/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 Find the quadratic function \(f(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 33

Find the quadratic function \(f(x)=a x^{2}+b x+c\) that goes through \((2,0)\) and has a local maximum at \((0,1)\).

Short Answer

Expert verified
The quadratic function is \(f(x) = -\frac{1}{4}x^2 + 1\).

Step by step solution

01

Set Up the General Form of the Quadratic Function

The general form of the quadratic function is given by: \(f(x) = ax^2 + bx + c\).
02

Use the Point (2,0)

Substitute the point (2,0) into the quadratic function to get one equation:\(0 = a(2)^2 + b(2) + c\) which simplifies to \(4a + 2b + c = 0\).
03

Use the Point (0,1)

Substitute the point (0,1) into the quadratic function to get another equation:\(1 = a(0)^2 + b(0) + c\) which simplifies to \(c = 1\).
04

Use the Local Maximum Condition at (0,1)

The local maximum at \((0,1)\) means that the derivative of the quadratic function is 0 at \(x=0\). Compute the derivative: \(f'(x) = 2ax + b\).Substitute \(x=0\):\(0 = 2a(0) + b\) which simplifies to \(b = 0\).
05

Solve the System of Equations

Substitute \(b=0\) and \(c=1\) into the equation from Step 2:\(4a + 2(0) + 1 = 0\), which simplifies to \(4a + 1 = 0\).Solve for \(a\):\(4a = -1\)\(a = -\frac{1}{4}\).
06

Write the Quadratic Function

Now that the values of \(a\), \(b\), and \(c\) are known, substitute them back into the general form of the quadratic function:\(f(x) = -\frac{1}{4}x^2 + 0x + 1\) which simplifies to \(f(x) = -\frac{1}{4}x^2 + 1\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Equations
A system of equations is a set of multiple equations that one needs to solve together. For this quadratic function, we derive:
  • 4a + 2b + c = 0 (from point (2, 0)).
  • c = 1 (from point (0, 1)).
  • b = 0 (from the local maximum condition).
We solve these equations step by step:
  • Substitute b = 0 and c = 1 into the first equation to find a.
  • Simplify to get the specific values of a, b, and c.
This process ultimately leads to the final quadratic function.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the minimum of \(Q=\) \(x^{2}+y^{2}\) if \(x+y=6\).

Sketch the following curves, indicating all relative extreme points and inflection points. $$ y=1+3 x^{2}-x^{3} $$

Find the \(x\) -intercepts of the given function. $$ y=2 x^{2}+5 x+2 $$

Find the positive values of \(x\) and \(y\) that minimize \(S=x+y\) if \(x y=36\), and find this minimum value.

Find the dimensions of the rectangular garden of greatest area that can be fenced off (all four sides) with 300 meters of fencing.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.