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SL 2.4—Features of a graph - IB Questionbank | RevisionDojo

Practice SL 2.4—Features of a graph with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 2

A team of environmental scientists is studying the growth of a certain plant species, which can be modeled by the function

$f(x) = x^4 - 6x^2 + 5$

This function represents the plant's growth rate over time, where $x$ is the number of weeks since planting.

Find all real zeros of the function, which represent the weeks when the growth rate is zero.

[2]

Describe the end behavior of $f(x)$ as $x \to \infty$ .

[2]

Sketch the graph of the function $f(x) = x(x-2)(x+1)$ , marking the zeros and describing the behavior of $f(x)$ around each zero in the context of plant growth.

The zeros occur at $x = -1$ , $x = 0$ , and $x = 2$ .

For each zero:

Describe whether $f(x)$ changes from positive to negative or negative to positive
Explain what this means in terms of plant growth rate

Label all important features on your graph.

[4]

Question 2

SLPaper 1

Consider the function f(x) = ax² + bx + c. The graph of y = f(x) is shown in the diagram. The vertex of the graph has coordinates (0.5, -12.5). The graph intersects the x-axis at two points, (-2, 0) and (p, 0).

Find the value of p.

[2]

Find the value of a.

[2]

Find the value of b.

[2]

Find the value of c.

[2]

Question 3

SLPaper 2

The depth of water in a port is modelled by the function $d(t) = p\cos qt + 7.5$ , for $0 \leqslant t \leqslant 12$ , where $t$ is the number of hours after high tide.

At high tide, the depth is 9.7 metres.

At low tide, which is 7 hours later, the depth is 5.3 metres.

Find the value of $p$ .

[2]

Find the value of $q$ .

[2]

Use the model to find the depth of the water 10 hours after high tide.

[2]

Question 4

SLPaper 1

The graphs of $y=6-x$ and $y=1.5 x^{2}-2.5 x+3$ intersect at $(2,4)$ and $(-1,7)$ , as shown in the following diagrams. In diagram 1, the region enclosed by the lines $y=6-x, x=-1, x=2$ and the $x$ -axis has been shaded. In diagram 2, the region enclosed by the curve $y=1.5 x^{2}-2.5 x+3$ , and the lines $x=-1$ , $x=2$ and the $x$ -axis has been shaded.

Calculate the area of the shaded region in diagram 1.

[3]

Write down an integral for the area of the shaded region in diagram 2.

[1]

Calculate the area of this region.

[3]

Hence, determine the area enclosed between $y=6-x$ and $y=1.5 x^{2}-2.5 x+3$ .

[1]

Question 5

SLPaper 1

A water tank initially contains 5 liters of water. Water is being drained from the tank at a constant rate of 1 liter per minute. The function ${g(x)=5-x}$ represents the amount of water (in liters) left in the tank after ${x}$ minutes.

Determine the domain and range of g(x) in the context of this problem.

[2]

Identify any intercepts on the graph and interpret their meaning in the context of the problem.

[2]

Question 6

SLPaper 2

Linda owns a field, represented by the shaded region R. The plan view of the field is shown in the following diagram, where both axes represent distance and are measured in metres. The segments [AB], [CD] and [AD] respectively represent the western, eastern and southern boundaries of the field. The function, f(x), models the northern boundary of the field between points B and C and is given by f(x) = -x^2/50 + 2x + 30, for 0 ≤ x ≤ 70

Find f'(x).

[1]

Hence find the coordinates of the point on the field that is furthest north.

[5]

Write down the integral which can be used to find the area of the shaded region R.

[1]

Find the area of Linda's field.

[4]

Calculate the percentage error in Linda's estimate.

[1]

Suggest how Linda might be able to reduce the error whilst still using the trapezoidal rule.

[3]

Find the x-coordinate of point E for the largest area of the square foundation of building EFGH.

[1]

Find the largest area of the foundation.

[5]

Question 7

SLPaper 2

A biochemist is analyzing the decay of a radioactive substance, which can be modeled by the function

$f(x) = 100e^{-0.5x}$

Here, $f(x)$ represents the amount of substance remaining after $x$ hours.

Identify the initial value of $f(x)$ and the asymptote of the function, which indicates the minimum amount of substance that can remain.

[2]

A radioactive substance decays exponentially. Initially there are 100 grams of the substance. After 4 hours there are 70.7 grams remaining.

Determine the rate of decay and find the time at which $f(x) = 50$ , indicating when half of the substance has decayed.

[2]

Sketch the graph of the function $f(x)=2+3e^{-x}$ , labeling the initial value and the asymptote.

[3]

Question 8

SLPaper 1

A function is defined by $f(x)=2-\frac{12}{x+5}$ for $-7 \leq x \leq 7, x \neq -5$ .

Find the range of $f$ .

[3]

Find the value of $f^{-1}(0)$ .

[2]

Question 9

SLPaper 1

The height of a baseball after it is hit by a bat is modelled by the function

$h (t) = - 4.8 t^{2} + 21 t + 1.2$

where $h (t)$ is the height in metres above the ground and $t$ is the time in seconds after the ball was hit.

Write down the height of the ball above the ground at the instant it is hit by the bat.

[1]

Find the value of $t$ when the ball hits the ground.

[2]

State an appropriate domain for $t$ in this model.

[2]

Question 10

SLPaper 2

A landscape architect is designing a parabolic garden feature, represented by the function $f(x) = (x - 2)^2 - 3$

This function describes the height of the garden feature at different horizontal positions $x$ .

Identify the vertex of the function and determine the range of heights for the garden feature.

[2]

Describe the transformation applied to the basic function $f(x) = x^2$ to obtain the given function, which reflects the design changes.

[2]

Sketch the graph of the garden feature, labeling the vertex and any intercepts.

h(x) = -2(x - 3)^2 + 8

[3]

SL 2.4—Features of a graph Questionbank