Math / Geometry

Question11. a) Determine the equation of the following conic. b) If this circle is translated 9 units to the right and 5 units up, what is the equation of this conic? c) Horizontally stretch the circle in (b) by a factor of 2 , what is the equation of this conic? $(h, k)=\left(\frac{-6}{+9}, \frac{-2}{+5}\right) \rightarrow(3,3)$

Studdy Solution

STEP 1

What is this asking? We've got a circle chilling at the origin, and we're gonna mess with it by moving it around and stretching it, then we need to find its equation after each transformation! Watch out! Remember that horizontal stretches affect the x-coordinate, and translations shift the center of the circle.
Don't mix them up!

STEP 2

1. Find the original equation
2. Find the translated equation
3. Find the stretched equation

STEP 3

Alright, let's kick things off by looking at the graph!
We see a circle centered at $(0,0)$ , which is the origin.
That's awesome because it makes our starting equation simpler.

STEP 4

The radius looks like it's 3.
The circle goes out to $(3,0)$ , $(-3,0)$ , $(0,3)$ , and $(0,-3)$ , confirming our suspicion!

STEP 5

The general equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$ , where $(h,k)$ is the center and $r$ is the radius.
Since our center is $(0,0)$ and our radius is $3$ , our initial equation is $x^2 + y^2 = 3^2$ , which simplifies to $x^2 + y^2 = 9$ .
Boom!

STEP 6

Now, we're sliding this circle 9 units to the right and 5 units up.
This means our center is changing.

STEP 7

Moving right means we add to the x-coordinate of the center, and moving up means we add to the y-coordinate.
So, our new center is $(0 + 9, 0 + 5) = (9, 5)$ .

STEP 8

The radius doesn't change during a translation, so it's still 3.
Our translated equation is $(x-9)^2 + (y-5)^2 = 3^2$ , which simplifies to $(x-9)^2 + (y-5)^2 = 9$ .
Nice!

STEP 9

Time to stretch!
We're horizontally stretching the circle from step 2.2 by a factor of 2.
This only affects the x part of the equation.

STEP 10

A horizontal stretch by a factor of $2$ means we replace $x$ with $\frac{x}{2}$ in the equation from the previous step.
So, our equation becomes $\left(\frac{x}{2} - 9\right)^2 + (y-5)^2 = 9$ .

STEP 11

We can leave it like that, or we can rewrite it as $\left(\frac{x-18}{2}\right)^2 + (y-5)^2 = 9$ .
Fantastic!

STEP 12

a) The original equation is $x^2 + y^2 = 9$ . b) The translated equation is $(x-9)^2 + (y-5)^2 = 9$ . c) The stretched equation is $\left(\frac{x-18}{2}\right)^2 + (y-5)^2 = 9$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the original equation
2. Find the translated equation
3. Find the stretched equation

STEP 3

Alright, let's kick things off by looking at the graph!
We see a circle centered at $(0,0)$ , which is the origin.
That's awesome because it makes our starting equation simpler.

STEP 4

The radius looks like it's 3.
The circle goes out to $(3,0)$ , $(-3,0)$ , $(0,3)$ , and $(0,-3)$ , confirming our suspicion!

STEP 5

The general equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$ , where $(h,k)$ is the center and $r$ is the radius.
Since our center is $(0,0)$ and our radius is $3$ , our initial equation is $x^2 + y^2 = 3^2$ , which simplifies to $x^2 + y^2 = 9$ .
Boom!

STEP 6

Now, we're sliding this circle 9 units to the right and 5 units up.
This means our center is changing.

STEP 7

Moving right means we add to the x-coordinate of the center, and moving up means we add to the y-coordinate.
So, our new center is $(0 + 9, 0 + 5) = (9, 5)$ .

STEP 8

The radius doesn't change during a translation, so it's still 3.
Our translated equation is $(x-9)^2 + (y-5)^2 = 3^2$ , which simplifies to $(x-9)^2 + (y-5)^2 = 9$ .
Nice!

STEP 9

Time to stretch!
We're horizontally stretching the circle from step 2.2 by a factor of 2.
This only affects the x part of the equation.

STEP 10

A horizontal stretch by a factor of $2$ means we replace $x$ with $\frac{x}{2}$ in the equation from the previous step.
So, our equation becomes $\left(\frac{x}{2} - 9\right)^2 + (y-5)^2 = 9$ .

STEP 11

We can leave it like that, or we can rewrite it as $\left(\frac{x-18}{2}\right)^2 + (y-5)^2 = 9$ .
Fantastic!

STEP 12

a) The original equation is $x^2 + y^2 = 9$ . b) The translated equation is $(x-9)^2 + (y-5)^2 = 9$ . c) The stretched equation is $\left(\frac{x-18}{2}\right)^2 + (y-5)^2 = 9$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the original equation2. Find the translated equation3. Find the stretched equation

STEP 3

Alright, let's **kick things off** by looking at the graph!We see a circle centered at (0,0) (0,0) (0,0), which is the **origin**.That's awesome because it makes our starting equation simpler.

STEP 4

The **radius** looks like it's **3**.The circle goes out to (3,0)(3,0)(3,0), (−3,0)(-3,0)(−3,0), (0,3)(0,3)(0,3), and (0,−3)(0,-3)(0,−3), confirming our suspicion!

STEP 5

STEP 6

Now, we're **sliding** this circle **9 units to the right** and **5 units up**.This means our center is changing.

STEP 7

Moving **right** means we *add* to the *x*-coordinate of the center, and moving *up* means we *add* to the *y*-coordinate.So, our **new center** is (0+9,0+5)=(9,5)(0 + 9, 0 + 5) = (9, 5)(0+9,0+5)=(9,5).

STEP 8

The **radius doesn't change** during a translation, so it's still **3**.Our **translated equation** is (x−9)2+(y−5)2=32 (x-9)^2 + (y-5)^2 = 3^2 (x−9)2+(y−5)2=32, which simplifies to (x−9)2+(y−5)2=9 (x-9)^2 + (y-5)^2 = 9 (x−9)2+(y−5)2=9.Nice!

STEP 9

Time to **stretch**!We're horizontally stretching the circle from step 2.2 by a factor of **2**.This only affects the *x* part of the equation.

STEP 10

A horizontal stretch by a factor of 222 means we replace xxx with x2\frac{x}{2}2x​ in the equation from the previous step.So, our equation becomes (x2−9)2+(y−5)2=9 \left(\frac{x}{2} - 9\right)^2 + (y-5)^2 = 9 (2x​−9)2+(y−5)2=9.

STEP 11

We can leave it like that, or we can **rewrite** it as (x−182)2+(y−5)2=9 \left(\frac{x-18}{2}\right)^2 + (y-5)^2 = 9 (2x−18​)2+(y−5)2=9.Fantastic!

STEP 12

a) The original equation is x2+y2=9 x^2 + y^2 = 9 x2+y2=9. b) The translated equation is (x−9)2+(y−5)2=9 (x-9)^2 + (y-5)^2 = 9 (x−9)2+(y−5)2=9. c) The stretched equation is (x−182)2+(y−5)2=9 \left(\frac{x-18}{2}\right)^2 + (y-5)^2 = 9 (2x−18​)2+(y−5)2=9.

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1. Find the original equation
2. Find the translated equation
3. Find the stretched equation

Alright, let's kick things off by looking at the graph!
We see a circle centered at $(0,0)$ , which is the origin.
That's awesome because it makes our starting equation simpler.

The radius looks like it's 3.
The circle goes out to $(3,0)$ , $(-3,0)$ , $(0,3)$ , and $(0,-3)$ , confirming our suspicion!

Now, we're sliding this circle 9 units to the right and 5 units up.
This means our center is changing.

Moving right means we add to the x-coordinate of the center, and moving up means we add to the y-coordinate.
So, our new center is $(0 + 9, 0 + 5) = (9, 5)$ .

The radius doesn't change during a translation, so it's still 3.
Our translated equation is $(x-9)^2 + (y-5)^2 = 3^2$ , which simplifies to $(x-9)^2 + (y-5)^2 = 9$ .
Nice!

Time to stretch!
We're horizontally stretching the circle from step 2.2 by a factor of 2.
This only affects the x part of the equation.

A horizontal stretch by a factor of $2$ means we replace $x$ with $\frac{x}{2}$ in the equation from the previous step.
So, our equation becomes $\left(\frac{x}{2} - 9\right)^2 + (y-5)^2 = 9$ .

We can leave it like that, or we can rewrite it as $\left(\frac{x-18}{2}\right)^2 + (y-5)^2 = 9$ .
Fantastic!

a) The original equation is $x^2 + y^2 = 9$ . b) The translated equation is $(x-9)^2 + (y-5)^2 = 9$ . c) The stretched equation is $\left(\frac{x-18}{2}\right)^2 + (y-5)^2 = 9$ .