Math Problem Statement

What is the vertex of the function?

f(x)=4x2+8x+3

Solution

To find the vertex of a quadratic function of the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, we use the vertex formula:

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

For the given function f(x)=4x2+8x+3f(x) = 4x^2 + 8x + 3, we have:

  • a=4a = 4
  • b=8b = 8

Let's calculate the xx-coordinate of the vertex:

x=82(4)=88=1x = \frac{-8}{2(4)} = \frac{-8}{8} = -1

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

f(1)=4(1)2+8(1)+3=4(1)8+3=48+3=1f(-1) = 4(-1)^2 + 8(-1) + 3 = 4(1) - 8 + 3 = 4 - 8 + 3 = -1

So, the vertex is (1,1)(-1, -1).

The vertex of the function is (1,1)(-1, -1).

Would you like more details or have any questions?


Here are 5 related questions to expand your understanding:

  1. How do you find the axis of symmetry for a quadratic function?
  2. What is the significance of the vertex in a quadratic function?
  3. How does the coefficient of aa affect the direction of the parabola?
  4. Can you rewrite the quadratic function in vertex form?
  5. How do you find the intercepts of a quadratic function?

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 and graph the function.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Vertex of a Parabola

Formulas

Vertex formula: x = -b / 2a
Quadratic function: f(x) = ax^2 + bx + c

Theorems

Vertex Theorem for Quadratic Functions

Suitable Grade Level

Grades 9-11