The Function Y = − 2 ( X − 3 ) 2 + 4 Y=-2(x-3)^2+4 Y = − 2 ( X − 3 ) 2 + 4 Shows The Daily Profit (in Hundreds Of Dollars) Of A Hot Dog Stand, Where X X X Is The Price Of A Hot Dog (in Dollars). Find And Interpret The Zeros Of This Function.Select Two Answers: One For The Zeros And

Understanding the Function

The given function, $y=-2(x-3)^2+4$, represents the daily profit of a hot dog stand in hundreds of dollars. The variable $x$ denotes the price of a hot dog in dollars. To find the zeros of this function, we need to determine the values of $x$ at which the daily profit becomes zero.

Finding the Zeros

To find the zeros of the function, we set $y$ equal to zero and solve for $x$. This gives us the equation:

$$-2(x-3)^2+4=0$$

We can start by isolating the squared term:

$$-2(x-3)^2=-4$$

Dividing both sides by $-2$ gives us:

$$(x-3)^2=2$$

Taking the square root of both sides, we get:

$$x-3=\pm\sqrt{2}$$

Adding $3$ to both sides, we find the two possible values of $x$:

$$x=3\pm\sqrt{2}$$

Interpreting the Zeros

The two zeros of the function are $x=3+\sqrt{2}$ and $x=3-\sqrt{2}$. These values represent the prices at which the daily profit of the hot dog stand becomes zero.

• Price 1: $x=3+\sqrt{2}$

  At this price, the daily profit of the hot dog stand is zero. This means that if the hot dog stand sells hot dogs at a price of $3+\sqrt{2}$ dollars, it will not make any profit. This price is higher than the current price of $3$ dollars, indicating that the hot dog stand needs to increase its price to break even.

• Price 2: $x=3-\sqrt{2}$

  At this price, the daily profit of the hot dog stand is also zero. This means that if the hot dog stand sells hot dogs at a price of $3-\sqrt{2}$ dollars, it will not make any profit. This price is lower than the current price of $3$ dollars, indicating that the hot dog stand needs to decrease its price to break even.

Discussion

The zeros of the function provide valuable insights into the pricing strategy of the hot dog stand. By analyzing the zeros, we can determine the prices at which the hot dog stand will break even. This information can be used to make informed decisions about pricing and profit margins.

Conclusion

In conclusion, the zeros of the function $y=-2(x-3)^2+4$ represent the prices at which the daily profit of the hot dog stand becomes zero. By finding and interpreting the zeros, we can gain a deeper understanding of the pricing strategy of the hot dog stand and make informed decisions about pricing and profit margins.

Additional Insights

• Profit Maximization: To maximize profit, the hot dog stand should sell hot dogs at a price higher than the current price of $3$ dollars. This will result in a positive daily profit.
• Price Elasticity: The zeros of the function indicate that the price elasticity of demand is high. This means that small changes in price can result in significant changes in demand.
• Break-Even Analysis: The zeros of the function provide a break-even analysis for the hot dog stand. By selling hot dogs at the break-even price, the hot dog stand will not make any profit.

Real-World Applications

The concept of zeros in a function has numerous real-world applications in various fields, including:

• Business: Zeros in a function can represent the break-even points for a business, indicating the prices at which the business will not make any profit.
• Economics: Zeros in a function can represent the equilibrium points in a market, indicating the prices at which supply and demand are equal.
• Finance: Zeros in a function can represent the points at which an investment becomes profitable, indicating the prices at which the investment will yield a positive return.

Conclusion

Q: What are the zeros of the function $y=-2(x-3)^2+4$?

A: The zeros of the function are $x=3+\sqrt{2}$ and $x=3-\sqrt{2}$. These values represent the prices at which the daily profit of the hot dog stand becomes zero.

Q: What do the zeros of the function represent?

A: The zeros of the function represent the prices at which the daily profit of the hot dog stand becomes zero. This means that if the hot dog stand sells hot dogs at these prices, it will not make any profit.

Q: How can the zeros of the function be used in real-world applications?

A: The zeros of the function can be used in various real-world applications, including:

• Business: Zeros in a function can represent the break-even points for a business, indicating the prices at which the business will not make any profit.
• Economics: Zeros in a function can represent the equilibrium points in a market, indicating the prices at which supply and demand are equal.
• Finance: Zeros in a function can represent the points at which an investment becomes profitable, indicating the prices at which the investment will yield a positive return.

Q: What is the significance of the zeros of the function in the context of the hot dog stand?

A: The zeros of the function are significant in the context of the hot dog stand because they represent the prices at which the daily profit becomes zero. This information can be used to make informed decisions about pricing and profit margins.

Q: How can the zeros of the function be used to maximize profit?

A: To maximize profit, the hot dog stand should sell hot dogs at a price higher than the current price of $3$ dollars. This will result in a positive daily profit.

Q: What is the relationship between the zeros of the function and the price elasticity of demand?

A: The zeros of the function indicate that the price elasticity of demand is high. This means that small changes in price can result in significant changes in demand.

Q: How can the zeros of the function be used to perform a break-even analysis?

A: The zeros of the function can be used to perform a break-even analysis for the hot dog stand. By selling hot dogs at the break-even price, the hot dog stand will not make any profit.

Q: What are some common applications of the concept of zeros in a function?

A: Some common applications of the concept of zeros in a function include:

• Business: Zeros in a function can represent the break-even points for a business, indicating the prices at which the business will not make any profit.
• Economics: Zeros in a function can represent the equilibrium points in a market, indicating the prices at which supply and demand are equal.
• Finance: Zeros in a function can represent the points at which an investment becomes profitable, indicating the prices at which the investment will yield a positive return.

Q: How can the concept of zeros in a function be used to make informed decisions about pricing and profit margins?

A: The concept of zeros in a function can be used to make informed decisions about pricing and profit margins by analyzing the break-even points, price elasticity of demand, and equilibrium points in a market.

Conclusion

In conclusion, the zeros of the function $y=-2(x-3)^2+4$ provide valuable insights into the pricing strategy of the hot dog stand. By finding and interpreting the zeros, we can gain a deeper understanding of the pricing strategy and make informed decisions about pricing and profit margins. The concept of zeros in a function has numerous real-world applications in various fields, including business, economics, and finance.