Math Problem Statement

Determine the domain on which the following function is increasing.

Solution

The image shows a parabola, and the question asks to "Determine the domain on which the following function is increasing."

For a quadratic function in the form of a parabola (like the one in the image), the domain where the function is increasing refers to the interval of x-values where the slope of the curve is positive (the curve is going upward).

Key Points:

  • The parabola opens downward, which means it has a maximum point at its vertex.
  • The function increases to the left of the vertex.

To solve:

  1. Find the vertex of the parabola: This will give us the turning point where the function changes from increasing to decreasing.
  2. Determine the domain where the function is increasing: This will be for x-values less than the x-coordinate of the vertex.

If you provide the specific coordinates or more details about the graph, I can pinpoint the exact domain.

Would you like to provide further details or ask more questions related to this?

Related Questions:

  1. How do you find the vertex of a quadratic function algebraically?
  2. Can you explain the difference between the domain and the range of a function?
  3. What are the characteristics of a function that is increasing or decreasing?
  4. How does the concavity of a parabola affect where it's increasing or decreasing?
  5. How can the first derivative help in determining where a function increases or decreases?

Tip:

For a parabola, the x-coordinate of the vertex can be found using the formula x=b2ax = -\frac{b}{2a}, where the quadratic equation is in the form ax2+bx+cax^2 + bx + c.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Parabolas

Formulas

Vertex formula x = -b / 2a
Quadratic equation: ax^2 + bx + c = 0

Theorems

Properties of parabolas
Increasing/Decreasing Intervals for Quadratic Functions

Suitable Grade Level

Grades 9-11