3.05 Characteristics from a graph

Lesson

Practice

Lesson

Key characteristics

We're now going to identify some key characteristics of graphs of functions 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.
Absolute Maxima/Minima	The greatest/least 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 greatest/least value $y$`y` takes in a particular region. This appears on the graph as a turning point.
End Behavior	What happens to the $y$`y` values as the $x$`x` values get very large in the positive or negative direction?
Intervals where function is positive	The set of $x$`x`-values where the graph is above the $x$`x`-axis, so the function values are positive.
Intervals where function is negative	The set of $x$`x`-values where the graph is below the $x$`x`-axis, so the function values are negative.
Intervals where function is increasing	The set of $x$`x`-values where as we move from left to right the function values are increasing.
Intervals where function is increasing	The set of $x$`x`-values where as we move from left to right the function values are decreasing.

Key features visually

Practice questions

Question 1

Consider the adjacent graph:

Loading Graph...

State the coordinates of the $x$x-intercept in the form $\left(a,b\right)$(a,b).

Question 2

The graph of the function $y=f\left(x\right)$y=f(x) is shown below.

Loading Graph...

What are the $x$x-intercepts of the graph?
$\left(-4,0\right)$(−4,0)
A
The function does not have $x$x-intercepts.
B
$\left(0,1\right)$(0,1)
C
$\left(-\infty,0\right)$(−∞,0)
D
State the coordinates of the $y$y-intercept in the form $\left(a,b\right)$(a,b).
As $x$x becomes larger in the positive direction (i.e. $x$x approaches $\infty$∞), what happens to the corresponding $y$y-values?
They become very large in the negative direction
A
They become very large in the positive direction
B
As $x$x becomes larger in the negative direction (i.e. $x$x approaches $-\infty$−∞), what happens to the corresponding $y$y-values?
They approach zero
A
They become very large in the positive direction
B
They become very large in the negative direction
C

Outcomes

I.A.REI.10

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

I.F.IF.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior.

I.F.IF.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.