how to find the max of a function

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For a diverseness of reasons, you may need to be able to ascertain the maximum or minimum value of a selected quadratic function. You lot tin can find the maximum or minimum if your original part is written in general grade, ${\displaystyle f(x)=ax^{2}+bx+c}$ , or in standard form, ${\displaystyle f(x)=a(x-h)^{2}+thou}$ . Finally, you lot may also wish to use some basic calculus to ascertain the maximum or minimum of any quadratic function.

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4. four

   Notice the corresponding f(x) value. Insert the value of ten that you merely calculated into the part to observe the respective value of f(ten). This will be the minimum or maximum of the function.

5. five

   Written report your results. Review the question you have been asked. If you are asked for the coordinates of the vertex, y'all need to report both the ${\displaystyle x}$ and ${\displaystyle y}$ (or ${\displaystyle f(x)}$ ) values. If you are only asked for the maximum or minimum, you only need to report the ${\displaystyle y}$ (or ${\displaystyle f(x)}$ ) value. Refer back to the value of the ${\displaystyle a}$ coefficient to be sure if you have a maximum or a minimum.

   • For the first example, ${\displaystyle f(x)=x^{two}+10x-1}$ , the value of ${\displaystyle a}$ is positive, and then y'all will be reporting the minimum value. The vertex is at ${\displaystyle (-5,-26)}$ , and the minimum value is ${\displaystyle -26}$ .
   • For the 2nd case, ${\displaystyle f(x)=-3x^{two}+6x-four}$ , the value of ${\displaystyle a}$ is negative, so you lot will be reporting the maximum value. The vertex is at ${\displaystyle (1,-1)}$ , and the maximum value is ${\displaystyle -ane}$ .

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1. 1

   Write your quadratic function in standard or vertex form. The standard course of a general quadratic function, which can also be chosen the vertex form, looks similar this:^[4]

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3. three

   Identify the minimum or maximum value. When the function is written in standard form, finding the minimum or maximum value is every bit simple as stating the value of the variable ${\displaystyle k}$ . For the 2 case functions given higher up, these values are:

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1. 1

   Start with the general form. Write your quadratic function in general form, ${\displaystyle f(x)=ax^{two}+bx+c}$ . If necessary, you may need to combine like terms and rearrange to become the proper form.^[7]

   • Begin with the sample function ${\displaystyle f(10)=2x^{2}-4x+1}$ .

2. ii

   Use the power rule to detect the first derivative. Using basic first-twelvemonth calculus, y'all can find the first derivative of the general quadratic office to be ${\displaystyle f^{\prime }(x)=2ax+b}$ .^[8]

3. iii

   Set the derivative equal to nothing. Call up that derivative of a part tells yous the slope of the function at that selected bespeak. The minimum or maximum of a function occurs when the slope is zilch. Therefore, to find where the minimum or maximum occurs, set the derivative equal to zero. Continue with the sample problem from in a higher place:^[nine]

4. 4

   Solve for 10. Employ basic rules of algebra to rearrange the function and solve the value for x, when the derivative equals zero. This solution volition tell you the x-coordinate of the vertex of the part, which is where the maximum or minimum will occur.^[ten]

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6. half dozen

   Study your solution. The solution gives you lot the vertex of the maximum or minimum point. For this sample function, ${\displaystyle f(x)=2x^{2}-4x+1}$ , the vertex occurs at ${\displaystyle (one,-1)}$ . The coefficient ${\displaystyle a}$ is positive, so the office opens upwards. Therefore, the minimum value of the part is the y-coordinate of the vertex, which is ${\displaystyle -one}$ .^[12]

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• Question

  How do you tell if a parabola is maximum or minimum?

  

  Jake Adams
  Academic Tutor & Test Prep Specialist

  Jake Adams is an academic tutor and the possessor of Simplifi EDU, a Santa Monica, California based online tutoring business offering learning resources and online tutors for academic subjects K-College, Sat & ACT prep, and higher admissions applications. With over 14 years of professional tutoring experience, Jake is dedicated to providing his clients the very best online tutoring experience and access to a network of fantabulous undergraduate and graduate-level tutors from summit colleges all over the nation. Jake holds a BS in International Business and Marketing from Pepperdine Academy.

  

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  First solve for a. If the value of a is a positive number, y'all'll have an up-facing parabola and you lot'll need to find its minimum value. If a is a negative number, yous'll have a downwards-facing parabola and yous'll need to find its maximum value.

• Question

  How do yous tell if a parabola is upwards or down?

  

  Jake Adams
  Academic Tutor & Exam Prep Specialist

  Jake Adams is an academic tutor and the possessor of Simplifi EDU, a Santa Monica, California based online tutoring business offer learning resources and online tutors for bookish subjects Grand-Higher, Sabbatum & ACT prep, and college admissions applications. With over xiv years of professional tutoring experience, Jake is dedicated to providing his clients the very all-time online tutoring experience and admission to a network of excellent undergraduate and graduate-level tutors from top colleges all over the nation. Jake holds a BS in International Business concern and Marketing from Pepperdine University.

  

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  Yous can remember this concept past thinking about smiles and frowns. If someone is positive they smile, and if someone is negative, they pout. Similarly, a positive number will have an upward-facing parabola, and a negative number will have a downward-facing parabola.

• Question

  How do I graph a quadratic function?

  

  Start, create a data tabular array with multiple experimental values for x. Sub in those ten coordinates and get y coordinates. Plot these along the 10 and y axis and join the dots with a polish curve.

• Question

  Later on the value of y is found, how do we find the corresponding value of 10?

  

  Substitute the value of y found into the original(given) equation and solve for 10.

• Question

  What are prerequisite skills for this topic?

  

  A bones agreement of differentiation and how quadratic graphs work, as well as gradients, will be required before attempting this.

• Question

  What if the vertex form you accept has no coefficient in the forepart of the parentheses?

  

  You would demand to rewrite the square in the equation in order to put it in vertex course.

• Question

  In the method iv in example 1, why did he utilize c-b^2/2a? Why not 4a?

  

  Because he was finding the minimum value. If he were finding max value, he would use 4a.

• Question

  What if the power is 3?

  

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• The parabola's axis of symmetry is x = h.

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Commodity Summary X

To detect the maximum or minimum value of a quadratic function, outset with the full general class of the function and combine any similar terms. For case, if you're starting with the function f(x) = 3x + 2x - 10^2 + 3x^two + four, you would combine the 10^2 and x terms to simplify and terminate upwardly with f(ten) = 2x^2 + 5x + 4. Now figure out which management the parabola opens past checking if a, or the coefficient of x^2, is positive or negative. If it's positive, the parabola opens upward. If it'southward negative, the parabola opens downward. In the function f(x) = 2x^ii + 5x + 4, the coefficient of 10^ii is positive, so the parabola opens up. Next, find the x value of the vertex by solving -b/2a, where b is the coefficient in front of x and a is the coefficient in front of x^2. In the function f(10) = 2x^2 + 5x + iv, b = 5 and a = 2. Therefore, you would divide -5 past 2 times ii, or iv, and get -ane.25. Finally, plug the x value into the role to find the value of f(x), which is the minimum or maximum value of the function. The function f(10) = 2x^2 + 5x + 4 would become f(-1.25) = 2(-1.25)^ii + 5(-1.25) + iv, or f(-ane.25) = 0.875. If the parabola opens upwards, your respond will be the minimum value. If the parabola opens downward, your respond is the maximum value. In this example, since the parabola opens upward, f(-1.25) = 0.875 is the minimum value of the role. If y'all want to learn how to use standard or vertex form for your formula, keep reading the article!

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