IB Maths: Analysis and Approaches HL 166711

Different forms of the equation of a straight line.

Gradient; intercepts.

Lines with gradients $m_1$ and $m_2$

Parallel lines $m_1=m_2$.

Perpendicular lines $m_1×m_2=-1$.

• $y=mx+c$ (gradient-intercept form).
• $ax+by+d=0$ (general form).
• $y−y_1=m(x−x_1)$ (point-gradient form).

Calculate gradients of inclines such as mountain roads, bridges, etc.

Concept of a function, domain, range and graph.

Function notation, for example $f(x)$, $v(t)$, $C(n)$.

The concept of a function as a mathematical model.

Example: $f(x)=\sqrt{2−x}$, the domain is $x≤2$, range is $f(x)≥0$.

A graph is helpful in visualizing the range.

Informal concept that an inverse function reverses or undoes the effect of a function.

Inverse function as a reflection in the line $y=x$, and the notation $f^{-1}(x)$.

Example: Solving $f(x)=10$ is equivalent to finding $f^{−1}(10)$.

Students should be aware that inverse functions exist for one to one functions; the domain of $f^{−1}(x)$ is equal to the range of $f(x)$.

The graph of a function; its equation $y=f(x)$.

Students should be aware of the difference between the command terms “draw” and “sketch”.

Creating a sketch from information given or a context, including transferring a graph from screen to paper.

Using technology to graph functions including their sums and differences.

All axes and key features should be labelled.

This may include functions not specifically mentioned in topic 2.

Determine key features of graphs.

Maximum and minimum values; intercepts; symmetry; vertex; zeros of functions or roots of equations; vertical and horizontal asymptotes using graphing technology.

Finding the point of intersection of two curves or lines using technology.

Composite functions.

$(f∘g)(x)=f(g(x))$

Identity function. Finding the inverse function $f^{-1}(x)$.

$(f∘f^{−1})(x)=(f^{−1}∘f)(x)=x$

The existence of an inverse for one-to-one functions.

Link to: concept of inverse function as a reflection in the line $y=x$ (SL 2.2).

The quadratic function $f(x)=ax^2+bx+c$: its graph, $y$-intercept $(0,c)$. Axis of symmetry.

The form $f(x)=a(x−p)(x−q)$, $x$-intercepts $(p,0)$ and $(q,0)$.

The form $f(x)=a(x−h)^2+k$, vertex $(h,k)$.

A quadratic graph is also called a parabola.

Link to: transformations (SL 2.11).

Candidates are expected to be able to change from one form to another.

Solution of quadratic equations and inequalities.

The quadratic formula.

Using factorization, completing the square (vertex form), and the quadratic formula.

Solutions may be referred to as roots or zeros.

The discriminant $Δ=b^2−4ac$ and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.

Example: For the equation $3kx^2+2x+k=0$, find the possible values of $k$, which will give two distinct real roots, two equal real roots or no real roots.

The reciprocal function f$(x)=\dfrac{1}{x}, x≠0$: its graph and self-inverse nature.

Rational functions of the form $f(x)=\dfrac{ax+b}{cx+d}$ and their graphs.

Equations of vertical and horizontal asymptotes.

Sketches should include all horizontal and vertical asymptotes and any intercepts with the axes.

Link to: transformations (SL2.11).

Vertical asymptote: $x=-\dfrac{d}{c}$;

Horizontal asymptote: $y=\dfrac{a}{c}$.

Exponential functions and their graphs:

$f(x)=a^x, a>0, f(x)=e^x$

Logarithmic functions and their graphs:

$f(x)=\log_a{x}, x>0, f(x)=\ln{x}, x>0$.

Link to: financial applications of geometric sequences and series (SL 1.4).

Relationships between these functions:

$a^x=e^{x\ln{a}}; \log_a{a^x}=x, a, x>0, a≠1$

Exponential and logarithmic functions as inverses of each other.

Solving equations, both graphically and analytically.

Example: $e^{2x}−5e^x+4=0$.

Link to: function graphing skills (SL 2.3).

Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.

Examples: $e^x=\sin{x}; x^4+5x-6=0$.

Applications of graphing skills and solving equations that relate to real-life situations.

Link to: exponential growth (SL 2.9)

Transformations of graphs.

Translations: $y=f(x)+b; y=f(x−a)$.

Reflections (in both axes): $y=−f(x); y=f(−x)$.

Vertical stretch with scale factor $p$: $y=pf(x)$.

Horizontal stretch with scale factor $\dfrac{1}{q}$: $y=f(qx)$.

Students should be aware of the relevance of the order in which transformations are performed.

Dynamic graphing packages could be used to investigate these transformations.

Composite transformations.

Example: Using $y=x^2$ to sketch $y=3x^2+2$

Link to: composite functions (SL2.5).

Not required at SL: transformations of the form $f(ax+b)$.

Polynomial functions, their graphs and equations; zeros, roots and factors.

The factor and remainder theorems.

Sum and product of the roots of polynomial equations.

For the polynomial equation: $\displaystyle\sum_{r=0}^{n}a_rx^r=0$,

the sum is $\dfrac{−a_{n−1}}{a_n}$

the product is $\dfrac{(−1)^na_0}{a_n}$

Link to: complex roots of quadratic and polynomial equations (AHL 1.14).

Rational functions of the form

$f(x)=\dfrac{ax+b}{cx^2+dx+e}$, and

$f(x)=\dfrac{ax^2+bx+c}{dx+e}$

The reciprocal function is a particular case.

Graphs should include all asymptotes (horizontal, vertical and oblique) and any intercepts with axes.

Dynamic graphing packages could be used to investigate these functions.

Link to: rational functions (SL 2.8).

Odd and even functions.

Even: $f(-x)=f(x)$

Odd: $f(-x)=-f(x)$

Includes periodic functions.

Finding the inverse function, $f^{-1}(x)$,

including domain restriction.

Self-inverse functions.

Solutions of $g(x)≥f(x)$, both graphically and analytically.

Graphical or algebraic methods for simple polynomials up to degree 3. Use of technology for these and other functions.

The graphs of the functions, $y=|f(x)|$ and

$y=f(|x|)$, $y=\dfrac{1}{f(x)}$, $y=f(ax+b)$, $y=[f(x)]^2$.

Dynamic graphing packages could be used to investigate these transformations.

Solution of modulus equations and inequalities.

Example: $|3x\arccos{(x)}|>1$