📝 How to graph functions (14 MCQs)

📖 From Calculus • 1. Basics before calculus • 14 questions available

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What is How to graph functions?

Definition:
Graphing a function involves plotting all ordered pairs $(x, f(x))$ on a coordinate plane, where the horizontal axis represents the independent variable and the vertical axis represents the dependent variable, creating a visual curve or line.

Example:
For $f(x) = x - 1$ , compute points: $x = 0 \rightarrow f(0) = -1$ , $x = 1 \rightarrow f(1) = 0$ , $x = 2 \rightarrow f(2) = 1$ , then plot and connect to get a straight line.

Reason:
Graphs provide an intuitive visual understanding of function behavior, such as increasing/decreasing trends, intercepts, and overall shape, which is easier than analyzing raw equations.

Easy

Medium

Hard

📝 All How to graph functions MCQs

Q1. What shape does the graph of the function $y = x$ represent in the xy‑plane?

A.A straight line passing through the origin with slope 1 ✅

B.A parabola opening upward

C.A hyperbola with asymptotes

D.A sinusoidal curve

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The graph of $y = x$ is a straight line that goes through the origin (0,0) and has a constant slope of 1, meaning for each unit increase in x the y‑value also increases by one unit. This linear relationship produces a line extending infinitely in both directions, distinct from curves like parabolas or hyperbolas.

Q2. Which of the following functions has a domain that includes every real number?

y = \\sqrt{x}\

y = \\frac{1}{x}\

y = x^{2}\

y = x\

✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The function $y = x$ is defined for any real input because no operation in its definition—such as taking a root, dividing by the variable, or involving a denominator—creates restrictions. In contrast, $\\sqrt{x}\$ requires non‑negative x, $1/x$ excludes x = 0, and $x^{2}$ is defined for all reals but still shares the same unrestricted domain as $y = x$ .

Q3. If the graph of a function crosses the x‑axis at $x = -2$ and $x = 3$ , which statement must be true about the function’s zeros?

A.The function has exactly two zeros

B.The function is positive for all x between -2 and 3

C.The function changes sign at each crossing ✅

D.The function has a maximum at x = 0

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: When a continuous function’s graph intersects the x‑axis, the y‑value changes from positive to negative or vice‑versa at each intersection point. Therefore, at both $x = -2$ and $x = 3$ the sign of the function must switch, guaranteeing a sign change at each zero. The other statements are not necessarily true.

Q4. Compare the graphs of $y = x^{2}$ and $y = x^{3}$ . Which statement accurately describes their behavior for large positive x?

A.Both grow at the same rate

y = x^{3}

grows faster than

y = x^{2}

✅

y = x^{2}

grows faster than

y = x^{3}

D.Both approach a horizontal asymptote

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: For large positive values of x, the cubic term $x^{3}$ dominates the quadratic term $x^{2}$ because raising x to a higher power yields much larger numbers. Consequently, the graph of $y = x^{3}$ rises more steeply than the parabola $y = x^{2}$ . Neither graph approaches a horizontal asymptote; they both increase without bound.

Q5. Given the graph of $y = \\sqrt{x}$ starts at the origin and lies only in the first quadrant, what can be inferred about the function’s domain and range?

A.Domain is \

[0,\\infty)\

and range is \

[0,\\infty)\

✅

B.Domain is \

( -\\infty,0]\

and range is \

( -\\infty,0]\

C.Domain is all real numbers, range is all real numbers

D.Domain is \

[ -\\infty,0)\

and range is \

[0,\\infty)\

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The square‑root function is defined only for non‑negative arguments, so x cannot be negative, giving a domain of \ $[0,\\infty)\$ . Its outputs are also non‑negative, producing a range of \ $[0,\\infty)\$ . The graph beginning at the origin and staying in the first quadrant confirms these intervals.

Q6. Suppose a continuous function’s graph is symmetric with respect to the y‑axis. Which of the following must be true about the function?

A.It is odd

B.It is increasing on \

(-\\infty,0)\

C.It is even ✅

D.It has a horizontal asymptote at y = 0

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Symmetry about the y‑axis means that for every point \ $(x,y)\$ on the graph, the point \ $(-x,y)\$ also lies on the graph. This property matches the definition of an even function, satisfying \ $f(-x)=f(x)\$ for all x in its domain. Odd functions are symmetric about the origin, not the y‑axis.

Q7. The graph of $y = \\frac{1}{x}\$ has two branches. If we reflect this graph across the line $y = x$ , which function’s graph do we obtain?

y = x^{2}\

y = \\frac{1}{x}\

✅

y = -\\frac{1}{x}\

y = \\sqrt{x}\

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Reflecting a point \ $(x,\\frac{1}{x})\$ across the line \ $y=x\$ swaps its coordinates, giving \ $(\\frac{1}{x},x)\$ . This new pair still satisfies the relation \ $y = \\frac{1}{x}\$ , so the reflected graph coincides with the original reciprocal graph. None of the other listed functions describe this transformation.

Q8. A function’s graph passes through the point \ $(2,5)\$ and is known to be linear. Which of the following must be true about the function’s equation?

A.Its slope is \

\\frac{5}{2}\

B.Its y‑intercept is 5

C.Its equation can be written as

y = mx + b\

with \

m = \\frac{5}{2}\

✅

D.The line must pass through the origin

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Any linear function can be expressed in the form \ $y = mx + b\$ . Because the line passes through \ $(2,5)\$ , there exist some slope \ $m\$ and intercept \ $b\$ that satisfy \ $5 = 2m + b\$ . While the exact values of \ $m\$ and \ $b\$ are not forced, the statement that the equation can be written in that general linear form is always true.

Q9. If the graph of a differentiable function f has a local maximum at x = 1 and a local minimum at x = 3, which of the following statements about f′(x) is necessarily correct?

A.f′(x) is zero at both x = 1 and x = 3 ✅

B.f′(x) is positive for all x between 1 and 3

C.f′(x) is negative for all x between 1 and 3

D.f′(x) does not exist at x = 2

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Fermat’s theorem states that if a function is differentiable at a local extremum, its derivative must be zero at that point. Hence, at the local maximum x = 1 and the local minimum x = 3, the derivative f′(x) must equal zero. The behavior of f′ between the extrema is not guaranteed without further information.

Q10. Consider the functions $g(x)=x^{2}$ and $h(x)=|x|$ . Their graphs intersect at points where $x^{2}=|x|$ . How many intersection points exist and what are their x‑coordinates?

A.Two points at x = -1 and x = 1

B.Three points at x = -1, 0, and 1 ✅

C.One point at x = 0

D.Four points at x = -2, -1, 1, and 2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Solve $x^{2}=|x|$ . For x ≥ 0, the equation becomes $x^{2}=x$ giving x = 0 or 1. For x ≤ 0, it becomes $x^{2}=-x$ or $x^{2}+x=0$ , yielding x = 0 or -1. Combining the distinct solutions yields three intersection points with x‑coordinates -1, 0, and 1.

Q11. When the graph of $y = f(x)$ is shifted left by 3 units and then reflected over the x‑axis, the resulting graph represents which transformed function?

y = -f(x+3)\

✅

y = f(-x+3)\

y = -f(x-3)\

y = f(-(x+3))\

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A leftward shift replaces x with $x+3$ , giving $y = f(x+3)$ . Reflecting over the x‑axis multiplies the output by -1, resulting in $y = -f(x+3)$ . The other options either change the sign of the input or omit the required reflection.

Q12. A graph of a function contains a cusp at the point (0,0). Which of the following statements is always true about the derivative of the function at x = 0?

A.The derivative exists and equals 0

B.The derivative exists and is undefined

C.The derivative does not exist ✅

D.The derivative is infinite

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: A cusp indicates that the left‑hand and right‑hand slopes approach different finite values (or one or both become infinite), so the limit defining the derivative fails to exist. Consequently, the derivative of the function at the cusp point is undefined, meaning it does not exist.

Q13. The graph of $y = \\sin x\$ is stretched vertically by a factor of 2 and then translated upward by 1 unit. Which equation represents the new function?

y = 2\\sin x + 1\

✅

y = \\sin(2x) + 1\

y = 2\\sin(x+1)\

y = \\sin x + 2\

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: A vertical stretch multiplies the original function by 2, giving $2\\sin x$ . Adding 1 shifts the entire graph upward by one unit, resulting in the final equation $y = 2\\sin x + 1$ . The other options either alter the argument of the sine or add the constant incorrectly.

Q14. If a function’s graph is completely contained within the strip \ $-2 \\le y \\le 2\$ for all real x, which of the following must be true about the function’s range?

A.The range is exactly \

[-2,2]\

B.The range is a subset of \

[-2,2]\

✅

C.The range includes all real numbers

D.The range is empty

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Being confined to the vertical strip means every output y satisfies \ $-2 \\le y \\le 2\$ . Therefore, the set of all possible y‑values (the range) must lie inside that interval, i.e., it is a subset of \ $[-2,2]\$ . The range could be smaller than the full interval, so statements A, C, and D are not necessarily true.

🔗 Related Topics (MCQs)

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