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SL 2.3—Graphing - IB Questionbank | RevisionDojo

Practice SL 2.3—Graphing with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. 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 $f(x)=1-x^{2}$ , for $-2 \leq x \leq 2$ .

Find the values of $x$ for which $f(x)=0$ .

[2]

Sketch the graph of $f$ on the grid provided below.

[3]

Question 2

SLPaper 1

Consider the function $f(x)=\frac{3 x+2}{x-2}$ , where $x \in \mathbb{R}, x \neq 2$ . The graph of $y=f(x)$ has a vertical asymptote and a horizontal asymptote.

State the equation of the vertical asymptote.

[1]

State the equation of the horizontal asymptote.

[1]

On the set of axes below, sketch the graph of $y=f(x)$ .

Clearly show the asymptotes and any points where the graph intersects the axes.

[3]

Hence, solve the inequality $0<\frac{3 x+2}{x-2} \leq 1$ .

[2]

Question 3

SLPaper 1

Consider the quadratic function $h(t)=a t^{2}+c$ , where $t, a, c \in \mathbb{R}$ . The point (2, 7) lies on the graph of $h$ .

Find the values of $a$ and $c$ , given that the vertex of the parabola is at ( 0,3 ). marks]

[3]

The graph of $f(t)=t^{2}$ is transformed to obtain the graph of $h$ . Describe the transformation in terms of scaling and translation.

[3]

On the axes below, sketch the graph of $y=h(t)$ for $-3 \leq t \leq 3$ and $0 \leq y \leq 10$ . Clearly show the vertex and the point $(2,7)$ .

[3]

Question 4

SLPaper 1

Consider the function $f(x)=4-x^{2}$ , for $-3 \leq x \leq 3$ .

Find the values of $x$ for which $f(x)=0$ .

[2]

Sketch the graph of $f$ on the grid provided below.

[3]

Question 5

SLPaper 2

A quadratic model $f(x)=ax^2+bx+c$ is fitted to three measured points $P(1,0.5+\ln 6)$ , $Q(3,1.5+\ln 9)$ , $R(6,\ln 5)$ .

Determine $a,b,c$ (numerically). Give your values correct to 3 significant figures.

[5]

Write $f(x)$ in vertex form $f(x)=a(x-h)^2+k$ . Hence state the vertex and the range of $f$ . Give $h,k$ to 3 d.p.

[4]

Let $g(x)=\ln(x+2)+1$ . Solve $f(x)=g(x)$ for $x\ge 0$ to 3 d.p., and justify the number of solutions.

[6]

Solve $f(x)\ge g(x)$ for $x\ge 0$ .

[3]

Question 6

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+9),\quad x>-9,\ k>0.$ It is given that $(f\circ g)(1)=7$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites and their maximal domains.

[4]

Solve $(x+9)^k=\ln(9+e^{kx})$ for $x\ge0$ (3 d.p.) and justify uniqueness.

[7]

Find $(f\circ g)^{-1}(y)$ and state domain/range.

[4]

Question 7

SLPaper 2

Consider the function $f(x)=\sin (x)+\frac{x}{2}$ , for $x(0<x<2 \pi)$ .

Find the coordinates of the local maximum of the graph of $y=f(x)$ , giving the $x$ tothreedecimalplaces.

[4]

On the axes below, sketch the graph of $y=f(x)$ for $0<x<2 \pi$ , showing clearly the local maximum, any axis intercepts, and the behavior as $x \rightarrow 0^{+}$ and $x \rightarrow 2 \pi^{-}$ .

[4]

Hence, or otherwise, solve the inequality $f(x)>1$ for $0<x<2 \pi$ , giving your answer

[3]

Question 8

SLPaper 2

A rational function has vertical asymptote $x=4$ and horizontal asymptote $y=1$ . It passes through $(0,-1)$ and $(5,9)$ .

Show $f(x)=1+\dfrac{k}{x-4}$ and determine $k$ . Hence express $f(x)=\dfrac{ax+b}{cx+d}$ with integers.

[4]

Determine domain, range, intercepts, asymptotes, and the sign of $f(x)-1$ on $(- \infty,4)$ , $(4,\infty)$ .

[5]

Solve $f(x)=g(x)$ to 3 d.p. and justify one solution in each of $(-1,4)$ and $(4,\infty)$ .

[6]

Describe the transformations from $y=1/x$ to $y=f(x)$ .

[2]

Question 9

SLPaper 2

A rational function has vertical asymptote $x=4$ and horizontal asymptote $y=7$ . It passes through the points $(0,5)$ and $(6,11)$ .

Show that the function can be written in the form $f(x)=7+\frac{k}{x-4}$ for some constant $k$ , and determine $k$ . Hence write $f(x)$ explicitly as $\dfrac{ax+b}{cx+d}$ with integer coefficients.

[4]

Determine all key features of $f$ : domain, range, intercepts, asymptotes, and the sign of $f(x)-7$ on $(- \infty,4)$ and $(4,\infty)$ .

[5]

Solve $f(x)=g(x)$ to three decimal places and justify exactly one solution in each of $(-1,4)$ and $(4,\infty)$ .

[6]

Describe transformations mapping $y=1/x$ to $y=f(x)$ .

[2]

Question 10

SLPaper 2

A quadratic model $f(x)=ax^2+bx+c$ is fitted to three measured points $P(1,1+\ln 2)$ , $Q(3,e)$ , $R(6,\ln 2)$ .

Determine $a,b,c$ (numerically). Give your values correct to 3 significant figures.

[5]

Write $f(x)$ in vertex form $f(x)=a(x-h)^2+k$ . Hence state the vertex and the range of $f$ . Give $h,k$ to 3 d.p.

[4]

Let $g(x)=\ln(x+2)+1$ . Solve $f(x)=g(x)$ for $x\ge 0$ to 3 d.p., and justify the number of solutions.

[6]

Solve $f(x)\ge g(x)$ for $x\ge 0$ .

[3]

SL 2.3—Graphing Questionbank