📝 Vertical and horizontal stretches compressions (13 MCQs)

📖 From Calculus • 1. Basics before calculus • 13 questions available

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What is Vertical and horizontal stretches compressions?

Definition:
Vertical stretches/compressions multiply outputs by a factor $a$ : $y = a f(x)$ stretches if $|a|>1$ and compresses if $0<|a|<1$ ; horizontal stretches/compressions multiply inputs: $y = f(bx)$ compresses if $|b|>1$ and stretches if $0<|b|<1$ .

Example:
For $f(x)=\sin x$ , $y=2\sin x$ doubles amplitude (vertical stretch), and $y=\sin(2x)$ halves period (horizontal compression).

Reason:
These transformations adjust the scale of functions to fit data or emphasize features, crucial in engineering and physics for scaling signals.

Easy

Medium

Hard

📝 All Vertical and horizontal stretches compressions MCQs

Q1. If the graph of $y = f(x)$ is stretched vertically by a factor of 3, what are the new coordinates of the point $(2,4)$ on the transformed graph?

A.-212 ✅

B.-64

C.-21.33

D.-612

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Multiplying the function by 3 scales every y‑value by 3 while leaving x unchanged. The original point (2,4) becomes (2,3·4) = (2,12). The x‑coordinate stays the same, confirming the vertical stretch factor of 3. Thus the transformed graph passes through (2,12), illustrating a vertical stretch.

Q2. Starting from $y = |x|$ , applying the transformation $y = 2|x-1|$ shifts the vertex and scales the graph. Which of the following points on the original graph corresponds to the point $(3,4)$ on the transformed graph?

A.-21

B.-22 ✅

C.-32

D.-42

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The transformation $y = 2|x-1|$ shifts the vertex to (1,0) and stretches vertically by factor 2. To find the original point that maps to (3,4), set $2|x-1| = 4$ ⇒ $|x-1| = 2$ ⇒ $x = 3$ or $-1$ . The corresponding y‑value on the original graph $y = |x|$ is $|2| = 2$ , giving the original point (2,2).

Q3. Which statement correctly compares the effect of multiplying the independent variable by $0.5$ versus multiplying the dependent variable by $0.5$ on the graph of $y = f(x)$ ?

A.Both produce a vertical compression by factor 2.

B.Multiplying x by 0.5 stretches horizontally by factor 2, while multiplying y by 0.5 compresses vertically by factor 2. ✅

C.Both produce a horizontal stretch by factor 2.

D.Multiplying x by 0.5 compresses horizontally by factor 2, while multiplying y by 0.5 stretches vertically by factor 2.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Multiplying x by 0.5 replaces x with 0.5x, which makes the graph twice as wide, i.e., a horizontal stretch by factor 2. Multiplying y by 0.5 replaces y with 0.5y, compressing the graph vertically by factor 2. Therefore the first operation stretches horizontally, while the second compresses vertically.

Q4. Which of the following transformations is equivalent to the graph of $y = -3\,f(2x)$ ?

A.Reflect across the x‑axis and compress horizontally by factor 2, then stretch vertically by factor 3.

B.Compress horizontally by factor 2, reflect across the y‑axis, then stretch vertically by factor 3.

C.Reflect across the x‑axis, stretch vertically by factor 3, then compress horizontally by factor 2.

D.Compress horizontally by factor 2, stretch vertically by factor 3, then reflect across the x‑axis. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The expression $y = -3\,f(2x)$ first compresses the graph horizontally because the argument $2x$ makes the graph twice as narrow. Next, the factor 3 stretches the graph vertically by 3. Finally, the leading minus sign reflects the graph across the x‑axis. Hence the order is horizontal compression, vertical stretch, then x‑axis reflection.

Q5. Multiplying a function $f(x)$ by $-1$ results in which geometric transformation?

A.Reflection across the y‑axis

B.Reflection across the x‑axis ✅

C.Vertical stretch by factor 1

D.Horizontal compression by factor 1

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Multiplying the entire function by –1 changes each y‑value to its negative while leaving x unchanged. This produces a mirror image of the original graph across the x‑axis, known as a reflection about the x‑axis. No horizontal or vertical scaling occurs beyond the sign change.

Q6. The function $g(x)=f(-2x+4)$ can be described as a sequence of transformations applied to $f(x)$ . Which ordered list is correct?

A.Reflect across the y‑axis, stretch horizontally by factor ½, then shift right 4 units.

B.Shift left 4 units, reflect across the y‑axis, then stretch horizontally by factor ½.

C.Stretch horizontally by factor ½, reflect across the y‑axis, then shift right 4 units.

D.Shift right 2 units, reflect across the y‑axis, then compress horizontally by a factor of 2. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Rewrite $-2x+4$ as $-2(x-2)$ . This indicates a rightward shift of 2 units, followed by a reflection across the y‑axis (the negative sign), and finally a horizontal compression by a factor of 2 (the coefficient 2). The ordered description matches option D.

Q7. A transformed graph of $y = f(x)$ contains the points $(-4, 6)$ and $(2, -3)$ . If the transformation is $y = -\frac{1}{2} f(3x - 6)$ , what are the corresponding original points on $y = f(x)$ ?

A.(-18, -12) and (0, 6) ✅

B.(-18, 12) and (0, -6)

C.(-9, -12) and (1, 6)

D.(-6, -12) and (2, 6)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For each transformed point $(x_t, y_t)$ we have $y_t = -\frac12 f(3x_t-6)$ . Solving gives $f(3x_t-6) = -2y_t$ . For $(-4,6)$ : $3(-4)-6 = -18$ and $f(-18) = -12$ , yielding $(-18,-12)$ . For $(2,-3)$ : $3·2-6 = 0$ and $f(0)=6$ , yielding $(0,6)$ . These are the original points.

Q8. For the transformation $y = f\left(\frac{1}{3}x\right)$ , what is the factor by which the graph of $y = f(x)$ is stretched or compressed horizontally?

A.Stretched by factor 3 ✅

B.Compressed by factor 3

C.Stretched by factor

\frac{1}{3}

D.Compressed by factor

\frac{1}{3}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: When the argument of $f$ is multiplied by $1/3$ , the input changes more slowly, causing the graph to expand horizontally. The horizontal stretch factor is the reciprocal of the constant, i.e., $1 ÷ (1/3) = 3$ . Thus the graph is stretched horizontally by a factor of 3.

Q9. If a function $h(x)$ is symmetric about the origin and we define $k(x)= -2\,h(-x)$ , which symmetry, if any, does the graph of $k(x)$ retain?

A.Symmetry about the x‑axis only

B.Symmetry about the y‑axis only

C.Symmetry about the origin ✅

D.No symmetry

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: An origin‑symmetric function satisfies $h(-x) = -h(x)$ . Substituting into $k(x) = -2\,h(-x)$ yields $k(x) = -2(-h(x)) = 2\,h(x)$ . This is merely a vertical scaling, which preserves the original origin symmetry. Consequently $k(x)$ remains symmetric about the origin.

Q10. What term describes the effect of multiplying the independent variable $x$ by a constant greater than 1?

A.Vertical stretch

B.Horizontal compression ✅

C.Vertical compression

D.Horizontal stretch

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Multiplying the independent variable $x$ by a constant greater than 1 makes the graph change faster horizontally, which compresses the graph horizontally. The factor of compression equals the constant itself, so the effect is a horizontal compression.

Q11. According to Table 0.2.4, what is the geometric effect of multiplying the function $f(x)$ by a constant $c$ where $0<c<1$ ?

A.Stretches the graph vertically by factor

c

B.Compresses the graph vertically by factor

1/c

C.Compresses the graph vertically by factor

c

✅

D.Stretches the graph horizontally by factor

c

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: When $0<c<1$ , the factor $c$ reduces each y‑value, shrinking the graph toward the x‑axis. This is a vertical compression by the factor $c$ , as stated in Table 0.2.4. The graph’s shape is unchanged, only its height is reduced.

Q12. Given the graph of $y = f(x)$ passes through $(1,5)$ . After applying the transformation $y = 4 f\left(\frac{x}{2}\right)$ , through which point does the new graph pass?

A.-220 ✅

B.-210

C.-120

D.-0.52

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The original point $(1,5)$ satisfies $f(1)=5$ . In the transformed equation $y = 4\,f(x/2)$ , we need $x/2 = 1$ , giving $x = 2$ . Substituting yields $y = 4·5 = 20$ . Hence the transformed graph passes through $(2,20)$ .

Q13. Compare the combined effect of the transformations $y = f(-x)$ and $y = -f(x)$ . Which statement is true?

A.Both produce a reflection across the y‑axis.

B.Both produce a reflection across the x‑axis.

C.The first reflects across the y‑axis, the second across the x‑axis. ✅

D.The first reflects across the x‑axis, the second across the y‑axis.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The transformation $y = f(-x)$ replaces $x$ with its negative, reflecting the graph across the y‑axis. The transformation $y = -f(x)$ multiplies the output by –1, reflecting across the x‑axis. Therefore the first operation is a y‑axis reflection, and the second is an x‑axis reflection.

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