Math Problem Statement

Determine the equation of the quadratic function shown below in factored form.

Solution

To determine the equation of the quadratic function shown in the graph in factored form, let's analyze the given information:

  1. Intercepts:

    • The graph intersects the x-axis at x=2x = 2 and x=6x = 6. These are the roots of the quadratic equation.
  2. Factored Form:

    • The general factored form of a quadratic equation is: y=a(xr1)(xr2)y = a(x - r_1)(x - r_2) where r1r_1 and r2r_2 are the roots (or x-intercepts) of the equation. Here, r1=2r_1 = 2 and r2=6r_2 = 6, so the equation becomes: y=a(x2)(x6)y = a(x - 2)(x - 6)
  3. Vertex Point:

    • From the graph, the vertex is visible at (4,4)(4, -4). The vertex helps us determine the value of aa.
  4. Substitute the Vertex into the Equation:

    • Substituting x=4x = 4 and y=4y = -4 into the equation y=a(x2)(x6)y = a(x - 2)(x - 6): 4=a(42)(46)-4 = a(4 - 2)(4 - 6) Simplify: 4=a(2)(2)-4 = a(2)(-2) 4=4a-4 = -4a a=1a = 1
  5. Final Equation:

    • Substituting a=1a = 1 back into the equation: y=(x2)(x6)y = (x - 2)(x - 6)
    • Expand to verify: y=x26x2x+12=x28x+12y = x^2 - 6x - 2x + 12 = x^2 - 8x + 12

Conclusion:

The equation of the quadratic function in factored form is: y=(x2)(x6)y = (x - 2)(x - 6)

Let me know if you have any questions!


Here are five related questions to deepen your understanding:

  1. How do we determine the vertex of a parabola from its graph?
  2. What role does the value of aa play in the shape of a parabola?
  3. How would the equation change if the parabola opened upwards instead of downwards?
  4. Can you find the standard form of a quadratic equation directly from the graph?
  5. How would you determine the axis of symmetry of a parabola based on its roots?

Tip: Always verify the final equation by substituting key points from the graph!

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Factored Form
Vertex Form

Formulas

y = a(x - r1)(x - r2)
Substitution to find 'a' using vertex

Theorems

Vertex property of parabolas
Factored form of quadratic equations

Suitable Grade Level

Grades 8-10