Understanding the Leading Coefficient Test in Polynomial Functions

[lLovePhysics]

What does the leading coefficient test determine? For example, if you test the function $$f(x)=2x^{2}+3x+1$$ does the leading coefficient test tell you that the leading coefficient is 2?

Also, why does the leading coefficient determine whether the graph eventuall rises or falls? and why does it determine *only* the right and left behavior of the grpah? I just don't get it...

I only know that the leading coefficient's degree determines whether the graph is even or odd and therefore where it falls/rises.

 

[lLovePhysics]

Ohhh, does it mean whether the leading coefficient is negative or positive?

 

[symbolipoint]

This kind of question is one reason why schools today use graphing calculators in some Mathematics courses. Students could test a few related functions each using different leading coefficients and observe the effects.

For such a quadratic function, a coefficient greater than 1 and positive will stretch the function vertically; and increases as you go to the left toward infinity, and increases as you go to the right toward infinity.
If the coefficient is less than 1 and positive, then it is shrunk vertically; and decreases as x progresses to the left and decreases as x goes to the right. Actually, this explanation needs some modification, since for coefficient <0, there may be a maximum, and for coefficient >0, there should be a minimum. Did I confuse you?

 

[HallsofIvy]

If y= ax²+ bx+ c, then y= a(x- x₀)²+ d for some values of x₀ and d. While it might be very difficult to determine x₀ and d, a is exactly the leading coefficient of the polynomial. What does that tell you?

 

[symbolipoint]

If y= ax²+ bx+ c, then y= a(x- x₀)²+ d for some values of x₀ and d. While it might be very difficult to determine x₀ and d, a is exactly the leading coefficient of the polynomial. What does that tell you?

HallsOfIvy is trying to tell you that you can find out on your own what effect the coefficient "a" has on the function. My explanation, although a bit awkward, is also the answer. The college algebra and Pre-Calculus textbooks treat this topic very well. You should check this in one of those books.