3.05 Review: Characteristics from a graph
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:
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.
What are the $x$x-intercepts of the graph?
$\left(-4,0\right)$(−4,0)
AThe function does not have $x$x-intercepts.
B$\left(0,1\right)$(0,1)
C$\left(-\infty,0\right)$(−∞,0)
DState 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
AThey become very large in the positive direction
BAs $x$x becomes larger in the negative direction (i.e. $x$x approaches $-\infty$−∞), what happens to the corresponding $y$y-values?
They approach zero
AThey become very large in the positive direction
BThey become very large in the negative direction
C