How Do I Get Vertex Form, Initial Profit, And Maximum Profit In An Ordered Pair Without A Graphic Calculator?

Mar 12, 2025 by ADMIN 110 views

How to Find Vertex Form, Initial Profit, and Maximum Profit in an Ordered Pair without a Graphic Calculator

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Introduction

In the real analysis category, finding the vertex form, initial profit, and maximum profit in an ordered pair is a crucial concept. This article will guide you through the process of finding these values without using a graphic calculator. We will use the given profit function, $P(x) = -0.02x^2 + 15x - 300$ , to demonstrate the steps involved.

Understanding the Profit Function

The profit function is a quadratic function, which can be written in the general form:

P(x) = ax^2 + bx + c

where $a$ , $b$ , and $c$ are constants. In this case, the profit function is:

P(x) = -0.02x^2 + 15x - 300

Finding the Vertex Form

The vertex form of a quadratic function is given by:

P(x) = a(x - h)^2 + k

where $(h, k)$ is the vertex of the parabola. To find the vertex form, we need to complete the square.

Step 1: Factor out the coefficient of $x^2$

First, we factor out the coefficient of $x^2$ , which is $-0.02$ .

P(x) = -0.02(x^2 - 750x) - 300

Step 2: Add and subtract the square of half the coefficient of $x$

Next, we add and subtract the square of half the coefficient of $x$ , which is $(-750/2)^2 = 562500$ .

P(x) = -0.02(x^2 - 750x + 562500 - 562500) - 300

Step 3: Simplify the expression

Now, we simplify the expression by combining like terms.

P(x) = -0.02(x^2 - 750x + 562500) + 11250 - 300

P(x) = -0.02(x - 375)^2 + 11025

Finding the Initial Profit

The initial profit is the profit at $x = 0$ . To find the initial profit, we substitute $x = 0$ into the profit function.

P(0) = -0.02(0)^2 + 15(0) - 300

P(0) = -300

Finding the Maximum Profit

The maximum profit is the profit at the vertex of the parabola. To find the maximum profit, we substitute $x = 375$ into the profit function.

P(375) = -0.02(375)^2 + 15(375) - 300

P(375) = -0.02(140625) + 56250 - 300

P(375) = -2812.5 + 56250 - 300

P(375) = 53537.5

Writing the Maximum Profit in an Ordered Pair

The maximum profit is written in an ordered pair as $(375, 53537.5)$ .

Conclusion

In this article, we have demonstrated how to find the vertex form, initial profit, and maximum profit in an ordered pair without using a graphic calculator. We used the given profit function, $P(x) = -0.02x^2 + 15x - 300$ , to illustrate the steps involved. By completing the square and simplifying the expression, we were able to find the vertex form of the profit function. We then used this vertex form to find the initial profit and maximum profit. Finally, we wrote the maximum profit in an ordered pair as $(375, 53537.5)$ .

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Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by:

P(x) = a(x - h)^2 + k

where $(h, k)$ is the vertex of the parabola.

Q: How do I find the vertex form of a quadratic function?

A: To find the vertex form, you need to complete the square. This involves factoring out the coefficient of $x^2$ , adding and subtracting the square of half the coefficient of $x$ , and simplifying the expression.

Q: What is the initial profit, and how do I find it?

A: The initial profit is the profit at $x = 0$ . To find the initial profit, you substitute $x = 0$ into the profit function.

Q: What is the maximum profit, and how do I find it?

A: The maximum profit is the profit at the vertex of the parabola. To find the maximum profit, you substitute $x = h$ into the profit function, where $h$ is the x-coordinate of the vertex.

Q: How do I write the maximum profit in an ordered pair?

A: The maximum profit is written in an ordered pair as $(h, P(h))$ , where $h$ is the x-coordinate of the vertex and $P(h)$ is the maximum profit.

Q: Can I use a graphic calculator to find the vertex form, initial profit, and maximum profit?

A: While a graphic calculator can be a useful tool, it is not necessary to find the vertex form, initial profit, and maximum profit. You can use the steps outlined in this article to find these values without a graphic calculator.

Q: What if I have a quadratic function in the form $P(x) = ax^2 + bx + c$ ? How do I find the vertex form?

Q: Can I use this method to find the vertex form, initial profit, and maximum profit for any quadratic function?

A: Yes, this method can be used to find the vertex form, initial profit, and maximum profit for any quadratic function in the form $P(x) = ax^2 + bx + c$ .

Q: What if I have a quadratic function with a negative leading coefficient? How do I find the vertex form?

A: If you have a quadratic function with a negative leading coefficient, you can still find the vertex form by completing the square. However, you may need to use a different method to find the x-coordinate of the vertex.

Q: Can I use this method to find the vertex form, initial profit, and maximum profit for a quadratic function with a complex coefficient?

A: Yes, this method can be used to find the vertex form, initial profit, and maximum profit for a quadratic function with a complex coefficient.

Q: What if I have a quadratic function with a coefficient of 0? How do I find the vertex form?

A: If you have a quadratic function with a coefficient of 0, you can still find the vertex form by completing the square. However, you may need to use a different method to find the x-coordinate of the vertex.

Q: Can I use this method to find the vertex form, initial profit, and maximum profit for a quadratic function with a fractional coefficient?

A: Yes, this method can be used to find the vertex form, initial profit, and maximum profit for a quadratic function with a fractional coefficient.

Q: What if I have a quadratic function with a coefficient of 1? How do I find the vertex form?

A: If you have a quadratic function with a coefficient of 1, you can still find the vertex form by completing the square. However, you may need to use a different method to find the x-coordinate of the vertex.

Q: Can I use this method to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of -1?

A: Yes, this method can be used to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of -1.

Q: What if I have a quadratic function with a coefficient of 0 in the $x^2$ term? How do I find the vertex form?

A: If you have a quadratic function with a coefficient of 0 in the $x^2$ term, you can still find the vertex form by completing the square. However, you may need to use a different method to find the x-coordinate of the vertex.

Q: Can I use this method to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of 0 in the $x$ term?

A: Yes, this method can be used to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of 0 in the $x$ term.

Q: What if I have a quadratic function with a coefficient of 0 in the constant term? How do I find the vertex form?

A: If you have a quadratic function with a coefficient of 0 in the constant term, you can still find the vertex form by completing the square. However, you may need to use a different method to find the x-coordinate of the vertex.

Q: Can I use this method to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of 0 in the $x^2$ term and a coefficient of 0 in the $x$ term?

A: Yes, this method can be used to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of 0 in the $x^2$ term and a coefficient of 0 in the $x$ term.

Q: What if I have a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant term? How do I find the vertex form?

A: If you have a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant term, you can still find the vertex form by completing the square. However, you may need to use a different method to find the x-coordinate of the vertex.

Q: Can I use this method to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant term?

A: Yes, this method can be used to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant term.

Q: What if I have a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant term, and the function is equal to 0? How do I find the vertex form?

A: If you have a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant term, and the function is equal to 0, you can still find the vertex form by completing the square. However, you may need to use a different method to find the x-coordinate of the vertex.

Q: Can I use this method to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant term, and the function is equal to 0?

Q: What if I have a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant term, and the function is equal to 0, and the function has a fractional coefficient? How do I find the vertex form?

A: If you have a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant term, and the function is equal to 0, and the function has a fractional coefficient, you can still find the vertex form by completing the square. However, you may need to use a different method to find the x-coordinate of the vertex.

Q: Can I use this method to find the vertex form, initial profit, and maximum profit for a quadratic function with a coefficient of 0 in the $x^2$ term, a coefficient of 0 in the $x$ term, and a coefficient of 0 in the constant