\begin{gathered} 4p(y-k)=(x-h)^2 \\ \\ \text{Where;} \\ (h,k)\text{ is the vertex} \\ |p|\text{ is the focal length} \end{gathered}$

Rewriting the equation in standard form,

$\begin{gathered} x^2=6y \\ 6y=x^2 \\ 4(\frac{3}{2})(y-k)=(x-h)^2 \\ \text{putting (h,k)=(0,0)} \\ 4(\frac{3}{2})(y-0)=(x-0)^2 \\ Comparing\text{to the standard form;} \\ p=\frac{3}{2} \end{gathered}$

Since the parabola is symmetric around the y-axis, the focus is a distance p from the center (0,0)

Hence,

$\begin{gathered} Focus\text{ is;} \\ (0,0+p) \\ =(0,0+\frac{3}{2}) \\ =(0,\frac{3}{2}) \end{gathered}$

Therefore, the focus is;

$(0,\frac{3}{2})$

Part C:

A parabola is the locus of points such that the distance to a point (the focus) equals the distance to a line (directrix)

Using the standard equation of a parabola;

$\begin{gathered} 4p(y-k)=(x-h)^2 \\ \\ \text{Where;} \\ (h,k)\text{ is the vertex} \\ |p|\text{ is the focal length} \end{gathered}$

Rewriting the equation in standard form,

$\begin{gathered} x^2=6y \\ 6y=x^2 \\ 4(\frac{3}{2})(y-k)=(x-h)^2 \\ \text{putting (h,k)=(0,0)} \\ 4(\frac{3}{2})(y-0)=(x-0)^2 \\ Comparing\text{to the standard form;} \\ p=\frac{3}{2} \end{gathered}$

Since the parabola is symmetric around the y-axis, the directrix is a line parallel to the x-axis at a distance p from the center (0,0).

Hence,

$\begin{gathered} Directrix\text{ is;} \\ y=0-p \\ y=0-\frac{3}{2} \\ y=-\frac{3}{2} \end{gathered}$

Therefore, the directrix is;

$y=-\frac{3}{2}$

3 0

1 year ago

A cosine function is graphed below. Use the drop-down menus to describe the graph. The amplitude of the graph is __ . The equati

Tresset [83]

Amplitude:4

Equation of Midline: 2

Period of function:3

Function shifted left:0.5

Function shifted up: 2

4 0

3 years ago

Read 2 more answers

17,358 divided blank equals 789

Otrada [13]

1= 22.08
2= 25704
3= 587
4= 610

6 0

3 years ago

Use the information given in the diagrams to show that

Vlad1618 [11]

Answer:

d = k·sin(2θ)·sin(α)/(sin(θ)·sin(β))

Step-by-step explanation:

The Law of Sines tells us that sides of a triangle are proportional to the sine of the opposite angle. This can be used along with a trig identity to demonstrate the required relation.

__

<h3>top triangle</h3>

The law of sines applied to the top triangle is ...

BC/sin(A) = AC/sin(θ)

Triangle ABC is isosceles, so the base angles at B and C are congruent. Then the angle at vertex A is ...

∠A = 180° -θ -θ = 180° -2θ

A trig identity tells us the sine of an angle is equal to the sine of its supplement. That means the sine of angle A is ...

sin(A) = sin(180° -2θ) = sin(2θ)

and our above Law of Sines equation tells us ...

BC = sin(A)/sin(θ)·AC = k·sin(2θ)/sin(θ)

__

<h3>bottom triangle</h3>

The law of sines applied to the bottom triangle is ...

DC/sin(B) = BC/sin(D)

d/sin(α) = BC/sin(β)

Multiplying by sin(α) we have ...

d = BC·sin(α)/sin(β)

__

Using our expression for BC gives the desired relation:

d = k·sin(2θ)·sin(α)/(sin(θ)·sin(β))

7 0

1 year ago

3 (w+4) + 2=27

pshichka [43]

Answer:

Step-by-step explanation:

3w + 12 + 2w = 27

5w + 12 = 27

5w = 15

w= 3

3 0

3 years ago