1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Pie
1 year ago
13

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

Mathematics
1 answer:
son4ous [18]1 year ago
3 0

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}

You might be interested in
SOMEONE PLS HELP ME WITH THIS: Jack Thomas has an fudge sundae with 450 calories. How long would it take him to burn off the cal
KATRIN_1 [288]

Answer:

It would take him 450 minutes to burn off the calories by sleeping

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Determine the intercepts of the line
abruzzese [7]
−
4
x
+
7
=
2
Step 1: Subtract 7 from both sides.
−
4
x
+
7
−
7
=
2
−
7
−
4
x
=
−
5
Step 2: Divide both sides by -4.
−
4
x
−
4
=
−
5
−
4
x
=
5
4
6 0
3 years ago
Y > 2 A: [2,[infinity]] B: (2,[infinity]] C: (2,[infinity]) D: [2,[infinity])
Ulleksa [173]

Answer:

B: (2,[infinity]]

Step-by-step explanation:

because you will take all numbers bigger than 2 but 2 is not included

7 0
3 years ago
How do you do this prolem the correct way and how do you do the floors
creativ13 [48]
There is nothing there to work with
7 0
3 years ago
Type you answer in
Bas_tet [7]

Answer:

Step-by-step explanation:

4 0
3 years ago
Other questions:
  • If f(x)= -3x+3, then f^-1(x)=
    13·1 answer
  • What is the equation for cx+4y=8t when your solving for x
    8·1 answer
  • 1)A system of equations is shown below:
    11·2 answers
  • Volume formula of sphere
    9·2 answers
  • Solve for X if MN =81<br> A)9<br> B)10<br> C)11<br> D)12
    11·1 answer
  • A seamstress used 4 meters of ribbon to make 20 bows. She used __ of a meter for each bow.
    11·1 answer
  • I really don’t get this math how am i supposed to figure out which side is which?
    12·2 answers
  • I really need help with this one! please help me!
    6·1 answer
  • What is 100s x 2546?
    13·1 answer
  • Solve for x where xe(0,2pi). state exact answers.
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!