Dilations on a Graph

CALCULATOR

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Dilation on a Graph Calculator

Step 1. Identify the center of dilation.

Origin

(0, 0)

Different Point

(x, y)

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INTRO

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A dilation is a transformation that changes the size of an image without changing its shape or proportions.

It's like how when Ant-Man gets bigger or smaller, each body part changes size by the same amount, so he looks the same - just bigger or smaller than before. Dilating shapes is similar 👯‍♀️.

When we dilate shapes on a grid, we need to know the center of dilation and the scale factor.

To help us understand these terms, let’s imagine we have a movable screen and a fixed projector that can’t be moved:

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✨ Drag to move the projection ✨

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The center of dilation is like the fixed projector: it’s a fixed point from which the image is drawn. The scale factor determines how much smaller or larger the dilated image will be.

If the scale factor is , we can imagine keeping the distance between the projector and the screen the same, which will make the new image the same as the original.

Check out our

Calculator

or explore our

Lesson

and

Practice

sections to learn more about dilations on a graph and test your understanding.

You can also use the Quick Links menu on the left to jump to a section of your choice.

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KEY STEPS

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How to Perform Dilations

Step 1. Identify the center of dilation.

Imagine this as the fixed location of the projector.

Origin

(0, 0)

Different Point

(x, y)

Step 2. Identify the original points of the polygon.

Imagine this as the original image before the screen is moved.

original points = (x_{1}, y_{1}), (x_{2}, y_{2}), . . ., (x_{n}, y_{n})

Step 3. Identify the scale factor .

If the scale factor is , we can imagine keeping the distance between the projector and the screen the same, which will make the new image the same as the original.

Step 4. Multiply each original point of the polygon by the scale factor to get the new points.

original points \times scale factor (x_{original}, y_{original}) \times scale factor = new points = (x_{new}, y_{new})

Original Point	$\times$	Scale Factor	$=$	New Point
$(x_{1}, y_{1})$	$\times$	$s$	$=$	$(x_{1} \times s, y_{1} \times s)$
$(x_{2}, y_{2})$	$\times$	$s$	$=$	$(x_{2} \times s, y_{2} \times s)$
$. . .$	$\times$	$s$	$=$	$. . .$
$(x_{n}, y_{n})$	$\times$	$s$	$=$	$(x_{n} \times s, y_{n} \times s)$

Step 5. Plot the new points to get the dilated shape.

LESSON

— Dilations Centered at the Origin

PRACTICE

— Dilations Centered at the Origin

LESSON

— Dilations NOT Centered at the Origin

PRACTICE

— Dilations NOT Centered at the Origin

CONCLUSION

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Original Point	$-$	Center of Dilation	$=$	Difference
$(x_{1}, y_{1})$	$-$	$(x_{c}, y_{c})$	$=$	$(x_{1} - x_{c}, y_{1} - y_{c})$
$(x_{2}, y_{2})$	$-$	$(x_{c}, y_{c})$	$=$	$(x_{2} - x_{c}, y_{2} - y_{c})$
$. . .$	$-$	$(x_{c}, y_{c})$	$=$	$. . .$
$(x_{n}, y_{n})$	$-$	$(x_{c}, y_{c})$	$=$	$(x_{n} - x_{c}, y_{n} - y_{c})$

New Difference	$+$	Center of Dilation	$=$	New Point
$((x_{1} - x_{c}) s, (y_{1} - y_{c}) s)$	$+$	$(x_{c}, y_{c})$	$=$	$((x_{1} - x_{c}) s + x_{c}, (y_{1} - y_{c}) s + y_{c})$
$((x_{2} - x_{c}) s, (y_{2} - y_{c}) s)$	$+$	$(x_{c}, y_{c})$	$=$	$((x_{2} - x_{c}) s + x_{c}, (y_{2} - y_{c}) s + y_{c})$
$. . .$	$+$	$(x_{c}, y_{c})$	$=$	$. . .$
$((x_{n} - x_{c}) s, (y_{n} - y_{c}) s)$	$+$	$(x_{c}, y_{c})$	$=$	$((x_{n} - x_{c}) s + x_{c}, (y_{n} - y_{c}) s + y_{c})$

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QUICK LINKS

QUICK LINKS

Dilations on a Graph

Dilation on a Graph Calculator

Step 1. Identify the center of dilation.

Step 2. Identify the original points of the polygon.

Step 3. Identify the scale factor .

Step 4.

Closer

Farther

How to Perform Dilations

Step 1. Identify the center of dilation.

Step 2. Identify the original points of the polygon.

Step 3. Identify the scale factor .

Step 4. Multiply each original point of the polygon by the scale factor to get the new points.

Step 5. Plot the new points to get the dilated shape.

Step 4. Find the difference between the $x$ and $y$ values of each original point and the center of dilation .

Step 5. Multiply each difference by the scale factor.

Step 6. Add the new difference to the center of dilation to get the new points.

Step 7. Plot the new points to get the dilated shape.

QUICK LINKS

QUICK LINKS

Dilations on a Graph

Dilation on a Graph Calculator

Step 1. Identify the center of dilation.

Step 2. Identify the original points of the polygon.

Step 3. Identify the scale factor .

Step 4.

Closer

Farther

How to Perform Dilations

Step 1. Identify the center of dilation.

Step 2. Identify the original points of the polygon.

Step 3. Identify the scale factor .

Step 4. Multiply each original point of the polygon by the scale factor to get the new points.

Step 5. Plot the new points to get the dilated shape.

Step 4. Find the difference between the x and y values of each original point and the center of dilation .

Step 5. Multiply each difference by the scale factor.

Step 6. Add the new difference to the center of dilation to get the new points.

Step 7. Plot the new points to get the dilated shape.

Step 4. Find the difference between the $x$ and $y$ values of each original point and the center of dilation .