If f(x) has zeros at x=a, and x=b, the x coordinate of the turning point between x= a and x=b is 1/2(a+b).
PLease help i don't understand this!
is it some times true or always true?? that's the question
Turning point is the point at which the slope of the graph changes direction, i.e. from positive to negative or vice versa.
This happens when dy/dx=0.
In the case of a quadratic function,
f(x)=ax²+bx+c,
dy/dx=2ax+b=0, or this happens when
x=-b/2a
It turns ou5 that the real zeroes of the quadratic function are at
x1,x2=(-b±sqrt(b²-4ac))/2a
and (x1+x2)/2 = -b/2a.
So, yes, the "turning point" where the function is a maximum/minimum happens at the average of the two zeroes. However, this is true only for the case of the quadratic equations, and is not generally true for all functions.
but what if its not a quadratic function?.. isn't it possible to have a function that has two zeros but is not a quadratic.. for example the first zero is passed by a cubic kind of curve that is connected to a straight line going to the other zero? so would it be quintic, so does the same thing apply?
And is the question always true, sometimes true, or never true?
"is not generally true for all functions. "
means that it is possible that dy/dx=0 at the average of two roots, but in general it is not true.
Example when it is true:
sin(x)=0 at x=0 and x=π.
dsin(x)/dx=0 at x=π/2
Example when it is not true:
y=x(x-1)(x-2)=x³-3x²+2x
y(0)=y(1)=y(2)=0
dy/dx=0 at x=1±(√3)/3
To understand why the x-coordinate of the turning point between x = a and x = b is given by 1/2(a + b), let's break it down step by step:
1. Turning point:
In mathematical terms, a turning point refers to the highest or lowest point on a curve. It is the point where the function changes its direction from increasing to decreasing or vice versa. If a function has zeros at x = a and x = b, it means that f(a) = 0 and f(b) = 0. This implies that the graph of the function intersects the x-axis at both points.
2. Symmetry:
A special property of functions with a turning point is that they often exhibit symmetry. In this case, because the function has zeros at x = a and x = b, the turning point is equidistant from these two points. In other words, the x-coordinate of the turning point is located right in the center between a and b.
3. Midpoint formula:
The midpoint formula is a mathematical formula that calculates the midpoint between two given points. It states that the coordinates of the midpoint M between two points (x₁, y₁) and (x₂, y₂) can be found using the following formula:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
In this case, the x-coordinate of the turning point is the midpoint between a and b. Applying the midpoint formula, we can substitute x₁ with a and x₂ with b:
M = ((a + b)/2, y)
Since we're only interested in the x-coordinate, we can ignore the "y" part of the formula.
4. Simplification:
To simplify the expression, we can remove the unnecessary "y" term. Therefore, the x-coordinate of the turning point between x = a and x = b is:
x = (a + b)/2
Hence, 1/2(a + b) is the x-coordinate of the turning point between x = a and x = b. It represents the midpoint between the two zeros of the function.