Skills Checklist | IB Math AAHL

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Skill Checklist

Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.

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📖 = included in formula booklet • 🚫 = not in formula booklet

Track your progress:

Don't know

Working on it

Confident

📖 = included in formula booklet • 🚫 = not in formula booklet

Skill Checklist

Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.

18 Skills Available

Track your progress:

Don't know

Working on it

Confident

📖 = included in formula booklet • 🚫 = not in formula booklet

Track your progress:

Don't know

Working on it

Confident

📖 = included in formula booklet • 🚫 = not in formula booklet

Linear Transformations

8 skills

Vertical translation of graphs y=f(x)+b

SL 2.11

The graph of y=f(x)+b can be obtained from the graph of y=f(x) by a vertical translation b units upwards (if b<0, the transformation may also be called a translation ∣b∣ units down).

Horizontal translation

SL 2.11

The graph of y=f(x−a) can be obtained from the graph of y=f(x) by a horizontal translation a units to the right (if a<0, the transormation may also be called a translation ∣a∣ units to the left).

Translation by a vector

SL 2.11

If a point P is translated by a vector (ab), apply a translation a units to the right and b units up:

P(x,y)P′(x−a,y+b).

Vertical scaling of graphs y=af(x)

SL 2.11

The graph of y=pf(x) can be obtained from the graph of y=f(x) by a vertical stretch with scale factor p.

Horizontal scaling

SL 2.11

The graph of y=f(qx) can be obtained from the graph of y=f(x) by a horizontal stretch with scale factor q1.

Reflection in the x-axis

SL 2.11

The graph of y=−f(x) can be obtained from the graph of y=f(x) by a reflection in the x-axis.

Reflection in the y-axis

SL 2.11

The graph of y=f(−x) can be obtained from the graph of y=f(x) by a reflection in the y-axis.

Combined horizontal scale and translation f(ax+b)

AHL 2.16

The graph of y=f(ax+b)=f[a(x+ab)] can be obtained from the graph of y=f(x) by a horizontal translation b units to the left, followed by a horizontal stretch with scale factor a1.

OR a horizontal stretch with scale factor a1 followed by a horizontal translation ab units to the left.

Rational functions

4 skills

The reciprocal function 1/x

SL AA 2.8

The reciprocal function is defined by f(x)=x1.

Notice that f(x) is not defined for x=0. In fact, since x1 gets very large as x approaches 0, f(x) has a vertical asymptote at x=0.

And since for very large x, x1 approaches zero, there is also a horizontal asymptote y=0.

Notice also that x11=x, so f(x)=x1 is self-inverse.

Graphs of linear rational functions

SL AA 2.8

A linear rational function has the form

f(x)=cx+dax+b

When the denominator is zero the graph will have a vertical asymptote:

cx+d=0⇒x=−cd🚫

And as x gets very large, the +b and +d can be ignored:

y=f(x)≈cxax=ca🚫

So there is a horizontal asymptote at y=ca.

Rationals with oblique asymptotes

AHL 2.13

When a rational function is of the form

f(x)=dx+eax2+bx+c

there is a vertical asymptote at

x=−de🚫

By performing polynomial division, we can find the oblique asymptote of f(x), which has the equation

y=dax+C

for some constant C determined during the polynomial division.

Rationals with quadratic denominator

AHL 2.13

When a rational function is of the form

f(x)=cx2+dx+eax+b

There will be vertical asymptotes when the quadratic cx2+dx+e=0.

The horizontal asymptote will simply be y=0 since cx2 dominates ax when x is very large.

Additionally, there will be an x-intercept at x=−ab, when the numerator changes sign.

Note that if the numerator and denominator share a root (ie x=−ab is a root of the denominator), then there will only be one asymptote and a "hole" on the x-axis. This has never shown up on exams.

Further Transformations

4 skills

Graphing f(|x|)

AHL 2.16

To obtain the graph of f(∣x∣), you can think of it as:

For x≥0: ∣x∣=x, so f(∣x∣)=f(x).
For x<0: ∣x∣=−x, so f(∣x∣)=f(−x).

Practically, you take the portion of y=f(x) for x≥0 and reflect it across the y-axis to fill in the x<0 side. The right side of the original graph becomes the entire graph of f(∣x∣).

Graphing |f(x)|

AHL 2.16

To obtain the graph of ∣f(x)∣, take the graph of f(x) and:

Leave all points where f(x)≥0 as they are, since ∣f(x)∣=f(x) in that region.
Reflect any parts of the graph where f(x)<0 above the x-axis, because ∣f(x)∣=−f(x) whenever f(x) is negative.

Effectively, every negative y-value becomes positive, mirroring the portion of the curve below the x-axis to above it.

Notice the "sharp corners" where the function touches the x-axis.

Graphing [f(x)]^2

AHL 2.16

To sketch [f(x)]2 from f(x):

Square all y-values: for each x, the new y-value is (f(x))2.
Because y2≥0, the entire graph of [f(x)]2 is on or above the x-axis.
Points where f(x)=0 remain on the x-axis.
Any minima below the x-axis become maxima above the x-axis.
Any maxima below the x-axis become minima above the x-axis.
If ∣f(x)∣<1, squaring makes it closer to 0; if ∣f(x)∣>1, squaring makes it larger.

Negative parts of f(x) become positive when squared, so everything below the x-axis is reflected above it, with the magnitude of y-values adjusted according to the square.

Notice the "rounded corners" where the function touches the x-axis, as opposed to the sharp corners for ∣f(x)∣.

Graphing 1/f(x)

AHL 2.16

To sketch f(x)1 from f(x):

At every x-value, take the reciprocal of the original y-value: y→y1.
Points where f(x)=0 become vertical asymptotes, since 01 is undefined.
If ∣f(x)∣ is large, then f(x)1 is small (close to the x-axis). Thus, vertical asymptotes of f(x) become horizontal asymptotes at y=0. Conversely, if ∣f(x)∣ is small, then f(x)1 is large.
Local maxima and minima "flip":
- a maximum at (a,b) on f(x) becomes a minimum at (a,b1) on f(x)1;
- a minimum at (c,d) becomes a maximum at (c,d1). This happens because reciprocal values invert magnitudes.

Modulus & Inequalties

2 skills

Absolute value

AHL 2.16

The absolute value of x is defined as

∣x∣={x−x for x≥0 for x<0.

Solving inequalities of functions

AHL 2.15

Inequalities of the form

g(x)≥f(x)

can be solved either algebraically or with technology.

It is crucial to remember that when multiplying both sides of an inequality by a negative number, the inequality changes direction:

−x<1⇒x>−1.