Leading Coefficient Test...1

[nycmathguy]

Section 2.2
Question 20

Can you solve 20 as a guide for me to do a few more? What is the Leading Coefficient Test for anyway?

 

[MathLover1]

The leading coefficient test tells us that the graph rises or falls depending on whether the leading terms are positive or negative, so for left-hand behavior (negative numbers), you will need to look at both the coefficient and the degree of the component together.

Let’s look at the following examples of when x is negative:

Leading coefficient test:

2x^3would be a positive coefficient multiplied by a negative variable=>it is negative
2x^4 would be a positive coefficient multiplied by a positive variable=>it is positive
-5x^3 would be a negative coefficient multiplied by a negative variable=>it is positive
-5x^2 would be a negative coefficient multiplied by a positive variable=>it is negative

same procedure for right-hand behavior (positive numbers)

20.
f(x)= 2x^2-3x+1

left-hand behavior (negative numbers),

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive
right-hand behavior (positive numbers)

the coefficient and the degree of the component together greater than zero=>the graph rises

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive
the coefficient and the degree of the component together greater than zero=>the graph rises

as you can see, from the turning point (vertex) the graph goes up on both sides (left and right)

 

[nycmathguy]

The leading coefficient test tells us that the graph rises or falls depending on whether the leading terms are positive or negative, so for left-hand behavior (negative numbers), you will need to look at both the coefficient and the degree of the component together.

Let’s look at the following examples of when x is negative:

Leading coefficient test:

2x^3would be a positive coefficient multiplied by a negative variable=>it is negative
2x^4 would be a positive coefficient multiplied by a positive variable=>it is positive
-5x^3 would be a negative coefficient multiplied by a negative variable=>it is positive
-5x^2 would be a negative coefficient multiplied by a positive variable=>it is negative

same procedure for right-hand behavior (positive numbers)

20.
f(x)= 2x^2-3x+1

left-hand behavior (negative numbers),

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive
right-hand behavior (positive numbers)

the coefficient and the degree of the component together greater than zero=>the graph rises

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive
the coefficient and the degree of the component together greater than zero=>the graph rises

as you can see, from the turning point (vertex) the graph goes up on both sides (left and right)

You said:

Leading coefficient test:

2x^3would be a positive coefficient multiplied by a negative variable=>it is negative
2x^4 would be a positive coefficient multiplied by a positive variable=>it is positive
-5x^3 would be a negative coefficient multiplied by a negative variable=>it is positive
-5x^2 would be a negative coefficient multiplied by a positive variable=>it is negative

same procedure for right-hand behavior (positive numbers)

1. Can you provide a list for the right-hand behavior?

You said:

". . .the coefficient and the degree of the component together greater than zero=>the graph rises."

2. What do you mean by the coefficient and the degree of the component together greater 0?

 

[MathLover1]

left-hand behavior (negative numbers),

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive

2x^2 =>the coefficient is 2 (means positive number greater 0)
if x=-1=> then x^2=(-1)^2=1 (means positive number greater 0)
so, combined 2x^2=(-1)^2=2*1=2 (means positive number greater 0) and y value goes up, so the graph rises


right-hand behavior (positive numbers)
the coefficient and the degree of the component together greater than zero=>the graph rises and approaching infinity

2x^2 =>the coefficient is 2 (means positive number greater 0)
if x=1=> then x^2=1^2=1 (means positive number greater 0)
so, combined 2x^2=2^1^2=2*1=2 (means positive number greater 0) and y value goes up, so the graph rises and approaching infinity

imagine you drawing this parabola starting at vertex, the curve goes from vertex up from both sides

This graph is a second degree polynomial (i.e. the highest power of x is 2). Since 2 is even, the end behavior will be the same for the left and the right. The leading coefficient (the number in front of the highest power of x - which in this case is 2) is positive. This means both the left and right hand behavior will be approaching infinity. In proper mathematical speak - you could say that: as x approaches positive infinity, f(x) approaches positive infinity and as x approaches negative infinity, f(x) also approaches positive infinity.

 

[nycmathguy]

left-hand behavior (negative numbers),

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive

2x^2 =>the coefficient is 2 (means positive number greater 0)
if x=-1=> then x^2=(-1)^2=1 (means positive number greater 0)
so, combined 2x^2=(-1)^2=2*1=2 (means positive number greater 0) and y value goes up, so the graph rises


right-hand behavior (positive numbers)
the coefficient and the degree of the component together greater than zero=>the graph rises

2x^2 =>the coefficient is 2 (means positive number greater 0)
if x=1=> then x^2=1^2=1 (means positive number greater 0)
so, combined 2x^2=2^1^2=2*1=2 (means positive number greater 0) and y value goes up, so the graph rises

imagine you drawing this parabola starting at vertex, the curve goes from vertex up from both sides

Wouldn't it be easier just to graph the function? From the graph itself, we can see the rise and fall.

 

[nycmathguy]

The leading coefficient test tells us that the graph rises or falls depending on whether the leading terms are positive or negative, so for left-hand behavior (negative numbers), you will need to look at both the coefficient and the degree of the component together.

Let’s look at the following examples of when x is negative:

Leading coefficient test:

2x^3would be a positive coefficient multiplied by a negative variable=>it is negative
2x^4 would be a positive coefficient multiplied by a positive variable=>it is positive
-5x^3 would be a negative coefficient multiplied by a negative variable=>it is positive
-5x^2 would be a negative coefficient multiplied by a positive variable=>it is negative

same procedure for right-hand behavior (positive numbers)

20.
f(x)= 2x^2-3x+1

left-hand behavior (negative numbers),

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive
right-hand behavior (positive numbers)

the coefficient and the degree of the component together greater than zero=>the graph rises

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive
the coefficient and the degree of the component together greater than zero=>the graph rises

as you can see, from the turning point (vertex) the graph goes up on both sides (left and right)

You said:

(20)= 2x^2-3x+1

left-hand behavior (negative numbers)...

1. What made you say left-hand behavior from simply looking at the given function?

2. Is there another way to tackle questions such as this one?

 

[nycmathguy]

The leading coefficient test tells us that the graph rises or falls depending on whether the leading terms are positive or negative, so for left-hand behavior (negative numbers), you will need to look at both the coefficient and the degree of the component together.

Let’s look at the following examples of when x is negative:

Leading coefficient test:

2x^3would be a positive coefficient multiplied by a negative variable=>it is negative
2x^4 would be a positive coefficient multiplied by a positive variable=>it is positive
-5x^3 would be a negative coefficient multiplied by a negative variable=>it is positive
-5x^2 would be a negative coefficient multiplied by a positive variable=>it is negative

same procedure for right-hand behavior (positive numbers)

20.
f(x)= 2x^2-3x+1

left-hand behavior (negative numbers),

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive
right-hand behavior (positive numbers)

the coefficient and the degree of the component together greater than zero=>the graph rises

2x^2 would be a positive coefficient multiplied by a positive variable=>it is positive
the coefficient and the degree of the component together greater than zero=>the graph rises

as you can see, from the turning point (vertex) the graph goes up on both sides (left and right)

Can we determine the left-hand and right-hand behavior of a graph using the number line?

 

[MathLover1]

no

 

[MathLover1]

simply looking at the graph of the given function you can see what is happening if x->infinity or if x-> -infinity

in this case you can see if x-> + infinity that f(x) ->+ infinity too
and if x-> -infinity that f(x) -> +infinity

+ infinity means right side limit
- infinity means left side limit

here is good example:

 

[nycmathguy]

simply looking at the graph of the given function you can see what is happening if x->infinity or if x-> -infinity

in this case you can see if x-> + infinity that f(x) ->+ infinity too
and if x-> -infinity that f(x) -> +infinity

+ infinity means right side limit
- infinity means left side limit

here is good example:

Thanks for video clip. I may need to use video lessons to learn most of the topics in Section 2.2. I will not post new questions until further notice. Feeling a bit frustrated in this section thus far.

 

[MathLover1]

here are some videos for you:

 

[nycmathguy]

here are some videos for you:

Thanks. I will view each video tomorrow.