📝 Scale and units in graphing functions (12 MCQs)

📖 From Calculus • 1. Basics before calculus • 12 questions available

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What is Scale and units in graphing functions?

Definition:
Scale and units in graphing functions refer to the consistent spacing and labeling of axes, where each tick mark represents a specific quantity, and units (e.g., meters, seconds) clarify the real-world meaning of coordinates.

Example:
Graph $y = 5x$ with $x$ in hours and $y$ in miles, using 1 unit = 1 hour on x-axis and 1 unit = 5 miles on y-axis, then a point (2,10) means 10 miles in 2 hours.

Reason:
Proper scales and units prevent misinterpretation, ensure accurate readings, and allow comparison between different datasets or functions.

Easy

Medium

Hard

📝 All Scale and units in graphing functions MCQs

Q1. When graphing a circle, what is the visual effect of using a longer unit length on the x‑axis than on the y‑axis?

A.The circle appears stretched horizontally.

B.The circle appears stretched vertically.

C.The circle becomes an ellipse squashed vertically. ✅

D.The circle remains a perfect circle.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Because the horizontal units are longer, each unit on the x‑axis covers more distance than a unit on the y‑axis. Points that should be equally distant from the center in both directions are plotted farther apart horizontally, turning the true circle into an ellipse that is compressed vertically.

Q2. A graph of $y = x^{2}$ for $-10 \le x \le 10$ is drawn with equal unit lengths on both axes, but the y‑axis is then compressed uniformly by a factor $k$ so that the entire curve fits on the same paper size. Which value of $k$ is required?

A.k = 0.1

B.k = 0.5

C.k = 0.05

D.k = 0.2 ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The y‑values range from 0 to 100, while the x‑range spans 20 units. To make the vertical extent match the horizontal extent (20 units), the y‑axis must be scaled down by 20/100 = 0.2. Thus a compression factor of 0.2 allows the whole parabola to occupy the same paper height as its width.

Q3. If the unit length on the x‑axis is doubled while the y‑axis unit remains unchanged, how does the slope of a line originally having slope 2 appear on the new graph?

A.It appears as slope 1 ✅

B.It appears as slope 4

C.It appears unchanged

D.It appears as slope 0.5

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Doubling the horizontal unit length stretches the x‑axis, so a given horizontal change now covers twice as many units. The visual rise‑over‑run ratio therefore halves, turning the original slope of 2 into an apparent slope of 1 on the resized graph.

Q4. On a plot where 1 cm represents 2 seconds on the x‑axis and 1 cm represents 1 meter on the y‑axis, a point moving at constant speed 3 m/s is drawn. What is the visual slope (rise over run) of its trajectory?

A.6 ✅

B.3

C.1.5

D.0.5

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: In graph units, a 1 cm horizontal step corresponds to 2 seconds, and a 1 cm vertical step corresponds to 1 meter. At 3 m/s, in 2 seconds the point moves 6 meters, which is 6 cm vertically. Thus the rise/run ratio is 6 cm / 1 cm = 6, giving a visual slope of 6.

Q5. Two graphs display the same quadratic function $y = x^{2}$ over $[-5,5]$ . Graph A uses equal unit lengths on both axes; Graph B compresses the y‑axis by a factor of 0.5. Which statement about the area under the curve from $-5$ to $5$ is true?

A.Both graphs show the same numerical area.

B.Graph A shows twice the area of Graph B.

C.Graph B shows half the area of Graph A.

D.The areas cannot be compared because scaling changes units. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Area under a curve is measured in square units that depend on the scaling of each axis. When the y‑axis is compressed, the vertical dimension of each infinitesimal rectangle is halved, so the numerical area value changes. Therefore the two numerical areas are not directly comparable.

Q6. When plotting $y = \sin(x)$ for $0 \le x \le 4\pi$ , which scaling choice best preserves the wave shape while keeping the graph readable?

A.Equal units, 1 cm per radian on x and 1 cm per unit on y.

B.Stretch x‑axis by factor 2, keep y‑axis equal. ✅

C.Compress y‑axis by factor 0.5, keep x‑axis equal.

D.Compress both axes by factor 0.5.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The sine wave spans roughly 12.6 cm horizontally if 1 cm per radian is used, making the plot too wide. Stretching the x‑axis by a factor of 2 reduces the horizontal length to about 6.3 cm, preserving the wave’s proportions while keeping the vertical amplitude unchanged, so the shape remains clear.

Q7. A parabola $y = -x^{2} + 4$ is plotted with equal axis scales. If the y‑axis is stretched by factor 3, how does the visual prominence of the maximum point change compared to the minimum point of $y = x^{2} - 4$ plotted with the same stretch?

A.The maximum appears three times taller than the minimum appears deep. ✅

B.Both appear equally prominent.

C.The maximum appears less prominent than the minimum.

D.The stretch affects only the maximum, not the minimum.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Vertical stretching multiplies every vertical distance from the x‑axis by the same factor. The peak of the first parabola is 4 units above the axis; after stretching, it rises 12 units, making it appear three times taller. The depth of the second parabola is also multiplied by 3, so both features are amplified proportionally.

Q8. Why is it important to use equal unit lengths on both axes when comparing geometric shapes such as circles or squares?

A.It ensures the numerical values are identical.

B.It preserves the true geometric proportions of the shapes. ✅

C.It makes the graph easier to print.

D.It eliminates the need for a legend.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Equal axis scales mean that a unit of length represents the same physical distance in both directions. Consequently, shapes are drawn with their authentic proportions; a circle remains a circle and a square retains equal sides. Any distortion would misrepresent the geometry and lead to incorrect visual conclusions.

Q9. You need to graph the function $y = e^{x}$ for $-2 \le x \le 2$ . Which strategy best selects axis scales to display the rapid growth while keeping the graph interpretable?

A.Use equal unit lengths on both axes.

B.Compress the y‑axis by a factor of 0.2.

C.Choose a logarithmic scale for the y‑axis. ✅

D.Extend the x‑axis length twice as much as the y‑axis.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The exponential function grows dramatically, so a linear y‑scale would force most of the curve into a tiny region. A logarithmic y‑scale linearizes the growth, spreading the points more evenly across the plot while preserving the shape’s essential features, making the graph far more readable.

Q10. If a graph is vertically stretched by a factor $k$ , how does the derivative $dy/dx$ at any point change?

A.It is multiplied by

k

. ✅

B.It is divided by

k

C.It remains unchanged.

D.It is multiplied by

k^{2}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The derivative represents the instantaneous rate of change, which is the ratio of a small vertical change to a small horizontal change. Stretching the vertical direction multiplies the numerator of this ratio by $k$ while leaving the denominator unchanged, so the derivative itself is multiplied by $k$ .

Q11. Applying a scaling transformation that multiplies the x‑coordinate by $a$ and the y‑coordinate by $b$ to the circle $x^{2}+y^{2}=r^{2}$ yields which equation, and under what condition does the image remain a circle?

(x/a)^{2}+(y/b)^{2}=r^{2}

; remains a circle if

a=b

. ✅

(x a)^{2}+(y b)^{2}=r^{2}

; remains a circle if

a=b

(x/a)^{2}+(y/b)^{2}=r^{2}

; remains a circle for any

a,b

(x a)^{2}+(y b)^{2}=r^{2}

; remains a circle if

ab=1

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Replacing $x$ with $x/a$ and $y$ with $y/b$ reflects the scaling: the new coordinates satisfy $(x/a)^{2}+(y/b)^{2}=r^{2}$ . This describes an ellipse unless the scaling factors are equal, i.e., $a=b$ ; only then does the equation reduce to a circle of radius $r$ in the transformed coordinates.

Q12. Two plots show the same data set: Plot 1 uses equal axis scales; Plot 2 uses unequal scales that preserve the area under the curve. Which plot gives a more accurate visual comparison of instantaneous rates of change?

A.Plot 1, because equal scales preserve slopes. ✅

B.Plot 2, because area preservation maintains integrals.

C.Both are equally accurate.

D.Neither; a third method is required.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Instantaneous rates of change correspond to the slope of the curve at a point, which depends directly on the relative scaling of the axes. Equal scales keep the geometric slope proportional to the actual derivative, whereas unequal scales distort this ratio, making Plot 1 the more reliable visual tool for comparing rates.

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