Math Problem Statement

Find the vertex and axis of symmetry for the quadratic function f(x) = 2x^2 + 8x + 7.

Solution

The given function is a quadratic equation, and you are looking for the vertex and axis of symmetry.

Function:

f(x)=2x2+8x+7f(x) = 2x^2 + 8x + 7

1. Finding the Vertex:

For any quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c, the x-coordinate of the vertex is given by: x=b2ax = \frac{-b}{2a} For your equation, a=2a = 2 and b=8b = 8, so: x=82×2=84=2x = \frac{-8}{2 \times 2} = \frac{-8}{4} = -2

Now, substitute x=2x = -2 into the function to find the y-coordinate of the vertex: f(2)=2(2)2+8(2)+7=2(4)16+7=816+7=1f(-2) = 2(-2)^2 + 8(-2) + 7 = 2(4) - 16 + 7 = 8 - 16 + 7 = -1

So, the vertex is (2,1)(-2, -1).

2. Finding the Axis of Symmetry:

The axis of symmetry is the vertical line that passes through the x-coordinate of the vertex. Therefore, the axis of symmetry is: x=2x = -2

Summary:

  • Vertex: (2,1)(-2, -1)
  • Axis of Symmetry: x=2x = -2

Would you like a further breakdown or have any other questions? Here are some related questions to explore:

  1. How do you graph a quadratic function from its vertex and axis of symmetry?
  2. What does the value of aa represent in a quadratic function?
  3. How can you determine if the parabola opens upwards or downwards from the equation?
  4. How does completing the square help find the vertex of a quadratic equation?
  5. Can the vertex form of a quadratic equation provide more insight into its graph?

Tip: Always remember that the axis of symmetry divides the parabola into two symmetrical parts, and the vertex gives you the minimum or maximum point, depending on the direction the parabola opens!

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex Form
Axis of Symmetry

Formulas

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

Theorems

Vertex theorem for quadratic functions

Suitable Grade Level

Grades 8-10