SL 2.4—Features of a graph Questionbank

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 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 2

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 3

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 4

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 5

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 6

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]

Question 7

SLPaper 1

In an experiment, a number of fruit flies are placed in a container. The population of fruit flies, $P$ , increases and can be modelled by the function

$P(t) = 12 \times 3^{0.498t}, \quad t \geqslant 0$

where $t$ is the number of days since the fruit flies were placed in the container.

Find the number of fruit flies which were placed in the container.

[2]

Find the number of fruit flies that are in the container after 6 days.

[2]

The maximum capacity of the container is 8000 fruit flies.

Find the number of days until the container reaches its maximum capacity.

[2]

Question 8

SLPaper 2

Consider the function defined by: $f(x) = \frac{2x + 1}{x - 3}$

Calculate the $x$ - and $y$ -intercepts of the graph of $f(x)$ .

[2]

Determine the equations of the vertical and horizontal asymptotes of the function.

[2]

Sketch the graph of $f(x)$ , showing all key features including intercepts and asymptotes.

[4]

Question 9

SLPaper 2

A function $f$ is given by $f(x) = (2x + 2)(5 - x^2)$ .

The graph of the function $g(x) = 5^x + 6x - 6$ intersects the graph of $f$ .

Expand the expression for $f(x)$ .

[1]

Find $f'(x)$ .

[3]

Use your answer to part 2 to find the values of $x$ for which $f$ is increasing.

[3]

Find the exact value of each of the zeros of $f$ .

[3]

Write down the coordinates of the point of intersection.

[2]

Draw the graph of $f$ for $-3 \leqslant x \leqslant 3$ and $-40 \leqslant y \leqslant 20$ . Use a scale of 2 cm to represent 1 unit on the $x$ -axis and 1 cm to represent 5 units on the $y$ -axis.

[4]

Question 10

SLPaper 2

Consider the function $f(x)=x^2+x+\frac{50}{x}, \quad x \neq 0$.

Find $f(1)$.

[2]

Solve $f(x)=0$.

[2]

The graph of $f$ has a local minimum at point $\text{A}$.

Find the coordinates of $\text{A}$.

[2]