Reviewed by Florence Khitsane
Over what interval is the graph of f(x) = -(x + 8)² – 1 decreasing?
Answer: The graph of f(x) = -(x + 8)² – 1 is decreasing for all x > -8
The function f(x) = -(x + 8)² – 1 is a quadratic function that opens downward due to its negative leading coefficient. Understanding where a quadratic function is increasing or decreasing involves analyzing its vertex and the direction in which its parabola opens. Identifying the decreasing interval helps us understand how the function behaves over the domain.
Methods
Math Tutor Explanation Using the Vertex Method
The vertex of a parabola determines where the function changes from increasing to decreasing (or vice versa).
Step 1: Step 1: Identify the vertex of f(x) using the formula x = -b/(2a) for a quadratic equation
Step 2: Step 2: Since f(x) = -(x + 8)² – 1 is in vertex form, recognize that the vertex is at (-8, -1)
Math Tutor Explanation Using the Derivative Method
Finding where the derivative is negative will tell us where the function is decreasing.
Step 1: Step 1: Compute the derivative: f'(x) = -2(x + 8)
Step 2: Step 2: Set f'(x) < 0 to find decreasing intervals: -2(x + 8) < 0, which simplifies to x > -8
Step 1:
Step 2:
Math Tutor suggests: Master Quadratic Functions & Their Graphs
Strengthen your understanding of quadratic functions, intervals of increase and decrease, and graph analysis with these related exercises.
FAQ on Quadratic Function Behavior
How do you find if a quadratic is opening upward or downward?
If the leading coefficient (a) is positive, it opens upward; if negative, it opens downward.
What is the vertex of f(x) = -(x + 8)² – 1?
The vertex is at the point (-8, -1).
Why is the function decreasing for x > -8?
Because the parabola opens downward, it decreases to the right of its vertex at x = -8.
Can a quadratic function be decreasing everywhere?
No, for quadratic functions, the graph is always increasing on one side of the vertex and decreasing on the other.
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