The quadratic function $y=-10 x^2+160 x-430$ models a store's daily profit ($y$), in dollars, for selling T-shirts priced at $x$ dollars.
A. the daily profit the company would make from the T-shirts if it gave the T-shirts away for free
B. the greatest daily profit the company could make from selling the T-shirts
C. a selling price that would result in the company making no profit from the T-shirts
D. the selling price that would result in the company making the greatest daily profit from the T-shirts
Match each item with what it represents in this situation by entering the appropriate letter in each box.
$\square$ $x$-coordinate of the vertex of the function
$\square$ $y$-coordinate of the vertex of the function
$\square$ an $x$-intercept of the function
$\square$ a $y$-intercept of the function
Answer (2)
The x -coordinate of the vertex represents the selling price that maximizes profit, found using x = − 2 a b .
The y -coordinate of the vertex represents the maximum daily profit, calculated by substituting the x -coordinate of the vertex into the function.
The x -intercepts represent selling prices that result in no profit, found by solving the quadratic equation for x when y = 0 .
The y -intercept represents the profit when the selling price is zero, found by setting x = 0 in the function. The matches are: x-coordinate of the vertex: D, y-coordinate of the vertex: B, x-intercept: C, y-intercept: A. x -coordinate of the vertex: D, y-coordinate of the vertex: B, x-intercept: C, y-intercept: A
Explanation
Understanding the Problem We are given the quadratic function y = − 10 x 2 + 160 x − 430 which models the daily profit y of a store selling T-shirts at a price of x dollars. We need to match the given items with what they represent in this situation.
Finding the x-coordinate of the vertex The x -coordinate of the vertex represents the selling price that maximizes the daily profit. We can find the x -coordinate of the vertex using the formula x = − 2 a b , where a = − 10 and b = 160 . Thus, x = − 2 ( − 10 ) 160 = 20 160 = 8 . This corresponds to option D, the selling price that results in the company making the greatest daily profit.
Finding the y-coordinate of the vertex The y -coordinate of the vertex represents the greatest daily profit the company can make. We can find the y -coordinate by substituting x = 8 into the equation: y = − 10 ( 8 ) 2 + 160 ( 8 ) − 430 = − 10 ( 64 ) + 1280 − 430 = − 640 + 1280 − 430 = 210 . This corresponds to option B, the greatest daily profit the company could make from selling the T-shirts.
Finding the x-intercepts The x -intercepts represent the selling prices at which the company makes no profit ( y = 0 ). We can find the x -intercepts by setting y = 0 and solving for x : − 10 x 2 + 160 x − 430 = 0 . Using the quadratic formula, x = 2 a − b ± b 2 − 4 a c , where a = − 10 , b = 160 , and c = − 430 . Thus, x = 2 ( − 10 ) − 160 ± 16 0 2 − 4 ( − 10 ) ( − 430 ) = − 20 − 160 ± 25600 − 17200 = − 20 − 160 ± 8400 = − 20 − 160 ± 20 21 = 8 ± 21 . The two x -intercepts are approximately 8 + 21 ≈ 12.58 and 8 − 21 ≈ 3.42 . These values represent the selling prices that would result in the company making no profit. This corresponds to option C, a selling price that would result in the company making no profit from the T-shirts.
Finding the y-intercept The y -intercept represents the daily profit when the T-shirts are given away for free (i.e., x = 0 ). We can find the y -intercept by setting x = 0 in the equation: y = − 10 ( 0 ) 2 + 160 ( 0 ) − 430 = − 430 . This corresponds to option A, the daily profit the company would make from the T-shirts if it gave the T-shirts away for free.
Final Answer Therefore, the matches are as follows:
x -coordinate of the vertex of the function: D
y -coordinate of the vertex of the function: B
An x -intercept of the function: C
A y -intercept of the function: A
Examples
Understanding quadratic functions can help businesses optimize their pricing strategies. For instance, a local bakery can use a quadratic function to model the relationship between the price of a cake and the number of cakes sold daily. By finding the vertex of this function, the bakery can determine the optimal price that maximizes their daily revenue. Similarly, farmers can use quadratic functions to model crop yield based on the amount of fertilizer used, helping them find the amount of fertilizer that maximizes their harvest. These models enable informed decision-making, leading to increased profitability and efficiency.
The matches are: the x -coordinate of the vertex corresponds to selling price maximizing profit (D), the y -coordinate is the greatest daily profit (B), an x -intercept indicates no profit (C), and a y -intercept shows profit when T-shirts are free (A).
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