Given that f(x)=x^2-3x+3 and g(x)=x-1/4 slove for f(g(x)) when x=5

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

In the transformation ABC A'B'C', the preimage of B' is ____. . a0

Misha Larkins [42]

Answer:

Pre image of B' is B

Step-by-step explanation:

Given:

ABC is a triangle

A transformation is done on ABC so that the image is A'B'C'.

Note that transformations are of various types such as dilation, vertical shift, horizontal shift, rotation about a point, reflection on a line, etc.

In any type of transformation, corresponding vertices will be matched. In other words, A will become A', B will become B' and C will become C'.

Because of the property of the transformation to keep images similar and also transforming correspondingly the vertices we get preimage of B' would be nothing but B itself.

8 0

3 years ago

A store sells new video games for $55 each. Used video games sell for $12 each. Paul is buying 3 new video games and x

harina [27]

answer:

y = 165+12x should be the answer

4 0

3 years ago

Plz help! The table shows the overnight low temperatures for a four-day period. The low temperature on Wednesday was between th

daser333 [38]

We have been given a table of overnight low temperature for a four-day period. We are supposed to give a possible temperature for Wednesday.

We are told that the low temperature on Wednesday was between the second and third highest temperature.

Let us place our given temperatures from least to greatest order.

Sunday (-11), Thursday (-8), Friday (7), Sunday (18)

We can see that second and third highest temperatures are -8 and 7 respectively.

Any number between -8 and 7 will be a possible temperature for Wednesday. 3 can be a possible temperature for Wednesday as 3 is between -8 and 7.

7 0

3 years ago

What is the value of y when x=3

notsponge [240]

The value of Y will equal Zero

7 0

3 years ago