Solved on Jan 18, 2024

Describe the translation from $y=2(x-15)^{2}+3$ to $y=2(x-11)^{2}+3$ : 4 units to the left

STEP 1

Assumptions
1. The original graph is given by the equation $y = 2(x - 15)^2 + 3$ .
2. The translated graph is given by the equation $y = 2(x - 11)^2 + 3$ .
3. The translation only affects the $x$ -coordinate of the vertex of the parabola.
4. The $y$ -coordinate of the vertex remains the same due to the constant term $+3$ being unchanged.

STEP 2

Identify the vertex of the original parabola.
The vertex of a parabola in the form $y = a(x - h)^2 + k$ is at the point $(h, k)$ .

STEP 3

Find the vertex of the original parabola using the given equation $y = 2(x - 15)^2 + 3$ .
The vertex is at $(15, 3)$ .

STEP 4

Identify the vertex of the translated parabola.
The vertex of the translated parabola is also in the form $(h, k)$ from the equation $y = 2(x - h)^2 + k$ .

STEP 5

Find the vertex of the translated parabola using the given equation $y = 2(x - 11)^2 + 3$ .
The vertex is at $(11, 3)$ .

STEP 6

Determine the horizontal translation.
To find the horizontal translation, we compare the $x$ -coordinates of the vertices of the original and the translated parabolas.

STEP 7

Calculate the change in the $x$ -coordinate.
The change in $x$ is $11 - 15$ .

STEP 8

Compute the horizontal translation.
The horizontal translation is $11 - 15 = -4$ .

STEP 9

Interpret the horizontal translation.
A negative value for the horizontal translation indicates a shift to the left.

STEP 10

Determine the phrase that best describes the translation.
Since the translation is 4 units to the left, the correct phrase is "4 units to the left".
The translation from the graph $y=2(x-15)^{2}+3$ to the graph of $y=2(x-11)^{2}+3$ is best described as 4 units to the left.

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Describe the translation from $y=2(x-15)^{2}+3$ to $y=2(x-11)^{2}+3$ : 4 units to the left

STEP 1

STEP 2

Identify the vertex of the original parabola.
The vertex of a parabola in the form $y = a(x - h)^2 + k$ is at the point $(h, k)$ .

STEP 3

Find the vertex of the original parabola using the given equation $y = 2(x - 15)^2 + 3$ .
The vertex is at $(15, 3)$ .

STEP 4

Identify the vertex of the translated parabola.
The vertex of the translated parabola is also in the form $(h, k)$ from the equation $y = 2(x - h)^2 + k$ .

STEP 5

Find the vertex of the translated parabola using the given equation $y = 2(x - 11)^2 + 3$ .
The vertex is at $(11, 3)$ .

STEP 6

Determine the horizontal translation.
To find the horizontal translation, we compare the $x$ -coordinates of the vertices of the original and the translated parabolas.

STEP 7

Calculate the change in the $x$ -coordinate.
The change in $x$ is $11 - 15$ .

STEP 8

Compute the horizontal translation.
The horizontal translation is $11 - 15 = -4$ .

STEP 9

Interpret the horizontal translation.
A negative value for the horizontal translation indicates a shift to the left.

STEP 10

Determine the phrase that best describes the translation.
Since the translation is 4 units to the left, the correct phrase is "4 units to the left".
The translation from the graph $y=2(x-15)^{2}+3$ to the graph of $y=2(x-11)^{2}+3$ is best described as 4 units to the left.

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Describe the translation from y=2(x−15)2+3y=2(x-15)^{2}+3y=2(x−15)2+3 to y=2(x−11)2+3y=2(x-11)^{2}+3y=2(x−11)2+3: 4 units to the left

STEP 1

STEP 2

Identify the vertex of the original parabola.The vertex of a parabola in the form y=a(x−h)2+ky = a(x - h)^2 + ky=a(x−h)2+k is at the point (h,k)(h, k)(h,k).

STEP 3

Find the vertex of the original parabola using the given equation y=2(x−15)2+3y = 2(x - 15)^2 + 3y=2(x−15)2+3.The vertex is at (15,3)(15, 3)(15,3).

STEP 4

Identify the vertex of the translated parabola.The vertex of the translated parabola is also in the form (h,k)(h, k)(h,k) from the equation y=2(x−h)2+ky = 2(x - h)^2 + ky=2(x−h)2+k.

STEP 5

Find the vertex of the translated parabola using the given equation y=2(x−11)2+3y = 2(x - 11)^2 + 3y=2(x−11)2+3.The vertex is at (11,3)(11, 3)(11,3).

STEP 6

Determine the horizontal translation.To find the horizontal translation, we compare the xxx-coordinates of the vertices of the original and the translated parabolas.

STEP 7

Calculate the change in the xxx-coordinate.The change in xxx is 11−1511 - 1511−15.

STEP 8

Compute the horizontal translation.The horizontal translation is 11−15=−411 - 15 = -411−15=−4.

STEP 9

Interpret the horizontal translation.A negative value for the horizontal translation indicates a shift to the left.

STEP 10

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Describe the translation from $y=2(x-15)^{2}+3$ to $y=2(x-11)^{2}+3$ : 4 units to the left

Identify the vertex of the original parabola.
The vertex of a parabola in the form $y = a(x - h)^2 + k$ is at the point $(h, k)$ .

Find the vertex of the original parabola using the given equation $y = 2(x - 15)^2 + 3$ .
The vertex is at $(15, 3)$ .

Identify the vertex of the translated parabola.
The vertex of the translated parabola is also in the form $(h, k)$ from the equation $y = 2(x - h)^2 + k$ .

Find the vertex of the translated parabola using the given equation $y = 2(x - 11)^2 + 3$ .
The vertex is at $(11, 3)$ .

Determine the horizontal translation.
To find the horizontal translation, we compare the $x$ -coordinates of the vertices of the original and the translated parabolas.

Calculate the change in the $x$ -coordinate.
The change in $x$ is $11 - 15$ .

Compute the horizontal translation.
The horizontal translation is $11 - 15 = -4$ .

Interpret the horizontal translation.
A negative value for the horizontal translation indicates a shift to the left.