Math Problem Statement

Use the equivalent forms found below to identify the key characteristics of g(x): g(x) = 2(x-4)^2 + 2, g(x) = 2x^2 - 16x + 30, and g(x) = 2(x-3)(x-5).

Solution

Let's analyze the provided functions step by step to identify the characteristics of g(x)g(x). The three forms of the function g(x)g(x) are given as:

  1. g(x)=2(x4)2+2g(x) = 2(x - 4)^2 + 2
  2. g(x)=2x216x+30g(x) = 2x^2 - 16x + 30
  3. g(x)=2(x3)(x5)g(x) = 2(x - 3)(x - 5)

Y-Intercept:

The y-intercept occurs when x=0x = 0. Using the second form g(x)=2x216x+30g(x) = 2x^2 - 16x + 30, we can find the y-intercept: g(0)=2(0)216(0)+30=30g(0) = 2(0)^2 - 16(0) + 30 = 30 Thus, the y-intercept is (0,30)(0, 30).

Minimum Value:

The first form g(x)=2(x4)2+2g(x) = 2(x - 4)^2 + 2 is in vertex form, which is useful for determining the minimum or maximum. Since the coefficient of (x4)2(x - 4)^2 is positive, the parabola opens upwards, indicating that the vertex is a minimum point. The vertex occurs at x=4x = 4, and the minimum value of g(x)g(x) is: g(4)=2(44)2+2=2g(4) = 2(4 - 4)^2 + 2 = 2 So, the minimum value is 2.

Zeros:

To find the zeros, use the third form g(x)=2(x3)(x5)g(x) = 2(x - 3)(x - 5). Set g(x)=0g(x) = 0: 2(x3)(x5)=02(x - 3)(x - 5) = 0 This gives x=3x = 3 and x=5x = 5.

End Behavior:

The end behavior of a quadratic function depends on the leading term. In the second form g(x)=2x216x+30g(x) = 2x^2 - 16x + 30, the leading term is 2x22x^2, which dominates as x±x \to \pm \infty.

  • As xx \to -\infty, g(x)+g(x) \to +\infty because the parabola opens upwards (positive leading coefficient).
  • As x+x \to +\infty, g(x)+g(x) \to +\infty.

Summary of Characteristics:

  • Y-intercept: (0,30)(0, 30)
  • Minimum value: 2
  • Zeros: 3 and 5 (listed in increasing order)
  • End behavior:
    • As xx \to -\infty, g(x)+g(x) \to +\infty
    • As x+x \to +\infty, g(x)+g(x) \to +\infty

Would you like any further details or clarifications?

Here are 5 related questions for further exploration:

  1. How does the vertex form of a quadratic function help in identifying key features like the vertex and direction of the parabola?
  2. How can you confirm the minimum or maximum value of a quadratic function using derivatives?
  3. What does the discriminant tell you about the number of zeros of a quadratic function?
  4. How would the end behavior change if the leading coefficient were negative?
  5. Can you determine the axis of symmetry for g(x)g(x) using any of the forms?

Tip: For quadratic functions, the vertex form a(xh)2+ka(x - h)^2 + k directly gives the vertex at (h,k)(h, k), making it useful for finding the minimum or maximum point.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Factoring
Vertex Form
Intercepts
End Behavior

Formulas

Vertex form of a quadratic equation: a(x-h)^2 + k
Factored form of a quadratic equation: a(x - p)(x - q)
Standard form of a quadratic equation: ax^2 + bx + c
End behavior determined by leading term

Theorems

Quadratic formula
Properties of parabolas
Intercepts theorem

Suitable Grade Level

Grades 9-10