Math Problem Statement
Solution
To solve the given problem, we will proceed as follows:
(i) Find and
The function given is .
-
Finding :
-
Finding :
(ii) State the range of
To find the range of for , we will:
-
Complete the square to find the vertex: The vertex of the parabola is at and .
-
Evaluate at the endpoints of the interval :
- At :
- At :
Since the parabola opens upwards, the minimum value within occurs at the vertex , and we have evaluated the function at the endpoints of the interval to find the maximum values. Therefore, the range of for is:
Summary
(i) and
(ii) The range of for is .
Would you like more details on any of these steps, or do you have any questions?
Related Questions
- What is the vertex form of a quadratic function?
- How do you complete the square for a quadratic equation?
- How can you determine the range of a quadratic function on a given interval?
- What are the properties of parabolas that open upwards?
- How do you find the maximum and minimum values of a function on a closed interval?
- What is the significance of the vertex in a parabola?
- How do transformations affect the graph of a quadratic function?
- Can you explain how to solve quadratic equations by factoring?
Tip
When finding the range of a quadratic function within a specific interval, always evaluate the function at the endpoints and the vertex if it lies within the interval.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Vertex Form
Completing the Square
Range of Functions
Formulas
Quadratic function formula f(x) = ax^2 + bx + c
Vertex form of a quadratic function
Theorems
Properties of parabolas
Suitable Grade Level
Grades 10-12
Related Recommendation
Find the Range of the Quadratic Function y = x^2 + 4x + 9 Over [-6, 1]
Range of Quadratic Function: f(x) = -2x^2 + 16x - 33
Find the Axis of Symmetry, Domain, and Range of f(x) = 3(x - 1)^2 - 4
Solving the Quadratic Function f(x) = x² - 8x + 16
Vertex Form and Intercepts of Quadratic Function f(x) = -x² + 16x