Transformations — practice questions

Describe fully how the graph of $y = (x + 3)^2 - 7$ relates to the graph of $y = x^2$.

The graph of $y = 3^x$ is reflected in the $y$-axis and then translated 4 units up. Find the equation of the new graph, state its asymptote, and explain what happens to its behaviour as $x$ increases.

Show that reflecting $y = f(x)$ in the $x$-axis and then in the $y$-axis produces the same graph as rotating it $180°$ about the origin. Illustrate with $f(x) = x^3 - 2x^2$.

A car is worth $36 000 and loses 20% of its value each year. Write an equation modelling its value $V$ after $t$ years, and identify the family of graph.

A ball is thrown and its height is modelled by $h = 24t - 4.9t^2$ metres, where $t$ is in seconds. Find the maximum height and when it occurs, state the practical domain, and interpret the two $t$-intercepts.

A biologist counts bacteria: 200 at hour 0, 460 at hour 2, and 1058 at hour 4. Decide whether a linear or an exponential model fits better, justify your choice, build the model, and evaluate how far it can safely be used.