Features of a graph

A point where the function value is higher than all nearby values — the graph has a peak there. The function increases up to it and decreases after it.

Maximum point: both coordinates, e.g. (2, 9). Maximum value: just the y-value, e.g. 9. IB questions ask for either — read carefully.

The gradient is zero at every turning point. The tangent line is horizontal there.

Yes — local max/min are only local (in a neighbourhood). The global maximum is the highest point overall, which may be different from any local maximum.

Local maximum at (3, 8). The x-coordinate is 3 and the maximum value is 8. State both.

A coordinate pair: e.g. (−1, −5). Both x and y must be stated. Writing only x = −1 loses the second mark.

1. The minimum value is the y-coordinate. The point (−2, 1) tells you the minimum occurs at x = −2, and the minimum value is 1.

Look for a trough — a point where the graph changes from decreasing (falling) to increasing (rising). The curve dips down then comes back up.

1. Graph f(x) with an appropriate window. 2. Press 2nd → Calc → Maximum. 3. Move left of the peak: press Enter (left bound). 4. Move right of the peak: press Enter (right bound). 5. Press Enter again (guess). GDC shows coordinates.

x = 2.72 (3 s.f.). Then substitute into f to find y, e.g. y = f(2.72). State both coordinates.

IB markschemes award separate marks for each coordinate. Writing only x earns 0 marks for the y-coordinate. Always give both.

The local minimum. Run GDC Minimum with bounds around the other turning point to get its coordinates too.

After 3 seconds the ball reaches its highest point of 36 m above the ground.

Maximum profit of 8000 occurs when 500 units are produced. Producing more or fewer reduces profit.

State what the x-value represents (e.g. time, units) and what the y-value represents (e.g. height, profit) using the context's specific units. E.g. "After 3 hours, temperature reaches its peak of 36°C."

At n = 10 units, profit is at its lowest. The business loses the most money at this production level, and should either produce fewer or more units.

f is increasing on an interval if the output rises as you move left to right: whenever x₁ < x₂, we have f(x₁) < f(x₂). The graph goes upward.

The graph moves downward as you read from left to right — outputs fall as inputs increase.

Increasing — the function rises up to the maximum, then begins decreasing after it.

Inequalities (e.g. 1 < x < 4) and interval notation (e.g. (1, 4)) are both accepted. Write whichever matches the question's phrasing.

f is increasing on −2 < x < 1 (or [−2, 1]).

Increasing: x < 2 and x > 5. Decreasing: 2 < x < 5.

An inequality or interval notation including both endpoints. E.g. 2 ≤ x ≤ 5 or [2, 5]. The interval must refer to x-values (inputs), not y-values.

For x < 0. The parabola falls from left toward x = 0, then rises for x > 0. The minimum is at (0, 0).

"Increasing at a point" is meaningless. Increasing is a property of an interval, not a single point. Write "f is increasing for x > 3" or "f is increasing on (1, 3)".

The answer should be an interval of x-values, not y-values. Correct: e.g. "1 < x < 3." The y-values (4 to 9) are outputs, not the interval.

IB accepts both x < 2 and x ≤ 2 for the increasing interval up to a maximum. Either strict or inclusive inequalities are fine unless the question specifies.

Increasing everywhere — gradient is 3 > 0, so the output always rises as x increases. No turning points.

The temperature rises during the first 5 hours.

An interval of t-values (the input variable), e.g. "3 < t < 8 hours." Not y-values. Use the same variable as the context.

n = 200 is where the profit function has its local maximum — the production level giving the greatest profit.

State whether f is increasing or decreasing, and whether it approaches a fixed value (asymptote) or continues without bound. E.g. "f is decreasing and approaches y = 3."

A horizontal line y = k that the graph approaches as x → ∞ or x → −∞, but (usually) never reaches or crosses.

Exponential: y = a · bˣ. As x → −∞ (for b > 1) or x → ∞ (for 0 < b < 1), the output approaches 0.

Write it as a full equation: e.g. y = 3. Not just "3" — the y = must be included.

As x gets very large (or very negative), the output of f gets arbitrarily close to the asymptote value — but the curve never quite touches that line.

y = 5. As x → −∞, 3 · 2ˣ → 0, so f(x) → 5. The +5 shifts the asymptote up from y = 0 to y = 5.

Range is f(x) > 4. The function always stays above y = 4 (never equals it), so 4 is excluded from the range.

Horizontal asymptote y = 10. As x → ∞, 100 · 0.5ˣ → 0, so f(x) → 10 from above.

The function never reaches the asymptote value, so that value is excluded from the range. E.g. if asymptote y = 3 and function approaches from above, range is f(x) > 3.

A vertical line x = a where the function is undefined and its output grows to ±∞ as x approaches a from either side.

At x = 3 — the denominator is zero there, so the function is undefined. The graph blows up to ±∞ near x = 3.

x-intercept: the curve actually touches or crosses y = 0. Asymptote y = 0: the curve approaches y = 0 but never reaches it.

Set denominator = 0: 2x + 4 = 0 → x = −2. Vertical asymptote at x = −2.

How f(x) behaves as x → ∞ or x → −∞ — whether it grows, falls, or approaches a limiting value (asymptote).

As x → ∞, 0.5ˣ → 0, so f(x) → 0. The graph approaches the asymptote y = 0 from above and decreases toward it.

f(x) → ∞ as x → ∞. There is no horizontal asymptote — the function grows forever.

State whether f increases, decreases, or approaches a fixed value. If it approaches a value, give the equation of the asymptote. Use context language if relevant.