What are the increasing and decreasing intervals for the equation y=x^2-2x-15
You have a parabola opening upwards,
so the function is decreasing to the left of the vertex and increasing to the right of the vertex.
the x of the vertex = -b/(2a) = 2/2 = 1
so decreasing for x < 1
increasing for x>1
To determine the increasing and decreasing intervals for the equation y=x^2-2x-15, we need to find the critical points and examine the sign of the derivative.
Step 1: Find the derivative of the equation
Taking the derivative of y=x^2-2x-15, we get:
dy/dx = 2x - 2
Step 2: Set the derivative equal to zero and solve for x to find critical points
2x - 2 = 0
2x = 2
x = 1
Step 3: Determine the sign of the derivative for different intervals
To determine the sign of the derivative, we can use a sign chart. Plug in test values into the derivative expression to evaluate the sign in each interval.
Test x = 0: dy/dx = 2(0) - 2 = -2 (negative)
Test x = 2: dy/dx = 2(2) - 2 = 2 (positive)
Step 4: Analyze the sign chart to find increasing and decreasing intervals
From the sign chart, we can conclude:
- The derivative is negative (-) when x < 1, indicating a decreasing interval.
- The derivative is positive (+) when x > 1, indicating an increasing interval.
Therefore, the decreasing interval is (-∞, 1) and the increasing interval is (1, ∞).
To determine the increasing and decreasing intervals of the equation y = x^2 - 2x - 15, we need to analyze the behavior of the quadratic function.
First, let's find the critical points by taking the derivative of the function with respect to x. The derivative of y = x^2 - 2x - 15 is obtained by applying the power rule:
dy/dx = 2x - 2.
Next, we set the derivative equal to zero to find the critical points:
2x - 2 = 0.
Solving this equation, we get:
2x = 2,
x = 1.
So, x = 1 is the critical point of the quadratic function.
Now, we need to evaluate the sign of the derivative in different intervals to determine whether the function is increasing or decreasing.
For x < 1:
- Choose any value less than 1, e.g., x = 0.
- Substitute this value into the derivative: 2(0) - 2 = -2.
- Since the derivative is negative, the quadratic function is decreasing for values of x less than 1.
For x > 1:
- Choose any value greater than 1, e.g., x = 2.
- Substitute this value into the derivative: 2(2) - 2 = 2.
- Since the derivative is positive, the quadratic function is increasing for values of x greater than 1.
Therefore, the increasing interval is (1, +∞) and the decreasing interval is (-∞, 1).