Key Features of Graphs
Maximum and Minimum Values
Maximum and minimum values are points that represent the highest and lowest y-values of a function within a domain. These points are useful for solving optimisation problems.
ExampleConsider the quadratic function $f(x) = -x^2 + 4x + 5$. Using a graphing calculator, we can determine that the maximum point occurs at (2, 9). This means the highest y-value the function reaches is 9, occurring when x = 2.
TipWhen using technology to find max/min values, always check if they are global (absolute) or local (relative) extrema. Some functions may have multiple local extrema but only one global maximum or minimum.
Intercepts
Intercepts are the points where a graph crosses the $x$ or $y$-axis.
- X-intercepts (or roots) occur where $y = 0$
- Y-intercepts occur where $x = 0$
For the function $g(x) = x^2 - 4x + 3$, using a graphing calculator reveals:
- $x$-intercepts: (1, 0) and (3, 0)
- $y$-intercept: (0, 3)
Alternatively, it should be obvious that
$$g(0)=3$$
and
$$g(x) = (x-3)(x-1)$$
So the $x$ and $y$ intercepts can also be found analytically.
NoteIntercepts are particularly useful in analyzing the behavior of functions and solving equations graphically.
Symmetry
Symmetry in graphs can be of two types:
- Symmetry about the y-axis: $f(x) = f(-x)$.
- Symmetry of rotation by 180 degrees about the origin: $f(x) = -f(-x)$.
Identifying symmetry can simplify the analysis of functions and help in sketching graphs more accurately.
ExampleThe function $h(x) = x^4 - 2x^2$ is symmetric about the y-axis. This can be verified by graphing and observing that the left and right halves of the graph are mirror images of each other.
