4 Pull It All Together

BIG idea Equivalence and Function

The parameters a, b, c, h, and k in the standard and vertex forms of a quadratic function give information on how the graph of the function relates to the graph of the parent function y = x 2 . y equals , x squared , .

Standard form: y = a x 2 + b x + c y equals eh , x squared , plus b x plus c

Vertex form: y = a ( x − h ) 2 + k y equals eh . open x minus h close squared . plus k

Task 1

Refer to the two forms shown above.

1. What information do the parameters, or combinations of parameters, provide about the graph of the quadratic function?
2. Begin with standard form. Transform it to vertex form. What are the values of h and k in terms of a, b, and c?
3. Show how the Quadratic Formula follows from your result in part (b). Hint: Set the expression in your vertex form equal to 0. Then solve by factoring.

BIG idea Solving Equations and Inequalities

A problem may require different types of equation solving. You should know when and how to use a graphing calculator to help you with your work.

Task 2

You shoot an arrow at a target. The parabolic path of your arrow passes through the points shown in the table.

1. Find a quadratic function in standard form that models the path of your arrow. Hint: The three points are (x, y)-values that satisfy y = a x 2 + b x + c . y equals eh , x squared , plus b x plus c .
2. If the y-value represents height above the ground, for what value of x would your arrow hit the ground if you miss the target?
3. If the target bull's-eye is at x = 100 , x equals 100 comma  at what height should the bull's-eye be for your arrow to hit it?
4. If the target bull's-eye is at height y = 2.98, at what value of x should the bull's-eye be for the arrow to hit it?

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