# 1.2 Basic classes of functions  (Page 4/28)

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Now consider a cubic function $f\left(x\right)=a{x}^{3}+b{x}^{2}+cx+d.$ If $a>0,$ then $f\left(x\right)\to \infty$ as $x\to \infty$ and $f\left(x\right)\to \text{−∞}$ as $x\to \text{−∞}.$ If $a<0,$ then $f\left(x\right)\to \text{−∞}$ as $x\to \infty$ and $f\left(x\right)\to \infty$ as $x\to \text{−∞}.$ As we can see from both of these graphs, the leading term of the polynomial determines the end behavior. (See [link] (b).)

## Zeros of polynomial functions

Another characteristic of the graph of a polynomial function is where it intersects the $x$ -axis. To determine where a function $f$ intersects the $x$ -axis, we need to solve the equation $f\left(x\right)=0$ for .n the case of the linear function $f\left(x\right)=mx+b,$ the $x$ -intercept is given by solving the equation $mx+b=0.$ In this case, we see that the $x$ -intercept is given by $\left(\text{−}\mathit{\text{b}}\text{/}m,0\right).$ In the case of a quadratic function, finding the $x$ -intercept(s) requires finding the zeros of a quadratic equation: $a{x}^{2}+bx+c=0.$ In some cases, it is easy to factor the polynomial $a{x}^{2}+bx+c$ to find the zeros. If not, we make use of the quadratic formula.

$a{x}^{2}+bx+c=0,$

where $a\ne 0.$ The solutions of this equation are given by the quadratic formula

$x=\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}.$

If the discriminant ${b}^{2}-4ac>0,$ this formula tells us there are two real numbers that satisfy the quadratic equation. If ${b}^{2}-4ac=0,$ this formula tells us there is only one solution, and it is a real number. If ${b}^{2}-4ac<0,$ no real numbers satisfy the quadratic equation.

In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the $x$ -axis. In some instances, it is possible to find the $x$ -intercepts by factoring the polynomial to find its zeros. In other cases, it is impossible to calculate the exact values of the $x$ -intercepts. However, as we see later in the text, in cases such as this, we can use analytical tools to approximate (to a very high degree) where the $x$ -intercepts are located. Here we focus on the graphs of polynomials for which we can calculate their zeros explicitly.

## Graphing polynomial functions

For the following functions a. and b., i. describe the behavior of $f\left(x\right)$ as $x\to \text{±}\infty ,$ ii. find all zeros of $f,$ and iii. sketch a graph of $f.$

1. $f\left(x\right)=-2{x}^{2}+4x-1$
2. $f\left(x\right)={x}^{3}-3{x}^{2}-4x$
1. The function $f\left(x\right)=-2{x}^{2}+4x-1$ is a quadratic function.
1. Because $a=-2<0,\text{as}\phantom{\rule{0.2em}{0ex}}x\to \text{±}\infty ,f\left(x\right)\to \text{−∞.}$
2. To find the zeros of $f,$ use the quadratic formula. The zeros are
$x=\frac{-4±\sqrt{{4}^{2}-4\left(-2\right)\left(-1\right)}}{2\left(-2\right)}=\frac{-4±\sqrt{8}}{-4}=\frac{-4±2\sqrt{2}}{-4}=\frac{2±\sqrt{2}}{2}.$
3. To sketch the graph of $f,$ use the information from your previous answers and combine it with the fact that the graph is a parabola opening downward.
2. The function $f\left(x\right)={x}^{3}-3{x}^{2}-4x$ is a cubic function.
1. Because $a=1>0,\text{as}\phantom{\rule{0.2em}{0ex}}x\to \infty ,f\left(x\right)\to \infty .$ As $x\to \text{−∞},f\left(x\right)\to \text{−∞}.$
2. To find the zeros of $f,$ we need to factor the polynomial. First, when we factor $x$ out of all the terms, we find
$f\left(x\right)=x\left({x}^{2}-3x-4\right).$

Then, when we factor the quadratic function ${x}^{2}-3x-4,$ we find
$f\left(x\right)=x\left(x-4\right)\left(x+1\right).$

Therefore, the zeros of $f$ are $x=0,4,-1.$
3. Combining the results from parts i. and ii., draw a rough sketch of $f.$

I don't understand the formula
who's formula
funny
What is a independent variable
a variable that does not depend on another.
Andrew
solve number one step by step
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
Hasnain
how to prove 1-sinx/cos x= cos x/-1+sin x?
1-sin x/cos x= cos x/-1+sin x
Rochel
how to prove 1-sun x/cos x= cos x / -1+sin x?
Rochel
how to prove tan^2 x=csc^2 x tan^2 x-1?
divide by tan^2 x giving 1=csc^2 x -1/tan^2 x, rewrite as: 1=1/sin^2 x -cos^2 x/sin^2 x, multiply by sin^2 x giving: sin^2 x=1-cos^2x. rewrite as the familiar sin^2 x + cos^2x=1 QED
Barnabas
how to prove sin x - sin x cos^2 x=sin^3x?
sin x - sin x cos^2 x sin x (1-cos^2 x) note the identity:sin^2 x + cos^2 x = 1 thus, sin^2 x = 1 - cos^2 x now substitute this into the above: sin x (sin^2 x), now multiply, yielding: sin^3 x Q.E.D.
Andrew
take sin x common. you are left with 1-cos^2x which is sin^2x. multiply back sinx and you get sin^3x.
navin
Left side=sinx-sinx cos^2x =sinx-sinx(1+sin^2x) =sinx-sinx+sin^3x =sin^3x thats proved.
Alif
how to prove tan^2 x/tan^2 x+1= sin^2 x
Rochel
Salim
what is function.
what is polynomial
Nawaz
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Alif
a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
joe
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
Nawaz
what is hyperbolic function
find volume of solid about y axis and y=x^3, x=0,y=1
3 pi/5
vector
what is the power rule
Is a rule used to find a derivative. For example the derivative of y(x)= a(x)^n is y'(x)= a*n*x^n-1.
Timothy
how do i deal with infinity in limits?
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz