Graphical Methods — practice questions

State the vertex and the range of $y = (x - 4)^2 - 9$.

A parabola has $x$-intercepts at $x = -2$ and $x = 6$, and passes through $(0, -12)$. Find its equation in the form $y = ax^2 + bx + c$, and state its vertex.

A parabola $y = ax^2 + bx + c$ has its vertex at $(3, k)$ and passes through $(1, 0)$ and $(0, 5)$. Find $a$, $b$, $c$ and $k$, and explain how the axis of symmetry lets you find a second $x$-intercept without further calculation.

State the $y$-intercept, the asymptote and the range of $y = 5(2)^x$.

A cubic has equation $y = a(x + 3)(x - 1)(x - 4)$ and passes through $(0, 24)$. Find $a$, and describe the shape of the graph including its behaviour at each root.

Explain why the graph of $y = \log_2 x$ is the reflection of $y = 2^x$ in the line $y = x$, and use this to state the domain, range and asymptote of each without plotting either graph.

State the asymptotes, domain and range of $y = \dfrac{4}{x - 5}$.

A hyperbola has vertical asymptote $x = 3$, horizontal asymptote $y = -1$, and passes through $(5, 2)$. Find its equation and state its domain and range.

A rectangular pen of area 60 m² is to be built against a wall, so only three sides need fencing. Express the total length of fencing $L$ as a function of the side length $x$ perpendicular to the wall, identify the family of graph, and determine the practical domain and the minimum fencing needed.

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.