IB Maths AA SL – Topic 2: Functions

Express $f(x) = 2x^2 – 12x + 7$ in the form $a(x-h)^2 + k$, and hence state the coordinates of the vertex of the graph of $y = f(x)$.

The graph of $y = f(x)$, where $f(x) = x^2$, is transformed to give the graph of $y = -2f(x+1) – 3$.(a) Describe fully the sequence of transformations applied to $y = f(x)$.(b) Write the resulting function in expanded form $ax^2+bx+c$.

The function $f(x) = (x-4)^2 – 9$ has domain $x \in \mathbb{R}$. Write down the range of $f$.

The function $g(x) = \dfrac{3}{x-2} + 1$ is defined for $x eq 2$. Write down the equations of the vertical and horizontal asymptotes of the graph of $y = g(x)$.

Solve the equation $5^{2x-1} = 40$, giving your answer correct to three significant figures.

Given that $\log_a 2 = m$ and $\log_a 5 = n$, express $\log_a 20$ in terms of $m$ and $n$.

Consider the function $f(x) = \dfrac{2x+3}{x-1}$, $x eq 1$.(a) Write down the equation of the vertical asymptote.(b) Write down the equation of the horizontal asymptote.(c) Find the coordinates of the points where the graph of $y = f(x)$ crosses the $x$-axis and the $y$-axis.

Let $f(x) = 2x – 1$ and $g(x) = x^2 + 3$.(a) Find $f(g(2))$.(b) Find $g(f(x))$, giving your answer in expanded form.(c) Solve the equation $f(g(x)) = 15$.

The population of a colony of bacteria is modelled by $P(t) = 500e^{0.08t}$, where $t$ is the time in hours and $P$ is the number of bacteria.(a) Write down the initial population.(b) Find the population after $10$ hours, giving your answer to the nearest whole number.(c) Find the time taken for the population to double, giving your answer correct to one decimal place.

Let $f(x) = \dfrac{x+3}{2x-1}$, $x eq \dfrac{1}{2}$.(a) Find $f^{-1}(x)$.(b) State the domain of $f^{-1}$.

The function $f(x) = \ln(x-2) + 3$ is defined for $x > 2$. Find $f^{-1}(x)$ and state its domain and range.

The graph of $y = f(x)$ has a maximum point at $(2, 5)$.(a) State the coordinates of the maximum point on the graph of $y = 2f(x-3) + 4$.(b) State the coordinates of the corresponding point on the graph of $y = -f(2x)$.