Math Problem Statement

Which is the vertex of the quadratic function f(x) = 2x^2 - 4x - 6?

Solution

To find the vertex of the quadratic function f(x)=2x24x6f(x) = 2x^2 - 4x - 6, we can use the vertex formula for a parabola given by f(x)=ax2+bx+cf(x) = ax^2 + bx + c. The x-coordinate of the vertex is calculated using:

x=b2ax = -\frac{b}{2a}

For the given function f(x)=2x24x6f(x) = 2x^2 - 4x - 6:

  • a=2a = 2
  • b=4b = -4

Let's find the x-coordinate:

x=422=44=1x = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1

Now, substitute x=1x = 1 back into the function to find the y-coordinate:

f(1)=2(1)24(1)6=246=8f(1) = 2(1)^2 - 4(1) - 6 = 2 - 4 - 6 = -8

So, the vertex of the function is at (1,8)(1, -8).

The correct answer is:

  • (1,8)(1, -8)

Would you like more details or have any questions? Here are some related questions:

  1. How do you derive the vertex formula x=b2ax = -\frac{b}{2a}?
  2. What is the significance of the vertex in the graph of a parabola?
  3. How can you determine whether a parabola opens upward or downward?
  4. What role does the coefficient aa play in the shape of a quadratic function?
  5. How do you convert a quadratic function into vertex form?

Tip: The vertex form of a quadratic function, f(x)=a(xh)2+kf(x) = a(x - h)^2 + k, makes it easier to identify the vertex, where (h,k)(h, k) is the vertex.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Parabolas

Formulas

Vertex formula: x = -b/(2a)
Standard form of a quadratic: f(x) = ax^2 + bx + c

Theorems

Properties of parabolas
Vertex of a parabola

Suitable Grade Level

Grades 9-12