High points, low points and the far ends

Relative and absolute extrema, end behaviour, and one graph described feature by feature.

A walk that goes uphill and then downhill has a highest point at the moment it turns. A walk that goes downhill and then uphill has a lowest one. Those turns, and what the ground does far off in each direction, are the last two items on the examiner's list.

This lesson takes them one at a time and then puts the whole list together on a single function.

High points and low points

In the previous lesson the increasing and decreasing report for $f(x) = x^2 - x - 6$ turned around exactly once. The place where it turned is a low point. Run this and pick the smallest output out of the sweep.

A point that is lower than everything immediately around it is a relative minimum, sometimes called a local minimum. A point higher than everything around it is a relative maximum. If a point is the lowest or highest on the whole graph, not merely in its neighbourhood, it is an absolute minimum or absolute maximum.

The parabola above has an absolute minimum, because nothing anywhere is lower. A cubic like $x^3 - 3x$ has a relative maximum and a relative minimum but no absolute ones, since it runs off to $-\infty$ on the left and $+\infty$ on the right.

That cubic is worth looking at rather than imagining. Slide the marker along it.

End behaviour

The last feature is what happens far away from the origin. There is no way to see it on a small sketch, so ask for it directly.

The outputs grow past a thousand, then past a million, then past a billion, and on the left they do the same downwards. End behaviour is the description of where the outputs head as the inputs run off to the far left and the far right, and it is written with arrows.

One graph, every feature

An exam rarely asks for one feature. It shows a curve and asks for the lot, so here is the lot, worked out in full for $f(x) = x^3 - 3x$.

1. y-intercept. Put $x = 0$ into the rule: $f(0) = 0^3 - 3 \times 0 = 0$. The curve passes through $(0, 0)$.
2. x-intercepts. Solve $x^3 - 3x = 0$. Take the common factor out: $x(x^2 - 3) = 0$, so either $x = 0$ or $x^2 = 3$. That gives $x = 0$, $x = \sqrt{3} \approx 1.73$ and $x = -\sqrt{3} \approx -1.73$.
3. Turning points, the left one. The sweep put it at $x = -1$. Check: $f(-1) = -1 + 3 = 2$, while $f(-0.9) = -0.729 + 2.7 = 1.971$ and $f(-1.1) = -1.331 + 3.3 = 1.969$. Both neighbours are lower, so $(-1, 2)$ is a relative maximum.
4. Turning points, the right one. Check $x = 1$: $f(1) = 1 - 3 = -2$, while $f(0.9) = 0.729 - 2.7 = -1.971$ and $f(1.1) = 1.331 - 3.3 = -1.969$. Both neighbours are higher, so $(1, -2)$ is a relative minimum.
5. Increasing and decreasing. Left to right the curve climbs up to $x = -1$, falls from $x = -1$ to $x = 1$, then climbs again. So it increases on $(-\infty, -1)$ and on $(1, \infty)$, and decreases on $(-1, 1)$.
6. Absolute extrema. There are none. The outputs run off to $+\infty$ on the right and to $-\infty$ on the left, so no output is the largest and none is the smallest.
7. End behaviour. As $x \to \infty$, $f(x) \to \infty$. As $x \to -\infty$, $f(x) \to -\infty$.
8. Domain and range. Every real number can go in, and every real number comes out somewhere, so both are $(-\infty, \infty)$.

Notice the kind of object each answer is. Intercepts are numbers or points. Intervals of increase and decrease are intervals of inputs. Extrema are points, given as a pair. End behaviour is two arrow statements, one per end. Handing back the wrong kind of object is the commonest way to lose marks on a question you understood.