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IB Maths AA SL · Unit 2: Functions

IB Maths AA SL Graphing Functions Questions

Exam-style IB Maths AA SL graphing functions questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.

Practise Graphing Functions questions → AA SL formula booklet

What's examined in AA SL graphing functions

The question bank covers these graphing functions question types (number of questions in brackets):

Graphing Functions worked examples

Worked example 1: Finding intercepts and asymptotes algebraically · easy

Let $g(x) = \frac{4x - 5}{2x + 3}$ for $x \neq -1.5$. Find the exact equations of the vertical and horizontal asymptotes, and the exact coordinates of the $x$ and $y$-intercepts.

Solution

1. Vertical asymptote: set denominator zero — $\mathbf{x = -1.5}$.

2. Horizontal asymptote: ratio of leading coefficients — $\mathbf{y = 2}$.

3. $y$-intercept at $x = 0$: $g(0) = -\frac{5}{3}$, giving $\mathbf{\left(0, -\frac{5}{3}\right)}$.

4. $x$-intercept at numerator $= 0$: $4x = 5 \implies x = 1.25$, giving $\mathbf{(1.25, 0)}$.

Examiner tip: For any rational function $y = \frac{ax+b}{cx+d}$, the horizontal asymptote is simply $y = \frac{a}{c}$.

Worked example 2: Solving intersections graphically · medium

Consider $f(x) = 5 - x^2$ and $g(x) = e^{0.5x}$. Use your GDC to find the $x$-coordinates of their intersections to 3 s.f.

Solution

1. Access the GDC graph menu and input Y1 $= 5 - x^2$, Y2 $= e^{0.5x}$.

2. Set V-Window to a standard range (e.g. $-4 \le x \le 4$) and draw.

3. Use G-Solv $\to$ ISCT to find the first intersection: $\mathbf{x = -2.14}$ (3 s.f.).

4. Navigate right for the second: $\mathbf{x = 1.83}$ (3 s.f.).

Examiner tip: Always check your V-Window if intersections are not visible. Never try to solve mixed exponential-polynomial equations algebraically.

Worked example 3: Intersecting a function with its inverse · hard

Consider $f(x) = \ln(x+2)$ for $x > -2$. The graph of $y = f(x)$ intersects $y = f^{-1}(x)$ exactly twice. Find both intersection coordinates to 3 s.f.

Solution

1. Recall: $f(x)$ and $f^{-1}(x)$ always intersect on the line $y = x$.

2. Simplify the search to solving $\ln(x+2) = x$.

3. Input into GDC: Y1 $= \ln(x+2)$, Y2 $= x$.

4. Use ISCT to find first point: $\mathbf{(-1.84, -1.84)}$ (3 s.f.).

5. Navigate for second point: $\mathbf{(1.15, 1.15)}$ (3 s.f.).

Examiner tip: Never find the algebraic inverse first if you only need intersection points — solving $f(x) = x$ is much faster and bypasses complex algebraic manipulation.

Try these IB Maths AA SL graphing functions questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.

Question 1 · easy · 4 marks · Paper 2

Consider the quadratic function \(f(x) = 2x^2 - 5x - 12\).

  1. Use your graphic display calculator to find the coordinates of the \(x\)-intercepts.

  2. Find the coordinates of the vertex of the graph of \(f\).

Attempt it and see the mark scheme →

Question 2 · medium · 6 marks · Paper 2

The graph of the rational function \(f(x) = \frac{ax - 4}{3x + b}\) has a vertical asymptote at \(x = 2\) and a horizontal asymptote at \(y = -1\).

  1. Deduce the value of \(a\) and the value of \(b\).

  2. Find the coordinates of the \(x\)-intercept of the graph of \(f\).

Attempt it and see the mark scheme →

Question 3 · hard · 5 marks · Paper 2

Let \(f(x) = \sin(e^x)\) for \(0 \le x \le 1.5\). The angle \(x\) is measured in radians.

  1. Sketch the graph of \(f\) on the given domain.

  2. Find the exact \(x\)-intercept of the graph of \(f\).

Attempt it and see the mark scheme →

All 15 graphing functions questions with mark schemes →

FAQ

How many IB Maths AA SL graphing functions questions are there?

There are 15 exam-style graphing functions questions in the AA SL question bank (Paper 2: 15), graded 4 easy, 5 medium, 4 hard, 2 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is graphing functions on Paper 1 or Paper 2?

In the question bank these questions are set as Paper 2 questions.

Where can I get the mark schemes?

Open the AA SL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

More AA SL Unit 2 topics

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