Math Problem Statement
Solution
To find the range of the function for , let's analyze the function step-by-step.
Step 1: Rewrite the Function in Vertex Form
The function is a quadratic function, and it can be rewritten in vertex form to make it easier to analyze the minimum or maximum values.
-
Rewrite by completing the square:
This shows that the function can be expressed as , where the vertex is at .
Step 2: Determine the Minimum Value
Since is a parabola that opens upwards, its minimum value occurs at the vertex , where:
Step 3: Evaluate the Function at the Endpoints
Next, evaluate at the endpoints of the interval :
-
For :
-
For :
Step 4: Determine the Range
From the calculations above:
- The minimum value of is , occurring at .
- The maximum value of on the interval is , occurring at .
Thus, the range of for is:
Would you like further details on any specific part of the solution?
Here are some questions that expand on this solution:
- How can you find the range of a quadratic function for a different interval?
- What if the function opened downwards? How would the range change?
- How does completing the square help in finding the vertex of a quadratic function?
- What is the significance of the vertex form of a quadratic function?
- How would the process change if the function had a different coefficient for ?
Tip: Always check the endpoints of the interval when finding the range of a function restricted to a closed interval.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Function Range
Formulas
Vertex form of a quadratic function
Range of a function on a closed interval
Theorems
Completing the square for quadratic functions
Suitable Grade Level
Grades 9-12
Related Recommendation
Finding the Range of a Quadratic Function on a Given Interval
Find the Range of the Quadratic Function (x + 2)^2 - 3
Finding the Range of the Function f(x) = x^2 + 1 with Domain (-2, 4)
Find the Range of the Function f(x) = x^2 - 2 for a Given Domain
Find the Range of Quadratic Function f(x) = x^2 - 2x - 3