2. Polynomials and Analysis of Nonlinear Functions

Lesson

One of the key characteristics of a graph are the intervals of increase, decrease and constant slope or rate of change. The intervals will always be in terms of the independent variable, usually $x$`x`. We can write these intervals in inequality or interval notation.

On the graph below, we can see what the different scenarios look like.

- If as values of $x$
`x`increase, the values of $f(x)$`f`(`x`) increase, then the function is increasing over that interval - If as the values of $x$
`x`increase, the values of $f(x)$`f`(`x`) decrease, then the function is decreasing over that interval - If as the values of $x$
`x`increase, the values of $f(x)$`f`(`x`) remain constant, then we can say the function is constant over that interval

We can also interpret this in terms of the rate of change of the variables. For example, acceleration is the rate of change of velocity with respect to time. If the acceleration is positive (so that the slope of the velocity with respect to time is positive) then the velocity is increasing, and if the acceleration is negative then the velocity is decreasing.

State the intervals of increase and decrease for the graph below.

**Think:** This is a parabola. To the left of the vertex the function is decreasing and to the right of the vertex the function is increasing. At the vertex, it is switching from decreasing to increasing, so is technically constant at that exact point.

**Do:** Let's write our answer in both inequality and interval notation. They will be in terms of the $x$`x`-values.

**Interval of decrease**: $x<0$`x`<0 which can be written as $x\in(-\infty,0)$`x`∈(−∞,0)

**Interval of increase**: $x>0$`x`>0 which can be written as $x\in(0,\infty)$`x`∈(0,∞)

**Reflect:** At $x=0$`x`=0 the function is neither increasing nor decreasing, so we use an open interval to exclude $0$0.

A company's daily revenue, $R$`R`, in thousand dollars over $12$12 months has been plotted below. We want to classify the entire domain of the function graphed below into regions which are increasing, or decreasing:

**Think:** While we do not know exactly what this function is, we have enough information in the graph to work this out. We will start by finding the points where it changes from increasing to decreasing or vice versa.

**Do:** First, notice points where the graph changes from increasing to decreasing or vice versa at $t=3$`t`=3 and $t=5$`t`=5. At $t=10$`t`=10, it flattens out, but does not actually change to decreasing, we will look at these kinds of points more in future studies.

**Intervals of increase:** $0`t`<3 and $5`t`<12 which can also be written as $t\in\left(0,3\right)\cup\left(5,12\right)$`t`∈(0,3)∪(5,12)

**Intervals of decrease:** $3`t`<5 which can also be written as $t\in\left(3,5\right)$`t`∈(3,5).

**Reflect:** Knowing the intervals when the function is increasing and decreasing lets us interpret the rate of change of the revenue. Since the function is always positive, the company is always taking in money.

This is true even when the function is decreasing. The rate of change is negative on the interval $\left(3,5\right)$(3,5), so the company is taking in less money during this period than they did at the start of month $3$3 but the rate of change is slow enough that the revenue never completely stops.

Increasing, decreasing and constant functions

A function can be described as either increasing, decreasing or constant. Some functions will be increasing, decreasing or constant only in specific intervals of the domain.

- A function is increasing (on an interval) if the rate of change is positive ($m>0$
`m`>0) - A function is decreasing (on an interval) if the rate of change is negative ($m<0$
`m`<0) - A function is constant (on an interval) if the rate of change is zero ($m=0$
`m`=0)

Consider the function $f\left(x\right)=\left(x+1\right)\left(x-3\right)$`f`(`x`)=(`x`+1)(`x`−3) drawn below.

Loading Graph...

What is the $x$

`x`-value of the point where $f\left(x\right)$`f`(`x`) changes from decreasing to increasing?What is the region of the domain where $f\left(x\right)$

`f`(`x`) is increasing?Write the answer in interval notation.

What is the region of the domain where $f\left(x\right)$

`f`(`x`) is decreasing?Write the answer in interval notation.

We're now going to identify some key characteristics of non-linear graphs 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 and $f(x)=0$f(x)=0. |
---|---|

$y$y-intercept |
The point where the graph crosses the $y$y-axis and $x=0$x=0. If the graph is a function, there will be only one of these. |

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. |

Slope | The steepness of the graph. The slope is zero when flat, more positive for 'uphill' slopes, and more negative for 'downhill' slopes. |

Extrema Behavior | Does the graph increase or decrease as $x\to\infty$x→∞ or $x\to-\infty$x→−∞? |

Questions might also ask about key features of a graph, but only in a particular region, by specifying a domain.

Consider the following graph.

**a)** What is the $y$`y`-intercept of the graph?

**b)** How many $x$`x`-intercepts does the graph have? What are they?

**c)** How many local maxima does the graph have? How many local minima does it have?

**d)** Is the graph increasing or decreasing as $x\to\infty$`x`→∞? How about as $x\to-\infty$`x`→−∞?

**e)** Does the graph have an absolute maximum? Does it have an absolute minimum?

**f)** In which domain will you find the greatest positive slope?

**(A)** $\left(-4,-2\right)$(−4,−2)

**(B)** $\left(-2,1\right)$(−2,1)

**(C)** $\left(1,2\right)$(1,2)

**(D)** $\left(2,6\right)$(2,6)

**a)** The graph crosses the $y$`y`-axis here.

Hence, the $y$`y`-intercept is $y=-2$`y`=−2.

**b) **The graph has five $x$`x`-intercepts at the following points. They are $x=-4$`x`=−4, $x=-2$`x`=−2, $x=1$`x`=1, $x=2$`x`=2 and $x=6$`x`=6.

Notice that $x=6$`x`=6 is still an intercept even though it only touches the axis rather than crossing through it.

**c)** The graph has three local maxima at the following points. They are the maximum values in the local region around them.

The graph has two local minima at the following points.

**d)** As $x\to\infty$`x`→∞ and $x\to-\infty$`x`→−∞ the graph is decreasing on both sides towards negative infinity.

**e)** The graph has an absolute maximum at the following point, since the graph never goes higher than this.

However, even though we have two local minima, there is no absolute minimum because the graph decreases without bound on either side as $x\to\infty$`x`→∞ and $x\to-\infty$`x`→−∞, so never finds a minimum.

**f)** The domains being referred to are these regions of the graph between the $x$`x`-intercepts (the hills and valleys).

The solid lines below show the parts of the graph in these regions that only have positive slopes.

We can see from this that the steepest positive slope occurs in the interval $\left(-4,-2\right)$(−4,−2). Hence, **(A)** is the correct answer.

Consider the adjacent graph:

Loading Graph...

What are the coordinates of the $x$

`x`-intercepts? State the coordinates in the form $\left(a,b\right)$(`a`,`b`), on the same line separated by a comma.State the coordinates of the $y$

`y`-intercept in the form $\left(a,b\right)$(`a`,`b`).As $x$

`x`→ +$\infty$∞, what does $y$`y`approach?$-\infty$−∞

A$\infty$∞

B$-\infty$−∞

A$\infty$∞

BAs $x$

`x`→ - $\infty$∞, what does $y$`y`approach?$-\infty$−∞

A$\infty$∞

B$-\infty$−∞

A$\infty$∞

B

Consider the adjacent graph:

Loading Graph...

State the coordinates of the $x$

`x`-intercept in the form $\left(a,b\right)$(`a`,`b`).How do the function values change as $x\to\infty$

`x`→∞?They become larger

AThey become smaller

BThey become larger

AThey become smaller

BWhat is the behavior of the function as $x\to-\infty$

`x`→−∞?They become larger

AThey become smaller

BThey become larger

AThey become smaller

B

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

Loading Graph...

State the coordinates of the $x$

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

`y`-intercept in the form $\left(a,b\right)$(`a`,`b`).What is the behavior of the function as $x\to\infty$

`x`→∞?Increasing

ADecreasing

BIncreasing

ADecreasing

B

An asymptote of a function is a straight line which the function values approach under certain conditions. There are three types of asymptotes: horizontal, vertical and oblique.

Horizontal asymptotes have equations of the form $y=c$`y`=`c`. They occur when the function approaches a constant value $c$`c` as $x$`x` tends towards positive or negative infinity.

Vertical asymptotes have equations of the form $x=c$`x`=`c`. They occur when the function values tend towards positive or negative infinity as $x$`x` approaches the constant value $c$`c`.

It is also possible for a function to have an oblique asymptote or slanted asymptote. Graphically, this means that the function approaches a straight line with a non-zero slope as $x$`x` tends towards positive or negative infinity.

As an example, here is the graph of a rational function:

Notice that the function values tend towards $\pm\infty$±∞ as the values of $x$`x` approach $1$1. This function has a vertical asymptote at $x=1$`x`=1, which has been displayed as a dashed line.

There is also a horizontal dashed line with equation $y=2$`y`=2, which is the horizontal asymptote of this function. We can see that the function values approach $2$2 as the values of $x$`x` tend towards positive and negative infinity.

Locating asymptotes

If the function values approach a constant as $x$`x` tends towards positive or negative infinity, then the function has a horizontal asymptote at that function value. Note that a function can have at most two horizontal asymptotes.

If the function values tend towards positive or negative infinity as $x$`x` approaches a certain value, then the function has a vertical asymptote at that $x$`x`-value.

A graph of the function $f\left(x\right)=-\frac{3}{x}$`f`(`x`)=−3`x` is shown below.

Loading Graph...

Complete the following statement.

If $x$

`x`is*positive*, then as the value of $x$`x`approaches zero the value of the function approaches $\editable{}$.Complete the following statement.

If $x$

`x`is*negative*, then as the value of $x$`x`approaches zero the value of the function approaches $\editable{}$.What is the equation of the vertical asymptote?

Complete the following statement.

If $x$

`x`is*positive*, then as the value of $x$`x`gets very large (approaching $\infty$∞) the value of the function approaches $\editable{}$.Complete the following statement.

If $x$

`x`is*negative*, then as the value of $x$`x`gets very small (approaching $-\infty$−∞) the value of the function approaches $\editable{}$.What is the equation of the horizontal asymptote?

A graph of the function $f\left(x\right)=4^x$`f`(`x`)=4`x` is shown below.

Loading Graph...

Complete the following statement.

As the value of $x$

`x`approaches $-\infty$−∞ the value of the function approaches $\editable{}$.What is the equation of the horizontal asymptote?

Determine if the following statement is true or false:

The function is continuous over the interval of all real numbers, $\left(-\infty,\infty\right)$(−∞,∞).

True

AFalse

BTrue

AFalse

B

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.