Functions, Graphs & Rates of Change

The function f(x) = (x - 4)/3 has an inverse. Find f⁻¹(x).

• Af⁻¹(x) = 3x + 12
• Bf⁻¹(x) = (x + 4)/3
• Cf⁻¹(x) = 3x + 4✓
• Df⁻¹(x) = 3x - 4

Answer: Writing y = (x − 4)/3 and rearranging to make x the subject gives 3y = x − 4 and then x = 3y + 4, so f⁻¹(x) = 3x + 4. A sign error when rearranging, or multiplying instead of adding the 4, produces the other results.

The function g(x) = 2x² + 1 for x ≥ 0 has an inverse. Find g⁻¹(x).

• Ag⁻¹(x) = √(2(x - 1))
• Bg⁻¹(x) = √((x + 1)/2)
• Cg⁻¹(x) = √((x - 1)/2)✓
• Dg⁻¹(x) = √(x - 1)/2

Answer: Writing y = 2x² + 1 and rearranging gives x² = (y − 1)/2, and taking the positive square root (since x ≥ 0) gives x = √((y − 1)/2), so g⁻¹(x) = √((x − 1)/2). Misplacing the bracket around the division, or a sign error on the 1, produces the other results.

Given f(x) = 2x + 1 and g(x) = x² - 3, find the composite function fg(x).

• A2x² - 6
• Bx² - 5
• C2x² - 5✓
• D2x² + 1

Answer: fg(x) means substitute g(x) into f, so fg(x) = f(x² − 3) = 2(x² − 3) + 1 = 2x² − 6 + 1 = 2x² − 5. Forgetting the final + 1, or applying only g's constant, produces the other results.

A curve has equation y = x² - 2x. Find the average rate of change of y between x = 1 and x = 4.

• A6
• B4
• C9
• D3✓

Answer: The average rate of change is the gradient of the chord joining (1, −1) and (4, 8), which is (8 − (−1))/(4 − 1) = 9/3 = 3. Forgetting to divide by the change in x, or a slip evaluating y at one of the endpoints, produces the other values.

A curve has equation y = x³. Estimate the instantaneous rate of change at x = 2 by finding the gradient of the chord between x = 2 and x = 2.1.

• A13.31
• B12.61✓
• C9.261
• D12.00

Answer: The gradient of the chord is the change in y divided by the change in x: y(2) = 8 and y(2.1) = 9.261, so the gradient is (9.261 − 8)/0.1 = 12.61. Using the true gradient of 12 from calculus instead of the chord estimate, or reporting y(2.1) itself, produces the other values.

A distance-time graph shows that between t = 2 and t = 5 seconds, a car's distance travelled increases from 18 m to 63 m. Find the average velocity over this interval.

• A9 m/s
• B45 m/s
• C15 m/s✓
• D22.5 m/s

Answer: Average velocity is the gradient of the chord on a distance–time graph: the change in distance is 63 − 18 = 45 m over a change in time of 5 − 2 = 3 s, giving 45/3 = 15 m/s. Forgetting to divide by the time interval, or a slip in the time difference, produces the other values.

A tangent drawn on a profit-time graph at t = 3 passes through the points (1, 4) and (5, 20). Find the instantaneous rate of change of profit at t = 3.

• A£16 per unit time
• B£5 per unit time
• C£8 per unit time
• D£4 per unit time✓

Answer: The instantaneous rate of change at t = 3 equals the gradient of the tangent line at that point, found from two points on it: (20 − 4)/(5 − 1) = 16/4 = £4 per unit time. Forgetting to divide by the change in t, or a slip in one of the coordinates, produces the other values.

As x becomes a large positive number on the curve y = x cubed, what happens to the value of y?

• Ay oscillates between positive and negative values
• By approaches a fixed maximum value
• Cy increases without bound as x grows✓
• Dy decreases towards zero

Answer: A positive cubic such as y = x cubed keeps rising as x rises, since cubing a large positive number gives an even larger positive number. There is no ceiling on this growth, so y increases without bound rather than levelling off. A reciprocal-style curve would tend to zero, and a curve with a fixed maximum would be capped, neither of which applies to a simple positive cubic.

A cubic graph with a positive x cubed term crosses the x-axis at x = -2, x = 1 and x = 3. Which equation could represent this curve?

• Ay = (x + 2)(x - 1)(x + 3)
• By = (x + 2)(x - 1)(x - 3)✓
• Cy = -(x + 2)(x - 1)(x - 3)
• Dy = (x - 2)(x + 1)(x + 3)

Answer: Each root of a cubic corresponds to a factor that equals zero at that x-value, so a root at x = -2 gives the factor (x + 2), a root at x = 1 gives (x - 1), and a root at x = 3 gives (x - 3). Multiplying these three factors together reproduces all three crossing points with a positive leading term. The other options place at least one root in the wrong position, or add a negative sign that would flip the graph and make the leading term negative.

The cubic curve y = x cubed minus 4x crosses the x-axis three times. What are the three roots?

• A-4, 0, 4
• B-2, 0, 2✓
• C-2, 0, 4
• D0, 2, 4

Answer: Factorising y = x cubed minus 4x gives x(x squared - 4), which factorises further into x(x - 2)(x + 2). Setting each factor to zero gives the roots x = 0, x = 2 and x = -2. The other sets of values do not satisfy the original equation when substituted back in.

What is the equation of the vertical asymptote of the graph y = 1/x?

• Ay = 0
• Bx = 1
• Cy = 1
• Dx = 0✓

Answer: The reciprocal function y = 1/x is undefined when x = 0, because dividing by zero has no value, so the curve approaches but never touches the vertical line x = 0. This vertical asymptote is different from the horizontal asymptote y = 0, which describes the curve's behaviour as x becomes very large. The lines x = 1 and y = 1 do not restrict the graph in this way at all.

The graph of y = 1/x is translated to give y = 1/x + 3. What is the equation of the new horizontal asymptote?

• Ay = 3✓
• Bx = 3
• Cy = 0
• Dy = -3

Answer: Adding 3 to the whole function shifts every point on the graph upward by 3 units, including the original horizontal asymptote y = 0. This moves the horizontal asymptote to y = 3, while the vertical asymptote x = 0 stays exactly where it was because only y-values changed. Writing the asymptote as x = 3 or y = -3 confuses the direction and size of the translation.

A point on the graph y = 1/x has x-coordinate 4. What is its y-coordinate?

• Ay = 4
• By = 0.4
• Cy = -0.25
• Dy = 0.25✓

Answer: Substituting x = 4 into y = 1/x gives y = 1/4, which equals 0.25. The option y = 4 simply repeats the x-coordinate rather than applying the reciprocal, y = -0.25 has the wrong sign for a positive x-value, and y = 0.4 comes from a slipped division. Only y = 0.25 satisfies the equation of the curve.

What is the y-intercept of the graph y = 5 to the power x?

• Ay = 0
• By = 5
• Cy = 25
• Dy = 1✓

Answer: Any positive number raised to the power zero equals 1, so substituting x = 0 into y = 5 to the power x gives y = 1. This is true for every exponential graph of the form y = k to the power x, regardless of the value of k. The values y = 5, y = 0 and y = 25 all come from substituting the wrong power into the expression.

The graphs y = 2 to the power x and y = 4 to the power x are both increasing exponential curves. Which statement correctly compares them for x > 0?

• Ay = 4 to the power x is a horizontal translation of y = 2 to the power x
• By = 4 to the power x increases more steeply than y = 2 to the power x✓
• Cboth graphs increase at exactly the same rate
• Dy = 2 to the power x increases more steeply than y = 4 to the power x

Answer: A larger base value in an exponential function produces faster growth for positive x, because each extra unit of x multiplies the output by a bigger factor. Since 4 is greater than 2, the curve y = 4 to the power x climbs more steeply than y = 2 to the power x once x is positive. The two curves are not translations of one another and do not share the same growth rate, since their bases are different.

Which equation represents an exponential graph that decreases as x increases?

• Ay = (1.5) to the power x
• By = 2 to the power x
• Cy = 3 to the power x
• Dy = (0.5) to the power x✓

Answer: An exponential graph y = k to the power x decreases as x increases only when the base k is between 0 and 1, because repeated multiplication by a fraction shrinks the value. Here k = 0.5 satisfies this condition, producing decay rather than growth. The other three options all have a base greater than 1, so each of those graphs rises as x increases instead of falling.

Using the symmetry of the sine graph and given that sin(30 degrees) = 0.500, what is the value of sin(150 degrees)?

• A0.866
• B0.500✓
• C1.000
• D-0.500

Answer: The sine graph is symmetrical about x = 90 degrees, so sin(150 degrees) equals sin(180 degrees - 150 degrees), which is sin(30 degrees). Since sin(30 degrees) = 0.500, it follows that sin(150 degrees) = 0.500 as well. The value -0.500 wrongly applies a sign change that only happens beyond 180 degrees, and 0.866 and 1.000 belong to entirely different angles.

The graph of y = cos(x) repeats its pattern at regular intervals as x increases. What is the period of y = cos(x)?

• A180 degrees
• B720 degrees
• C360 degrees✓
• D90 degrees

Answer: The cosine graph completes one full wave, from peak to peak, over an interval of 360 degrees, after which the pattern repeats identically. A period of 180 degrees would only capture half a wave, and 90 degrees would capture just a quarter, so neither reproduces the full shape. Doubling to 720 degrees does repeat the pattern, but it is not the smallest interval that does so.

For 0 degrees to 180 degrees, at which value of x is the graph of y = tan(x) undefined?

• A0 degrees
• B90 degrees✓
• C180 degrees
• D45 degrees

Answer: The tangent function is defined as sine divided by cosine, and cosine equals zero at 90 degrees, so tan(x) is undefined there because division by zero has no value. At 0 degrees and 180 degrees cosine is not zero, so tangent takes the finite value 0 at both, and at 45 degrees tangent equals 1. Only x = 90 degrees produces the vertical asymptote on the graph.

The graph of y = f(x) is translated to give y = f(x) + 4. In which direction does the graph move?

• Ato the right by 4 units
• Bto the left by 4 units
• Cupward by 4 units✓
• Ddown by 4 units

Answer: Adding a constant to the whole function increases every y-value by that amount, which moves the entire graph vertically rather than horizontally. Since 4 is added, the graph moves upward by 4 units, not downward. A horizontal shift would instead come from changing the input, such as replacing x with x - 4 or x + 4, which is not what has happened here.

The graph y = f(-x) is obtained from y = f(x) by which single transformation?

• Aa rotation of 180 degrees
• Ba reflection in the x-axis
• Ca translation of 1 unit left
• Da reflection in the y-axis✓

Answer: Replacing x with -x swaps the sign of every input before the function is applied, which flips the graph horizontally across the y-axis while leaving each y-value unchanged. A reflection in the x-axis would instead come from negating the whole output, written as -f(x). Neither a small translation nor a full rotation produces the same mirrored effect as this substitution.

The graph y = -f(x) is obtained from y = f(x) by which single transformation?

• Aa reflection in the y-axis
• Ba vertical stretch by scale factor 2
• Ca reflection in the x-axis✓
• Da translation downward by 1 unit

Answer: Negating the whole output of a function flips the sign of every y-value while the x-values stay fixed, which reflects the graph vertically across the x-axis. A reflection in the y-axis instead comes from negating the input, written as f(-x), which is a different transformation. There is no stretching or fixed vertical shift involved, since the size of the change depends on the original y-value, not a constant amount.

A distance-time graph forms a curve rather than a straight line. What does the gradient at any single point on this curve represent?

• Athe average speed over the whole journey
• Bthe instantaneous speed at that point✓
• Cthe total distance travelled
• Dthe time taken for the journey

Answer: On a distance-time graph, the gradient measures how quickly distance is changing with respect to time at that exact moment, which is the instantaneous speed. Total distance is read from the height of the curve rather than its steepness, and average speed over a journey compares the total distance with the total time rather than a single point. The time taken is an axis value, not something the gradient itself represents.

On a velocity-time graph, the area under the curve between t = 0 and t = 5 seconds is estimated using the trapezium rule. What does this area represent?

• Athe average velocity between 0 and 5 seconds
• Bthe distance travelled between 0 and 5 seconds✓
• Cthe acceleration between 0 and 5 seconds
• Dthe final velocity at 5 seconds

Answer: The area enclosed between a velocity-time graph and the time axis represents distance, because multiplying velocity by a small interval of time gives a small distance, and summing these strips over the interval builds up the total distance travelled. Acceleration is instead found from the gradient of the graph, not the area beneath it. Average velocity and final velocity are single values read from the curve, not the region it encloses.

A savings graph plots the amount in an account against time, and the curve gets steeper as time passes. What does this increasing gradient show?

• Athe savings are decreasing over time
• Bthe savings are growing at an increasing rate✓
• Cthe savings are growing at a constant rate
• Dthe interest rate is falling over time

Answer: A steeper gradient later on the curve means the amount in the account is rising by more for each extra unit of time than it was earlier, which shows the growth rate itself is increasing, typical of compound interest. A constant rate would instead produce a straight line rather than a curve that steepens. Since the account balance is rising rather than falling, the savings are not decreasing, and a rising balance with a steepening curve points to a growing, not falling, effective rate.

The curve y = x squared is used to estimate the area between the curve and the x-axis from x = 0 to x = 2, using the y-values at x = 0, 1 and 2, which are 0, 1 and 4, with two trapeziums of width 1. What is the estimated area?

• A3.5 square units
• B3 square units✓
• C2.5 square units
• D4 square units

Answer: Each trapezium's area is found by averaging its two parallel y-values and multiplying by the width, so the first strip from x = 0 to 1 gives (0 + 1) divided by 2, which is 0.5, and the second strip from x = 1 to 2 gives (1 + 4) divided by 2, which is 2.5. Adding these two strips together gives a total estimated area of 3 square units. The other values come from omitting a strip or averaging the y-values incorrectly.

A line has equation y = 4x - 2. What is the gradient of a line that is parallel to it?

• Agradient = -1/4
• Bgradient = 1/4
• Cgradient = 4✓
• Dgrad = -4

Answer: Parallel lines share exactly the same gradient, so a line parallel to y = 4x - 2 must also have gradient 4. A gradient of -4 would instead describe a line with the same steepness but sloping the opposite way, and 1/4 or -1/4 would come from mistakenly taking a reciprocal, which is the rule for perpendicular lines, not parallel ones.

A line has equation y = 2x + 5. What is the gradient of a line perpendicular to it?

• Agradient = -2
• Bgradient = 1/2
• Cgradient = -1/2✓
• Dgradient = 2

Answer: Perpendicular lines have gradients that multiply together to give -1, so the perpendicular gradient is the negative reciprocal of the original gradient. Since the original gradient is 2, its negative reciprocal is -1/2, and checking confirms 2 multiplied by -1/2 equals -1. The other options either keep the same sign as the original gradient or fail to take the reciprocal, so they do not satisfy this condition.

By completing the square, y = x squared - 6x + 5 can be written as y = (x - 3) squared - 4. What is the turning point of the curve?

• A(-3, -4)
• B(6, -4)
• C(3, 4)
• D(3, -4)✓

Answer: In completed-square form y = (x - a) squared + b, the turning point of the curve is always (a, b), because the squared term is smallest, equal to zero, exactly when x = a. Here a = 3 and b = -4, so the turning point is (3, -4). The other coordinate pairs mix up the sign of a or the value of b, which does not match the given completed-square expression.

By completing the square, y = 2x squared + 8x + 3 can be written as y = 2(x + 2) squared - 5. What is the turning point of the curve?

• A(-2, -5)✓
• B(-2, 5)
• C(2, -5)
• D(-4, -5)

Answer: In completed-square form y = k(x - a) squared + b, the turning point is (a, b), reached when the squared bracket equals zero. Writing 2(x + 2) squared as 2(x - (-2)) squared shows a = -2, and the constant term gives b = -5, so the turning point is (-2, -5). The other options either flip the sign of a or the sign of b, which contradicts the bracket and constant shown in the completed-square expression.