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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) = 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 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 at 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 $y=1.5 x^{2}-2.5 x+3$ 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 4

HLPaper 1

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

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 graph of the original and transformed functions over the interval $0 \leq x \leq 2\pi$ .

[2]

Describe the effect of the transformation on the amplitude and 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.

[2]

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]

SL 2.3—Graph of a function Questionbank