Interpreting key features of non-linear graphs

We're now going to identify some key characteristics of non-linear graphs without being given their equations.

We'll be looking for the following features of the graphs:

$x$x-intercepts	Points where the graph crosses the $x$x-axis.
$y$y-intercept	The point where the graph crosses the $y$y-axis. If the graph is a function, there will be only one of these.
Absolute Maxima/Minima	The highest/lowest value $y$y takes across the whole domain. This appears on the graph as a turning point. Sometimes, when the graph has no upper/lower limit, this will not exist.
Local Maxima/Minima	The highest/lowest value $y$y takes in a particular region. This appears on the graph as a turning point.
Slope/Slope	The steepness of the graph. The slope is zero when flat, more positive for 'uphill' slopes, and more negative for 'downhill' slopes.
Extrema Behaviour	Does the graph increase or decrease as $x\to\infty$x→∞ or $x\to-\infty$x→−∞?

Also recall what we covered about domains. Functions can be defined on their full natural domain (every value of $x$x for which the function is mathematically possible) or some user-specified domain.

Questions might also ask about key features of a graph, but only in a particular region, by specifying a domain.

Remember that domains can either be expressed using inequalities or intervals. $-3$−3$<$<$x$x$<$<$5$5 could be written as $\left(-3,5\right)$(−3,5), $x>18$x>18 could be written as $\left(18,\infty\right)$(18,∞), and $x<-11$x<−11 could be written as $\left(-\infty,-11\right)$(−∞,−11).

Examples

Question 1

Consider the following graph.

a) What is the $y$y-intercept of the graph?

b) How many $x$x-intercepts does the graph have? What are they?

c) How many local maxima does the graph have? How many local minima does it have?

d) Is the graph increasing or decreasing as $x\to\infty$x→∞? How about as $x\to-\infty$x→−∞?

e) Does the graph have an absolute maximum? Does it have an absolute minimum?

f) In which domain will you find the greatest positive slope?

(A) $\left(-4,-2\right)$(−4,−2)

(B) $\left(-2,1\right)$(−2,1)

(C) $\left(1,2\right)$(1,2)

(D) $\left(2,6\right)$(2,6)

SOLUTION

a) The graph crosses the $y$y-axis here.

Hence, the $y$y-intercept is $y=-2$y=−2.

b) The graph has five $x$x-intercepts at the following points. They are $x$x$=$=$-4$−4,$-2$−2,$1$1,$2$2,$6$6.

Notice that $x=6$x=6 is still an intercept even though it only touches the axis rather than crossing through it.

c) The graph has three local maxima at the following points. They are the maximum values in the local region around them.

The graph has two local minima at the following points. 

d) As $x\to\infty$x→∞ and $x\to-\infty$x→−∞ the graph is decreasing on both sides towards negative infinity.

e) The graph has an absolute maximum at the following point, since the graph never goes higher than this.

However, even though we have two local minima, there is no absolute minimum because the graph decreases without bound on either side as $x\to\infty$x→∞ and $x\to-\infty$x→−∞, so never finds a minimum.

f) The domains being referred to are these regions of the graph between the $x$x-intercepts (the hills and valleys).

The solid lines below show the parts of the graph in these regions that only have positive slopes.

We can see from this that the steepest positive slope occurs in the interval $\left(-4,-2\right)$(−4,−2). Hence, (A) is the correct answer.

Question 2

Question 3

Question 4