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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

$f(x) = x^4 - 26x^2 + 25$

This function represents the plant's growth rate over time, where $x \ge 0$ is the number of weeks since planting.

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

[2 marks]

[2]

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

[2 marks]

[2]

Sketch the graph of the function $f(x)$ for $x \ge 0$ , marking the zeros found in Part 1 and describing the behavior of the plant's growth rate around each zero.

[4 marks]

[4]

Question 2

SLPaper 1

Consider the function $f(x) = ax^2 + bx + c$ . 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 $(p, 0)$ .

Graph of f(x)

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 $y = 1.5x^{2} - 2.5x + 3$ intersect at $(-1, 7)$ and $(2, 4)$ , as shown in the following diagrams.

In Diagram 1, the region enclosed by the line $y = 6 - x$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 1

In Diagram 2, the region enclosed by the curve $y = 1.5x^{2} - 2.5x + 3$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 2

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 the shaded region in Diagram 2.

[3]

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

[1]

Question 5

SLPaper 1

A water tank initially contains 5 litres of water. Water is being drained from the tank at a constant rate of 1 litre per minute. The function $g(x)=5-x$ represents the amount of water (in litres) 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 $f(x) = -\frac{x^2}{50} + 2x + 30$ , for $0 \le x \le 70$ .

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]

Linda estimates the area of the field to be $4701.93\text{ m}^2$ using the trapezoidal rule. 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]

A building with a square foundation $EFGH$ is to be constructed in the field such that $E$ lies on $[AD]$ , $F$ lies on the northern boundary, and $G$ and $H$ lie on the eastern boundary $[CD]$ .

Find the $x$ -coordinate of point $E$ for the square foundation.

[1]

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

[2]

Sketch the graph of the function $g(x)=2+3e^{-x}$ , 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

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

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.

[3]

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

$f(x) = x^4 - 26x^2 + 25$

This function represents the plant's growth rate over time, where $x \ge 0$ is the number of weeks since planting.

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

[2 marks]

[2]

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

[2 marks]

[2]

Sketch the graph of the function $f(x)$ for $x \ge 0$ , marking the zeros found in Part 1 and describing the behavior of the plant's growth rate around each zero.

[4 marks]

[4]

Question 2

SLPaper 1

Graph of f(x)

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 $y = 1.5x^{2} - 2.5x + 3$ intersect at $(-1, 7)$ and $(2, 4)$ , as shown in the following diagrams.

In Diagram 1, the region enclosed by the line $y = 6 - x$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 1

In Diagram 2, the region enclosed by the curve $y = 1.5x^{2} - 2.5x + 3$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 2

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 the shaded region in Diagram 2.

[3]

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

[1]

Question 5

SLPaper 1

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.

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]

Linda estimates the area of the field to be $4701.93\text{ m}^2$ using the trapezoidal rule. 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]

A building with a square foundation $EFGH$ is to be constructed in the field such that $E$ lies on $[AD]$ , $F$ lies on the northern boundary, and $G$ and $H$ lie on the eastern boundary $[CD]$ .

Find the $x$ -coordinate of point $E$ for the square foundation.

[1]

Find the 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 equation of the horizontal asymptote of the function, which indicates the minimum amount of substance that can remain.

[2]

Sketch the graph of the function $g(x)=2+3e^{-x}$ , 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

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

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.

[3]

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

$f(x) = x^4 - 26x^2 + 25$

This function represents the plant's growth rate over time, where $x \ge 0$ is the number of weeks since planting.

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

[2 marks]

[2]

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

[2 marks]

[2]

Sketch the graph of the function $f(x)$ for $x \ge 0$ , marking the zeros found in Part 1 and describing the behavior of the plant's growth rate around each zero.

[4 marks]

[4]

Question 2

SLPaper 1

Graph of f(x)

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 $y = 1.5x^{2} - 2.5x + 3$ intersect at $(-1, 7)$ and $(2, 4)$ , as shown in the following diagrams.

In Diagram 1, the region enclosed by the line $y = 6 - x$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 1

In Diagram 2, the region enclosed by the curve $y = 1.5x^{2} - 2.5x + 3$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 2

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 the shaded region in Diagram 2.

[3]

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

[1]

Question 5

SLPaper 1

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.

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]

Linda estimates the area of the field to be $4701.93\text{ m}^2$ using the trapezoidal rule. 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]

A building with a square foundation $EFGH$ is to be constructed in the field such that $E$ lies on $[AD]$ , $F$ lies on the northern boundary, and $G$ and $H$ lie on the eastern boundary $[CD]$ .

Find the $x$ -coordinate of point $E$ for the square foundation.

[1]

Find the 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 equation of the horizontal asymptote of the function, which indicates the minimum amount of substance that can remain.

[2]

Sketch the graph of the function $g(x)=2+3e^{-x}$ , 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

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

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.

[3]

SL 2.4—Features of a graph Questionbank