Activity
Ashton designs movie sets. For one scene, he has to hang a banner across an archway that takes the shape of a parabola modeled by the quadratic equation y = −x2 + 6x. The banner is to be hung at an angle defined by the linear equation y − x = 4, as shown in the figure.

A rectangle named banner inside a parabola opens downward named archway.

Ashton needs to know the how long a banner to order. He can find the length using the distance formula if he knows the points on the archway at which the ends of the banner are to be attached.

Question 1
Using the graphing method, find the points at which the equation of the angle of the banner will intersect the equation of the archway.















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Question 2
Using the algebraic method, confirm whether the two points obtained by the graphing method are correct.















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Question 3
Using the distance formula
, find the distance between points A and B. How long a banner does Ashton need to order?















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Question 4
Find the values of x and y that satisfy 3x − 1 + 2y = 0 and 3x2 − y2 + 4 = 0. Use the algebraic method to find the solution, and then use the graphing method to confirm whether the system of equations has real roots.















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Self-Evaluation
How did you do? Rate your work on a scale of 1 to 5, with 5 as the highest score. Then write a brief evaluation of your work below. Note what you learned and what challenged you.















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