1. Functions

Lesson

We sometimes need to compare the behavior of two different functions, perhaps over a particular domain interval.

We may do this by looking at the graphs of the functions, by comparing tables of values of the functions or by examining the algebraic representations of the functions.

Questions we may wish to answer in comparing two functions include such things as

- Whether one function is always greater than or less than the other,
- Whether the graphs intersect and if so how often,
- Whether one or both functions are strictly increasing or decreasing on an interval,
- How the function values compare at selected points in the domain,
- How the slopes differ, and
- How the maximum or minimum values compare.

Consider the line $P$`P` given by the equation $y=-12+\frac{x}{10}$`y`=−12+`x`10, and the table of values for parabola $Q$`Q`.

Parabola $Q$`Q`:

$x$x |
$-8$−8 | $-7$−7 | $-6$−6 | $-5$−5 | $-4$−4 |
---|---|---|---|---|---|

$y$y |
$-1$−1 | $-4$−4 | $-5$−5 | $-4$−4 | $-1$−1 |

Line $P$

`P`is shown below. Draw the graph of the parabola $Q$`Q`.Loading Graph...How many times do $P$

`P`and $Q$`Q`intersect?Twice

AThree times

BZero times

COnce

DTwice

AThree times

BZero times

COnce

DWhich function has the higher function value at the $y$

`y`-intercept?$Q$

`Q`A$P$

`P`B$Q$

`Q`A$P$

`P`BWhich function has the higher function value at $x=-1$

`x`=−1?$Q$

`Q`A$P$

`P`B$Q$

`Q`A$P$

`P`B

The graph of the parabola $P$`P`, given by $y=\left(x-5\right)^2+5$`y`=(`x`−5)2+5 is shown below.

Graph the parabola $Q$

`Q`given by the equation $y=-2\left(x-7\right)\left(x-3\right)$`y`=−2(`x`−7)(`x`−3).Loading Graph...By referring to the graphs, which function has a maximum at $y=8$

`y`=8?$P$

`P`A$Q$

`Q`B$P$

`P`and $Q$`Q`CNeither $P$

`P`or $Q$`Q`D$P$

`P`A$Q$

`Q`B$P$

`P`and $Q$`Q`CNeither $P$

`P`or $Q$`Q`DBy referring to the graphs, which function has a minimum at $y=5$

`y`=5?$P$

`P`A$Q$

`Q`B$P$

`P`and $Q$`Q`CNeither $P$

`P`or $Q$`Q`D$P$

`P`A$Q$

`Q`B$P$

`P`and $Q$`Q`CNeither $P$

`P`or $Q$`Q`DComplete the following statements:

Parabola $P$

`P`has $\editable{}$ zero(s).Parabola $Q$

`Q`has $\editable{}$ zero(s).How many times do the parabolas intersect?

Once

ATwice

BThree times

CZero times

DOnce

ATwice

BThree times

CZero times

D

The graph of the exponential function $P$`P`, given by $y=4^{-x}$`y`=4−`x` is shown below.

Graph the line $Q$

`Q`given by the equation $y=8x+12$`y`=8`x`+12.Loading Graph...How many times do $P$

`P`and $Q$`Q`intersect?Zero times

ATwice

BThree times

COnce

DZero times

ATwice

BThree times

COnce

DWhich function has the higher function value at the $y$

`y`-intercept?$P$

`P`A$Q$

`Q`B$P$

`P`A$Q$

`Q`BWhich function has the higher function value at $x=1$

`x`=1?$P$

`P`A$Q$

`Q`B$P$

`P`A$Q$

`Q`B

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).