Determine the intervals on which the function is increasing, decreasing, and constant.
A parabola is shown facing down with a vertex of 0,1.
To determine the intervals on which the function is increasing, decreasing, or constant for a given parabola, we need to examine the behavior of the function's derivative.
Let's start by finding the equation of the given parabola. Since the vertex is given as (0,1), and the parabola is facing downward, we can write the equation in the form of:
f(x) = ax^2 + bx + c
Given the vertex (0,1), we know that x = 0 corresponds to the vertex, so substituting these values into the equation, we have:
f(0) = a(0)^2 + b(0) + c = 1
This tells us that the constant term c is equal to 1.
Next, we need to determine the values of a and b. For this, we can consider another point on the parabola. Let's say we have a point (x1, y1) on the parabola. Since the vertex is at (0,1), it would be helpful to choose another point symmetrically opposite to the vertex. This point would be (-x1, y1).
Substituting these values into the equation of the parabola, we get:
y1 = a(x1)^2 + b(x1) + 1 ..... (1)
y1 = a(-x1)^2 + b(-x1) + 1 ..... (2)
Simplifying equation (1) and (2), we have:
y1 = ax1^2 + bx1 + 1 ..... (3)
y1 = ax1^2 - bx1 + 1 ..... (4)
Equating equations (3) and (4), we get:
ax1^2 + bx1 + 1 = ax1^2 - bx1 + 1
Simplifying further, we have:
2bx1 = 2bx1
This tells us that b can be any value, as long as it is equal to its opposite (-b).
Therefore, the equation of the parabola can be written as:
f(x) = ax^2 + bx + 1
Now, let's determine the intervals on which the function is increasing, decreasing, or constant.
To find the critical points of the function, we need to find where the derivative of the function is equal to zero or undefined. Since we are dealing with a parabola that is symmetric, its derivative will be zero at the vertex.
The derivative of the function f(x) = ax^2 + bx + 1 can be found by using the power rule:
f'(x) = 2ax + b
Setting this derivative equal to zero and solving for x, we have:
2ax + b = 0
2ax = -b
x = -b / (2a)
Since the parabola has a vertex at (0,1), we know that x = 0 when the derivative is zero. Therefore, we have the equation:
0 = -b / (2a)
Simplifying, we find:
b = 0
This tells us that the function is decreasing to the left of the vertex (0,1) and increasing to the right of the vertex. Graphically, this represents the parabola facing downward with a vertex at (0,1).
Therefore, the function is decreasing in the interval (-β, 0) and increasing in the interval (0, β). The function is also constant at the vertex point (0,1).
To summarize:
- The function is decreasing on the interval (-β, 0).
- The function is constant at the vertex (0,1).
- The function is increasing on the interval (0, β).