Maximum Value of a Quadratic in Vertex Form

Directions: Create a quadratic equation with the greatest possible maximum value using the digits 1 through 9, no more than one time each.


Where is the best place to put the 9? How does each box affect the graph’s behavior?


There are many solutions but a 9 must go into the last term as the k-value. That affects its y-axis position while the the a and h terms affect its width and x-axis position.  Play with the sliders on this Desmos graph for a hands on way of understanding how the a, x, and k terms affect the quadratic:

Source: Robert Kaplinsky

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  1. What makes this problem a DOK 3? It seems too simple to be a DOK 3. Please help me understand. Thank you.

    • Robert Kaplinsky

      Hi Karen. I’m not going to win an award for a speedy response, but the reason it’s a DOK 3 is the optimization. It’s challenging enough to make a quadratic that (as an example) goes through a given point. However, to find the absolute maximum value requires much deeper understanding.

  2. I understand what Karen is saying. The constant at the end is the value that affects the maximum. So it must be 9. The other two numbers can be anything as the number in the parentheses affects the horizontal shift, and the constant in front of the parentheses affects how wide or narrow the parabola is. Unless you are allowing multiple digits in each box? As long as you know how each number affects the shape or location of the parabola, this is not too difficult a question.

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