# 3.05 Characteristics from a graph

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. The point where the graph crosses the $y$y-axis. 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. The greatest/least value $y$y takes in a particular region. This appears on the graph as a turning point. What happens to the $y$y values as the $x$x values get very large in the positive or negative direction? The set of $x$x-values where the graph is above the $x$x-axis, so the function values are positive. The set of $x$x-values where the graph is below the $x$x-axis, so the function values are negative. The set of $x$x-values where as we move from left to right the function values are 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

1. 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.

1. 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

$\left(-4,0\right)$(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
2. State the coordinates of the $y$y-intercept in the form $\left(a,b\right)$(a,b).

3. 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

They become very large in the negative direction

A

They become very large in the positive direction

B
4. 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

They approach zero

A

They become very large in the positive direction

B

They become very large in the negative direction

C

### Outcomes

#### A.REI.D.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).

#### F.IF.B.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; and periodicity. 'Linear and exponential, (linear domain)

#### F.IF.B.5'

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 'Linear and exponential, (linear domain)