Exercises: Quadratic, cubic, exponential and logarithmic graphs

Write $f(x) = x^2 - 8x + 11$ in the form $(x-p)^2 + q$ and state the coordinates of its turning point.

For $f(x) = 2^x$:

1. a) Complete a table of values for $x = -2, -1, 0, 1, 2, 3$.
2. b) State the equation of the horizontal asymptote.
3. c) State the coordinates of the point where the graph crosses the $y$-axis.

For $g(x) = \log_2 x$:

1. a) Find $g(1)$, $g(2)$, $g(8)$ and $g\!\left(\tfrac14\right)$.
2. b) State the domain of $g$ and the equation of its vertical asymptote.
3. c) State the point where the graph crosses the $x$-axis.

Let $f(x) = -2x^2 + 8x + 1$.

1. a) Write $f(x)$ in the form $a(x-p)^2 + q$.
2. b) State the turning point and whether it is a maximum or minimum.
3. c) Find the exact roots of $f(x) = 0$ using your completed-square form.

Let $f(x) = x^3 - 3x^2 - 4x + 12$.

1. a) Show that $x = 2$ is a root and factorise $f(x)$ fully.
2. b) State the points where the graph crosses the axes.
3. c) Describe the end behaviour as $x\to\pm\infty$ and sketch the graph, indicating its key points.

The number of bacteria in a sample at a lab in Maynooth is modelled by $N(t) = 500 \cdot 2^{t/3}$, where $t$ is the time in hours.

1. a) Find $N(0)$ and $N(6)$.
2. b) Find the time at which the population reaches 8,000.
3. c) Describe how the graph of $N$ relates to that of $y = 2^t$.

Let $f(x) = x^2 - 4x + 1$ and $g(x) = x - 3$.

1. a) Solve $f(x) = g(x)$ and hence find the points of intersection of their graphs.
2. b) Use the graphs to solve $f(x) < g(x)$.
3. c) Find the value of $k$ for which the line $y = x + k$ is a tangent to the graph of $f$.

Let $f(x) = 2^x$ and $g(x) = 8 - x$.

1. a) Show that the equation $f(x) = g(x)$ has a root between $x = 2$ and $x = 3$.
2. b) Explain, using the shapes of the graphs, why there is exactly one solution.
3. c) Use two steps of bisection to give an estimate of the solution, and state the interval in which it lies.

Let $f(x) = \ln(x-1) + 2$.

1. a) State the domain, the vertical asymptote and the $x$-intercept of $f$.
2. b) Find $f^{-1}(x)$ and state its range.
3. c) Describe the transformations mapping $y = \ln x$ to $y = f(x)$, and hence sketch $f$ and $f^{-1}$ on the same axes, stating the point where $f^{-1}$ crosses the $y$-axis.