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SL 2.3—Graph of a function - IB Questionbank | RevisionDojo

Practice SL 2.3—Graph of a function 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 2

Consider a quadratic function that has a $y$ -intercept of 1, one $x$ -intercept at 1, and the $x$ -coordinate of its vertex is 3. The equation of the quadratic function is in the form $y = ax^2 + bx + c$ .

Write down the value of $c$ .

[1]

Write down the second $x$ -intercept of the function.

[2]

Find the values of $a$ and $b$ .

[4]

Question 3

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 4

HLPaper 1

The function $f$ is defined by $f(x) = 2x^3 + 5$ , $-2 \le x \le 2$ .

Write down the range of $f$ .

[2]

Find an expression for $f^{-1}(x)$ .

[2]

Write down the domain and range of $f^{-1}$ .

[2]

Question 5

SLPaper 2

Let $f(x) = x^2 + 2x + 1$ and $g(x) = x - 5$, for $x \in \mathbb{R}$.

Find $(g \circ f)(x)$.

[2]

Find $f(8)$.

[2]

Solve $(g \circ f)(x) = 0$.

[3]

Question 6

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 7

SLPaper 2

A music producer is analyzing sound waves and transforms the function $f(x) = \sin(x)$ by reflecting it over the $x$ -axis and stretching it vertically by a factor of 2.

Write the equation of the transformed function.

[2]

Sketch the graphs of the original function and the transformed function over the interval $0 \leq x \leq 2\pi$ .

[2]

Describe the effect of the transformation on the amplitude and the period of the sine wave.

[2]

Question 8

HLPaper 1

Two schools are represented by points $A(2,20)$ and $B(14,24)$ on the graph below. A road, represented by the line $R$ with equation $-x+y=4$ , passes near the schools. An architect is asked to determine the location of a new bus stop on the road such that it is the same distance from the two schools.

Find the equation of the perpendicular bisector of $[AB]$ . Give your equation in the form $y=mx+c$ .

[5]

Determine the coordinates of the point on $R$ where the bus stop should be located.

[3]

Question 9

SLPaper 2

A civil engineer is designing a parabolic arch for a bridge, which can be expressed in the form $g(x) = b(x - q)(x - 4)$ . The graph of $g$ has an axis of symmetry at $x = 3$ and a $y$ -intercept at $(0, -8)$ .

Find the value of $q$ .

[2]

Find the value of $b$ .

[2]

The line $y = mx - 7$ is a tangent to the curve of $g$ . Find the value(s) of $m$ .

[3]

Question 10

SLPaper 2

Two towns, $P(-4, -3)$ and $Q(5, 9)$ , are connected by a road. Another road runs along the perpendicular bisector of the segment connecting towns P and Q.

Find the equation of this road.

[3]

Determine whether it passes through the point $R(2, 1)$ .

[2]

State whether the bisector is parallel or perpendicular to the $x$ -axis or $y$ -axis.

[1]

Paper

Level

Question 1

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 2

SLPaper 2

Write down the value of $c$ .

[1]

Write down the second $x$ -intercept of the function.

[2]

Find the values of $a$ and $b$ .

[4]

Question 3

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 4

HLPaper 1

The function $f$ is defined by $f(x) = 2x^3 + 5$ , $-2 \le x \le 2$ .

Write down the range of $f$ .

[2]

Find an expression for $f^{-1}(x)$ .

[2]

Write down the domain and range of $f^{-1}$ .

[2]

Question 5

SLPaper 2

Let $f(x) = x^2 + 2x + 1$ and $g(x) = x - 5$, for $x \in \mathbb{R}$.

Find $(g \circ f)(x)$.

[2]

Find $f(8)$.

[2]

Solve $(g \circ f)(x) = 0$.

[3]

Question 6

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 7

SLPaper 2

A music producer is analyzing sound waves and transforms the function $f(x) = \sin(x)$ by reflecting it over the $x$ -axis and stretching it vertically by a factor of 2.

Write the equation of the transformed function.

[2]

Sketch the graphs of the original function and the transformed function over the interval $0 \leq x \leq 2\pi$ .

[2]

Describe the effect of the transformation on the amplitude and the period of the sine wave.

[2]

Question 8

HLPaper 1

Find the equation of the perpendicular bisector of $[AB]$ . Give your equation in the form $y=mx+c$ .

[5]

Determine the coordinates of the point on $R$ where the bus stop should be located.

[3]

Question 9

SLPaper 2

Find the value of $q$ .

[2]

Find the value of $b$ .

[2]

The line $y = mx - 7$ is a tangent to the curve of $g$ . Find the value(s) of $m$ .

[3]

Question 10

SLPaper 2

Two towns, $P(-4, -3)$ and $Q(5, 9)$ , are connected by a road. Another road runs along the perpendicular bisector of the segment connecting towns P and Q.

Find the equation of this road.

[3]

Determine whether it passes through the point $R(2, 1)$ .

[2]

State whether the bisector is parallel or perpendicular to the $x$ -axis or $y$ -axis.

[1]

Paper

Level

Question 1

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 2

SLPaper 2

Write down the value of $c$ .

[1]

Write down the second $x$ -intercept of the function.

[2]

Find the values of $a$ and $b$ .

[4]

Question 3

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 4

HLPaper 1

The function $f$ is defined by $f(x) = 2x^3 + 5$ , $-2 \le x \le 2$ .

Write down the range of $f$ .

[2]

Find an expression for $f^{-1}(x)$ .

[2]

Write down the domain and range of $f^{-1}$ .

[2]

Question 5

SLPaper 2

Let $f(x) = x^2 + 2x + 1$ and $g(x) = x - 5$, for $x \in \mathbb{R}$.

Find $(g \circ f)(x)$.

[2]

Find $f(8)$.

[2]

Solve $(g \circ f)(x) = 0$.

[3]

Question 6

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 7

SLPaper 2

A music producer is analyzing sound waves and transforms the function $f(x) = \sin(x)$ by reflecting it over the $x$ -axis and stretching it vertically by a factor of 2.

Write the equation of the transformed function.

[2]

Sketch the graphs of the original function and the transformed function over the interval $0 \leq x \leq 2\pi$ .

[2]

Describe the effect of the transformation on the amplitude and the period of the sine wave.

[2]

Question 8

HLPaper 1

Find the equation of the perpendicular bisector of $[AB]$ . Give your equation in the form $y=mx+c$ .

[5]

Determine the coordinates of the point on $R$ where the bus stop should be located.

[3]

Question 9

SLPaper 2

Find the value of $q$ .

[2]

Find the value of $b$ .

[2]

The line $y = mx - 7$ is a tangent to the curve of $g$ . Find the value(s) of $m$ .

[3]

Question 10

SLPaper 2

Two towns, $P(-4, -3)$ and $Q(5, 9)$ , are connected by a road. Another road runs along the perpendicular bisector of the segment connecting towns P and Q.

Find the equation of this road.

[3]

Determine whether it passes through the point $R(2, 1)$ .

[2]

State whether the bisector is parallel or perpendicular to the $x$ -axis or $y$ -axis.

[1]

SL 2.3—Graph of a function Questionbank