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Quizzes > High School Quizzes > Mathematics

Transformations and Similarity Practice Quiz

Sharpen Your Skills on Transformations and Congruence

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art promoting a trivia quiz on mastering function transformations in high school algebra.

What does a vertical shift of a function involve?
Reflecting the graph over the y-axis
Adding a constant to the output moves the graph up or down
Multiplying the input by a constant
Changing the slope of the graph
Adding a constant to f(x) results in shifting the graph vertically. This transformation adjusts the y-values without affecting the shape of the graph.
Which algebraic modification represents a horizontal shift to the right by 3 units?
f(3x)
f(x + 3)
3f(x)
f(x - 3)
Replacing x with x - 3 shifts the graph to the right by 3 units. This modification affects the input and moves the graph laterally.
What effect does multiplying a function by 2 have on its graph?
It results in a vertical stretch, doubling the distance from the x-axis
It causes a horizontal stretch by a factor of 2
It shifts the graph upward by 2 units
It reflects the graph across the y-axis
Multiplying the entire function by a constant greater than 1 stretches the graph vertically. The y-values are scaled, making the graph taller without altering its horizontal position.
How can you reflect the graph of a function across the x-axis?
Add a negative constant to the function
Multiply the function by -1
Replace x with -x in the function
Multiply the input by -1
Multiplying f(x) by -1 reflects the graph over the x-axis by negating all y-values. This transformation flips the graph vertically.
What is the effect of replacing x with 2x in the function f(x)?
It causes a horizontal stretch by a factor of 2
It shifts the graph to the right
It vertically stretches the graph
It causes a horizontal compression by a factor of 2
Replacing x with 2x compresses the graph horizontally because the input values reach the same output values faster. This transformation reduces the width of the graph.
Given the function f(x) = x², what is the effect of transforming it to g(x) = (x - 2)² + 3?
The parabola reflects over the x-axis
The parabola shifts left by 2 units and down by 3 units
The parabola shifts right by 3 units and up by 2 units
The parabola shifts right by 2 units and up by 3 units
The term (x - 2)² indicates a horizontal shift to the right by 2, while the +3 indicates a vertical shift upward by 3. These are standard transformations for quadratic functions.
Which transformation reflects a function across the y-axis?
f(x) - x
f(x) + x
-f(x)
f(-x)
Replacing x with -x produces a mirror image of the graph over the y-axis. This transformation reverses the horizontal orientation of the graph.
Analyze the transformation: y = -2f(3x + 6) + 4. Which of the following lists the transformations in their proper sequence?
Compress horizontally by factor 1/3, shift left by 2, then reflect vertically and stretch vertically by factor 2 before shifting up by 4
Shift left by 2, compress horizontally by factor 1/3, reflect vertically, stretch vertically by factor 2, then shift up by 4
Reflect vertically, shift left by 2, compress horizontally by factor 1/3, stretch vertically by factor 2, then shift up by 4
Shift up by 4, then reflect vertically, compress horizontally by factor 1/3, shift left by 2, and finally stretch vertically by factor 2
Factor 3x + 6 can be rewritten as 3(x + 2), indicating a horizontal shift left by 2 and a horizontal compression by 1/3. The outer multiplication by -2 reflects the graph vertically and stretches it by a factor of 2, while the +4 shifts it upward.
Which transformation of f(x) = √x results in the function g(x) = √(x - 5)?
Vertical shift upward by 5 units
Horizontal shift right by 5 units
Vertical compression by a factor of 5
Horizontal shift left by 5 units
Replacing x with x - 5 in the function shifts the graph to the right by 5 units. This is a standard horizontal translation of the square root function.
What is the effect of multiplying a function f(x) by 1/2 to form g(x) = (1/2)f(x)?
Horizontal compression by a factor of 1/2
Vertical stretch by a factor of 1/2
Horizontal stretch by a factor of 1/2
Vertical compression by a factor of 1/2
Multiplying the function by 1/2 reduces all the y-values by half, resulting in a vertical compression. The shape of the graph remains the same, only its height is reduced.
Which transformation reflects a function over both the x-axis and the y-axis?
g(x) = f(-x)
g(x) = -f(x)
g(x) = -f(-x)
g(x) = -f(x - 1)
Replacing x with -x results in a reflection over the y-axis, and multiplying the entire function by -1 reflects it over the x-axis. Combining these two transformations as -f(-x) achieves both reflections.
If a function undergoes a horizontal stretch by a factor of 2, which of the following represents the transformed function?
f(2x)
2f(x)
f(x) + 2
f(x/2)
Replacing x with x/2 stretches the graph horizontally by a factor of 2, as it takes longer for the function to complete its cycle. This does not affect the vertical values of the function.
Determine the sequence of transformations applied to f(x) if the function is given as g(x) = -3f(x + 2) - 1.
Shift left by 2, reflect vertically, stretch vertically by a factor of 3, then shift down by 1
Reflect vertically, shift left by 2, then stretch vertically by a factor of 3 and shift down by 1
Stretch vertically by a factor of 3, shift left by 2, reflect vertically, then shift down by 1
Shift left by 2, stretch vertically by a factor of 3, then reflect horizontally and shift down by 1
The term (x + 2) indicates a horizontal shift to the left by 2. The -3 outside the function reveals a vertical reflection and a vertical stretch by 3, and the -1 indicates a downward shift. The correct sequence respects the order of transformations.
Two similar triangles have a scale factor of 3. If the area of the smaller triangle is 4, what is the area of the larger triangle?
9
24
36
12
The area of similar figures scales by the square of the scale factor. Since 3² = 9, the larger triangle's area is 9 times the smaller triangle's area: 9 x 4 = 36.
For two similar figures with corresponding side lengths in the ratio 2:3, what is the ratio of their areas?
2:3
4:9
6:9
3:2
The area ratio of similar figures is the square of the ratio of their corresponding side lengths. Squaring 2 and 3 gives 4 and 9 respectively, so the area ratio is 4:9.
Given f(x) = x², the transformed function is g(x) = 2(x - 3)² + 5. What is the correct order of transformations applied to f(x)?
Shift up by 5, vertical stretch by factor of 2, then shift right by 3
Shift right by 3, shift up by 5, then vertical stretch by factor of 2
Shift right by 3, vertical stretch by factor of 2, then shift up by 5
Vertical stretch by factor of 2, shift right by 3, then shift up by 5
The expression (x - 3)² indicates a horizontal shift to the right by 3. The multiplier 2 outside the squared term stretches the graph vertically, and the +5 shifts the graph upward. This sequence ensures the correct transformation order.
If g(x) = f(2(x + 1)) - 4 where f(x) is the base function, what is the sequence of transformations to obtain g(x) from f(x)?
Shift left by 1, then compress horizontally by a factor of 1/2, and finally shift down by 4
Compress horizontally by 1/2, shift left by 1, and then shift up by 4
Shift right by 1, then compress horizontally by 2, and shift down by 4
Shift left by 1, then stretch vertically by 2, and finally shift down by 4
By rewriting 2(x + 1), it is clear that the horizontal transformation involves a left shift by 1 and a horizontal compression by factor 1/2. The subtraction of 4 indicates a vertical shift downward. The correct sequence follows these steps.
Consider the transformation h(x) = -0.5f(4 - x). Which of the following correctly describes the sequence of transformations?
Apply a vertical compression by 0.5, reflect horizontally, then shift right by 4
Reflect vertically, shift left by 4, then apply a vertical compression by 0.5
Shift right by 4, reflect vertically, and then compress horizontally by 0.5
Reflect horizontally and shift right by 4, then reflect vertically and apply a vertical compression by 0.5
Writing 4 - x as -(x - 4) reveals a horizontal reflection followed by a shift to the right by 4 units. The multiplier -0.5 then reflects the graph vertically and compresses it by 0.5. This combination of steps is accurately described in the correct option.
A function is transformed by either shifting upward by k and then stretching vertically by 3, or by first stretching vertically by 3 and then shifting upward by 9. If both methods yield the same function, what is the value of k?
3
9
12
0
Shifting first gives 3(f(x) + k) = 3f(x) + 3k, which must equal 3f(x) + 9. Setting 3k equal to 9 shows that k is 3. Thus, the transformations are equivalent when k = 3.
For two similar triangles, if the side length of the smaller triangle is given by the function f(x) = 2x + 3 and the corresponding side in the larger triangle is obtained by scaling by 3 and then shifting by 4 units, which function represents the side length of the larger triangle?
6x + 7
2x + 7
6x + 13
6x + 4
Scaling f(x) by 3 yields 3(2x + 3) = 6x + 9. Adding the shift of 4 units results in 6x + 13. This transformation correctly represents the combined scaling and shifting process.
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Study Outcomes

  1. Analyze the effects of horizontal and vertical shifts in function graphs.
  2. Apply scaling techniques to modify the shape and size of graphs.
  3. Understand reflections and their impact on graph orientation.
  4. Solve equations by using combined transformations of functions.
  5. Evaluate problems by recognizing transformational patterns in algebraic functions.

Post Test: Transformations & Similarity Cheat Sheet

  1. Master Vertical and Horizontal Shifts - Think of your graph as a sliding puzzle: adding or subtracting a constant moves it up, down, left, or right with ease. For example, f(x) + k lifts your curve k units upward, while f(x − h) nudges it h units to the right. Explore shifts on OpenStax
  2. Understand Reflections Over Axes - Mirror magic happens when you multiply by - 1: - f(x) flips your graph over the x-axis, and f( - x) flips it over the y-axis. Spotting these reflections can make your sketches instantly symmetrical and satisfying. Discover reflections on OpenStax
  3. Grasp Vertical and Horizontal Stretches/Compressions - Stretch your imagination (and your graph!) by multiplying the output or input by a constant. If a > 1, af(x) elongates vertically; if 0 < a < 1, it squashes it down. See stretch/compress details on OpenStax
  4. Combine Multiple Transformations - Want to level up? Stack shifts, reflections, and stretches to see their combined effect. Practicing sequences helps you predict the final shape like a transformation detective. Practice combos at MathBits Notebook
  5. Identify Even and Odd Functions - Symmetry fans rejoice: even functions mirror across the y-axis (f( - x)=f(x)), while odd functions spin around the origin (f( - x)= - f(x)). Recognizing these patterns saves time when sketching or proving properties. Read about symmetry on OpenStax
  6. Apply Transformations to Real-World Problems - Graph transformations aren't just theory - they model everything from sound waves to population growth. Translating a story problem into shifts and stretches boosts your problem-solving superpowers. See real-world applications on OpenStax
  7. Practice with Interactive Tools - Nothing beats hands-on learning: use GeoGebra to drag and twist graphs in real time. Watching parameters change visually cements your intuition. Try it in GeoGebra
  8. Work Through Practice Problems - Repetition is your friend - tackle a variety of exercises to reinforce each transformation type. The more problems you solve, the more patterns you'll spot. Solve problems on GeeksforGeeks
  9. Memorize Key Transformation Formulas - Keep a cheat sheet handy with essentials like f(x)+k for vertical shifts or f(x - h) for horizontal shifts. Quick recall of these formulas speeds up both exams and homework sessions. Review formulas on OpenStax
  10. Understand the Order of Transformations - Order matters! Applying a shift before a stretch gives a different result than stretching then shifting. Mastering the correct sequence ensures your final graph is spot‑on. Learn the correct order on OpenStax
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