Graph Transformations Practice Questions

Graph transformations are essential tools in mathematics that allow us to manipulate the appearance and position of graphs on a coordinate plane. These transformations include shifting, reflecting, stretching, and rotating graphs. Understanding these transformations is crucial for analyzing functions and interpreting data visually.
In other words we can also say that Graph transformations refer to the various ways in which the graph of a function can be altered or modified. These changes do not affect the function's nature but adjust how it is presented. By applying different transformations, we can move graphs around, flip them, or change their size, making it easier to compare or contrast different functions.

For Example -
Consider this simple quadratic function: y = x².
Let's add 3 and create a new function that has a vertical shift.
Then, we have y = x² +3. We get the same basic parabolic shape,
but the whole graph is shifted up to 3 units.

In this article, we will look at some Graph Transformations Practice Questions -

Types of Graph Transformations

Some of the types of graph transformations are:

• Translation: Modifies the graph's height or width without changing its form or orientation.

• Reflection: To create a mirror image, flip the graph over one of the axes (the x- or y-axis).

• Dilation: Modifies the graph's steepness by multiplying the function by a constant, stretching or compressing it.

• Rotation: The graph's orientation on the plane is altered by turning it around a given point, often the origin.

How to Solve Graph Transformations Practice Questions

The first step in solving graph transformation questions is identifying the transformation type. Pay attention to whether the graph is shifted, flipped, stretched, or rotated. Recognizing these changes is key to correctly interpreting the transformation.

Once you have identified the transformation, apply it to the graph step by step. For example, if the graph is translated, determine the direction and distance of the shift. If it's a reflection, identify the axis over which the graph is flipped.

After applying the transformation, check your results by comparing the new graph to the original one. Ensure that the transformation was applied correctly and that the graph maintains the properties expected of the transformation.

For Example

Given the function y = |x|, apply a horizontal translation 2 units to the right and a vertical translation 3 units up.

• Original Function: y = |x|
• Horizontal Translation: y = |x - 2|
• Vertical Translation: y = |x - 2| + 3

The final transformed function is y = |x - 2| + 3.

Formulas/Concepts Related to Graph Transformations

A list of Formulas/Concepts Related to Graph Transformations:

Translation:

• Horizontal: y = f(x - h)
• Vertical: y = f(x) + k

Reflection:

• Over the x-axis: y = -f(x)
• Over the y-axis: y = f(-x)

Dilation:

• Vertical Stretch: y = a \cdot f(x) (where a > 1)
• Vertical Compression: y = a \cdot f(x) (where 0 < a < 10)

Rotation and Shearing: Usually involves trigonometric identities and matrices.

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Solved Questions on Graph Transformations

Here are some solved problems related to graph transformations:

Que 1: Reflect the function y = -2x + 3 over the x-axis.

Solution:

To reflect the function y=−2x+3 over the x-axis, we need to change the sign of the output y. Reflecting over the x-axis means that for every point (x,y) on the original line, the reflected point will be (x,−y).

Start with the original function: y=−2x+3

To reflect over the x-axis, replace y with −y:−y=−2x+3

Solve for y: y = 2x − 3

The reflection of the function y=−2x+3 over the x-axis is y=2x−3.

Que 2: Apply a vertical stretch of 2 to the function y = \sqrt{x}​.

Solution:

To apply a vertical stretch to a function, you multiply the output y by the stretch factor. In this case, the stretch factor is 2.

Start with the original function: y = \sqrt{x}.

Multiply the entire function by 2 to apply the vertical stretch: y = 2\sqrt{x}

This means that for every point (x, \sqrt{x}) on the original graph, the new point on the transformed graph will be (x, 2\sqrt{x}).

The transformed function after applying a vertical stretch of 2 is y = 2\sqrt{x}​.

Que 3: Translate the function y = x^3 4 units to the left.

Solution:

To translate a function horizontally, you replace x with x + h in the function, where h is the number of units you want to translate. A positive h translates the function to the left.

Start with the original function: 3y = x^3.

Replace x with x + 4 to translate the function 4 units to the left: 3y = (x + 4)^3

This shifts every point on the graph of y = x^3 to the left by 4 units.

The transformed function after translating 4 units to the left is 3y = (x + 4)^3.

Que 4: Reflect the function y = \ln(x) over the y-axis.

Solution:

To reflect a function over the y-axis, you replace x with -x in the function.

Start with the original function: ⁡y = \ln(x).

Replace x with -x to reflect it over the y-axis: y = \ln(-x)

This reflection changes the domain of the function, so it’s now defined for x < 0 instead of x > 0.

The transformed function after reflecting over the y-axis is ⁡y = \ln(-x).

Que 5: Apply a vertical compression by a factor of 1/3 to the function y = e^x.

Solution:

To apply a vertical compression, you multiply the function by the compression factor \frac{1}{3}.

Start with the original function: y = e^x.

Multiply the entire function by \frac{1}{3} to apply the vertical compression: y = \frac{1}{3}e^x

This compression reduces the height of the graph by a factor of 3, making it less steep.

The transformed function after applying a vertical compression by a factor of \frac{1}{3} is y = \frac{1}{3}e^x.

Que 6:Rotate the function y = x^2 90 degrees counterclockwise.

Solution:

Rotating a function by 90 degrees counterclockwise in the coordinate plane is a more advanced transformation and involves swapping and possibly negating the coordinates. However, a simpler interpretation of rotation in this context can be a reflection along a line.

The original function is y = x^2.

After a 90-degree counterclockwise rotation, the equation is swapped, and y and x are exchanged, with the new equation being x = -y^2. This represents a parabola that opens to the left instead of upwards.

To express the function in terms of y, it would be: y = -\sqrt{x} \quad \text{or} \quad y = \sqrt{-x}.

Que 7:Shear the function y = |x| horizontally by a factor of 2.

Solution:

A horizontal shear transformation changes the x-coordinates of points on the graph while leaving the y-coordinates unchanged.

Start with the original function: y = |x|.

To shear horizontally by a factor of 2, you replace x with \frac{x}{2}​: y = \left|\frac{x}{2}\right|

This transformation changes the slope of the V-shape graph of the absolute value function, making it less steep.

The transformed function after shearing horizontally by a factor of 2 is y = \left|\frac{x}{2}\right|.

Que 8: Reflect and translate the function y = \sin(x) over the x-axis and 2 units up.

Solution:

To reflect over the x-axis, change the sign of the function. To translate the function upwards, add the desired number of units to the function.

Start with the original function: y = \sin(x).

Reflect over the x-axis by multiplying the function by −1: y = -\sin(x)

Translate the function 2 units up by adding 2: y = -\sin(x) + 2

This transformation results in a sine wave that is flipped upside down and shifted 2 units upwards.

The transformed function after reflecting over the x-axis and translating 2 units up is y = -\sin(x) + 2.

Practice Questions on Graph Transformations

Q1: Translate the graph of y = x^2 two units up.

Q2: Reflect the graph of y = \sin(x) over the x-axis.

Q3: Dilate the graph of y = x^3 by a factor of 2 vertically.

Q4: Translate and then reflect the graph of y = |x| two units right and over the y-axis.

Q5: Dilate the graph of y = \cos(x) by a factor of 0.5 horizontally and shift it three units down.

Q6: Rotate the graph of y = \sqrt{x} 90 degrees counterclockwise about the origin.

Q7: Apply a series of transformations to the graph of y = \tan(x) involving a reflection over the y-axis, a horizontal dilation by a factor of 3, and a translation five units left.

Q8: Reflect the graph of ( y = \ln(x) ) over both axes and then rotate it 180 degrees. What is the resulting graph?

Q9: Combine a vertical stretch by a factor of 4 with a translation of seven units down on the graph of y = e^x.

Answer Key

1. The new equation is y = x^2 + 2. The graph of y = x^2 shifts upward by two units.
2. The new equation is y = -\sin(x). The graph of y = \sin(x) is flipped over the x-axis.
3. The new equation is y = 2x^3. The graph of y = x^3 is stretched vertically by a factor of 2.
4. After translation, the equation becomes y=∣x−2∣. Reflecting over the y-axis results in y=∣−(x−2)∣ or simplified as y=∣2−x∣.
5. New equation is y = \cos(2x) - 3. The graph is horizontally compressed by a factor of 0.5 and shifted downward by three units.
6. New graph after a 90-degree counterclockwise rotation about the origin is ( y = x^2 ).
7. The new equation is y = \tan\left(-\frac{x + 5}{3}\right). The graph is reflected over the y-axis, horizontally stretched by a factor of 3, and then translated five units to the left.
8. After reflecting over both axes and rotating 180 degrees, the resulting graph is ( y = -\ln(-x) ).
9. The new equation is y = 4e^x - 7. The graph of y = e^x is stretched vertically by a factor of 4 and shifted downward by seven units.

Conclusion

Graph transformations are of core math concepts, exhibiting behaviors of functions upon various manipulations. With these transformations, it will be easy to project and analyze how functions will behave under different scenarios, be it in theory or in applications.