**Functions** can be represented in a variety of ways, including equations, tables, and graphs. It is important to be able to compare functions whether they are represented in similar or different ways.

Useful information can usually be obtained by comparing characteristics or **key features** of the functions. Remember that key features include:

Domain and range

Intercepts

Zeros

End behavior

Relative maximum or minimum value(s)

Absolute maximum or minimum value(s)

Increasing, decreasing, and constant intervals

Asymptote(s)

Axis of symmetry

Recall the function families we have studied throughout Algebra 2:

List the following functions in order from left-most vertical asymptote to right-most vertical asymptote.

- f(x)=\dfrac{x}{x-3} + 5
- g(x)=\log{(x-5)} + 6
- h(x)=\dfrac{x}{x+4} - 3
- j(x)=\log{(x+2)} - 5

Worked Solution

Consider the following square root and logarithmic functions:

a

Compare the zeros of the functions.

Worked Solution

b

Compare the domain and range.

Worked Solution

c

Compare the absolute maxima or minima, if any.

Worked Solution

d

Compare the end behavior.

Worked Solution

Consider the function f(x) = -(x+2)^3 and the function g(x) represented by the table of values.

x | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|

g(x) | -27 | -8 | -1 | 0 | 1 | 8 | 27 |

a

Compare and contrast the intervals over which the functions are increasing or decreasing.

Worked Solution

b

Identify and compare the zeros.

Worked Solution

The graph of function f \left(x \right) shown follows the vertical height of Ameth's golf ball after being hit into the air and falling on the fairway:

The equation of function g(x) shown follows the vertical height of Massiel's golf ball. She did not get a good hit on the ball and, after a few seconds in the air, the ball hit a tree and ricocheted off into a sand trap where it got stuck.

g(x) = \begin{cases} 6 \sqrt[3] x, & 0 \leq x < 4 \\ -2.4x+19.125 , & 4 \leq x < 7.5 \\ \dfrac{1}{2} (x-9)^2, & 7.5 \leq x < 9 \end{cases}

a

Compare the intervals where each golf ball is increasing in height.

Worked Solution

b

Describe what g(x)=-2.4x+19.125 for 4\leq x\lt 7.5 represents in context.

Worked Solution

c

Determine whose ball goes higher.

Worked Solution

Idea summary

Real-world contexts can be modeled by various types of functions. We can compare different types of models for contextual situations using their key features:

Domain and range

Intercepts

Zeros

End behavior

Relative maximum or minimum value(s)

Absolute maximum or minimum value(s)

Increasing, decreasing, and constant intervals

Asymptote(s)

Axis of symmetry