Math Problem Statement

Identify which form is represented and the information given by that form for the provided functions.

Solution

Based on the provided image, it appears to be a mathematical question asking for the identification of forms for functions and the features that are most easily identifiable in each case. Here's an analysis:

  1. Function 5: f(x)=3x22x+6f(x) = 3x^2 - 2x + 6

    • Form: This is in standard form (quadratic polynomial form).
    • Feature most easily identifiable: The y-intercept can be identified directly as 6 because the constant term gives the value of f(0)f(0).
  2. Function 6: f(x)=(x4)(x2)f(x) = (x - 4)(x - 2)

    • Form: This is in factored form.
    • Feature most easily identifiable: The x-intercepts are most easily identifiable as x=4x = 4 and x=2x = 2, based on the factors.
  3. Function 7: f(x)=(x+3)24f(x) = (x + 3)^2 - 4

    • Form: This is in vertex form (a transformation of the standard quadratic function).
    • Feature most easily identifiable: The vertex is most easily identifiable as (3,4)(-3, -4), derived from the equation's structure.

Would you like further details on these concepts? Here are some related questions to expand understanding:

  1. How can you convert between factored form, vertex form, and standard form for quadratic equations?
  2. What is the significance of the vertex in a quadratic function?
  3. How do you determine the axis of symmetry from vertex form?
  4. What are the key steps to graph a quadratic function from its standard form?
  5. How can you use the discriminant to understand the nature of the roots in standard form?

Tip: To quickly identify features like the vertex, intercepts, or roots, match the function to the corresponding form and apply its defining characteristics.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Forms of Quadratic Functions

Formulas

Standard Form: f(x) = ax^2 + bx + c
Factored Form: f(x) = a(x - r1)(x - r2)
Vertex Form: f(x) = a(x - h)^2 + k

Theorems

Vertex Form Property
Factored Form Property

Suitable Grade Level

Grades 9-11