Solved on Dec 05, 2023

Find the equation of a quadratic relation transformed from $y=(x+6)^{2}+3$ with a vertical compression by $\frac{1}{2}$ , reflection along the $x$ -axis, and translations 1 unit down and 3 units right.

STEP 1

Assumptions
1. The original equation of the quadratic relation is $y = (x + 6)^2 + 3$ .
2. The transformations applied to the quadratic relation are: - Vertical compression by a factor of one half. - Reflection along the x-axis. - Translation 1 unit down. - Translation 3 units to the right.

STEP 2

First, we will apply the vertical compression by a factor of one half to the original equation. This means we will multiply the function by $\frac{1}{2}$ .
$y = \frac{1}{2}(x + 6)^2 + 3$

STEP 3

Next, we apply the reflection along the x-axis. This is achieved by multiplying the function by -1.
$y = -\frac{1}{2}(x + 6)^2 + 3$

STEP 4

Now, we will apply the translation 1 unit down. This is done by subtracting 1 from the function.
$y = -\frac{1}{2}(x + 6)^2 + 3 - 1$

STEP 5

Simplify the equation by combining like terms.
$y = -\frac{1}{2}(x + 6)^2 + 2$

STEP 6

Finally, we apply the translation 3 units to the right. This is done by replacing $x$ with $x - 3$ in the equation.
$y = -\frac{1}{2}(x - 3 + 6)^2 + 2$

STEP 7

Simplify the equation by combining the terms inside the parentheses.
$y = -\frac{1}{2}(x + 3)^2 + 2$
This is the equation of the quadratic relation after applying all the given transformations.
The final equation is: $y = -\frac{1}{2}(x + 3)^2 + 2$

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Find the equation of a quadratic relation transformed from $y=(x+6)^{2}+3$ with a vertical compression by $\frac{1}{2}$ , reflection along the $x$ -axis, and translations 1 unit down and 3 units right.

STEP 1

Assumptions
1. The original equation of the quadratic relation is $y = (x + 6)^2 + 3$ .
2. The transformations applied to the quadratic relation are: - Vertical compression by a factor of one half. - Reflection along the x-axis. - Translation 1 unit down. - Translation 3 units to the right.

STEP 2

First, we will apply the vertical compression by a factor of one half to the original equation. This means we will multiply the function by $\frac{1}{2}$ .
$y = \frac{1}{2}(x + 6)^2 + 3$

STEP 3

Next, we apply the reflection along the x-axis. This is achieved by multiplying the function by -1.
$y = -\frac{1}{2}(x + 6)^2 + 3$

STEP 4

Now, we will apply the translation 1 unit down. This is done by subtracting 1 from the function.
$y = -\frac{1}{2}(x + 6)^2 + 3 - 1$

STEP 5

Simplify the equation by combining like terms.
$y = -\frac{1}{2}(x + 6)^2 + 2$

STEP 6

Finally, we apply the translation 3 units to the right. This is done by replacing $x$ with $x - 3$ in the equation.
$y = -\frac{1}{2}(x - 3 + 6)^2 + 2$

STEP 7

Simplify the equation by combining the terms inside the parentheses.
$y = -\frac{1}{2}(x + 3)^2 + 2$
This is the equation of the quadratic relation after applying all the given transformations.
The final equation is: $y = -\frac{1}{2}(x + 3)^2 + 2$

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Find the equation of a quadratic relation transformed from y=(x+6)2+3y=(x+6)^{2}+3y=(x+6)2+3 with a vertical compression by 12\frac{1}{2}21​, reflection along the xxx-axis, and translations 1 unit down and 3 units right.

STEP 1

STEP 2

First, we will apply the vertical compression by a factor of one half to the original equation. This means we will multiply the function by 12 \frac{1}{2} 21​.y=12(x+6)2+3 y = \frac{1}{2}(x + 6)^2 + 3 y=21​(x+6)2+3

STEP 3

Next, we apply the reflection along the x-axis. This is achieved by multiplying the function by -1.y=−12(x+6)2+3 y = -\frac{1}{2}(x + 6)^2 + 3 y=−21​(x+6)2+3

STEP 4

Now, we will apply the translation 1 unit down. This is done by subtracting 1 from the function.y=−12(x+6)2+3−1 y = -\frac{1}{2}(x + 6)^2 + 3 - 1 y=−21​(x+6)2+3−1

STEP 5

Simplify the equation by combining like terms.y=−12(x+6)2+2 y = -\frac{1}{2}(x + 6)^2 + 2 y=−21​(x+6)2+2

STEP 6

Finally, we apply the translation 3 units to the right. This is done by replacing x x x with x−3 x - 3 x−3 in the equation.y=−12(x−3+6)2+2 y = -\frac{1}{2}(x - 3 + 6)^2 + 2 y=−21​(x−3+6)2+2

STEP 7

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Find the equation of a quadratic relation transformed from $y=(x+6)^{2}+3$ with a vertical compression by $\frac{1}{2}$ , reflection along the $x$ -axis, and translations 1 unit down and 3 units right.

First, we will apply the vertical compression by a factor of one half to the original equation. This means we will multiply the function by $\frac{1}{2}$ .
$y = \frac{1}{2}(x + 6)^2 + 3$

Next, we apply the reflection along the x-axis. This is achieved by multiplying the function by -1.
$y = -\frac{1}{2}(x + 6)^2 + 3$

Now, we will apply the translation 1 unit down. This is done by subtracting 1 from the function.
$y = -\frac{1}{2}(x + 6)^2 + 3 - 1$

Simplify the equation by combining like terms.
$y = -\frac{1}{2}(x + 6)^2 + 2$

Finally, we apply the translation 3 units to the right. This is done by replacing $x$ with $x - 3$ in the equation.
$y = -\frac{1}{2}(x - 3 + 6)^2 + 2$