/*! 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 76 Find the function \(y=a x^{2}+b ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 76

Find the function \(y=a x^{2}+b x+c\) whose graph contains the points \((1,-1),(3,-1),\) and (-2,14).

Short Answer

Expert verified
The function is \( y = x^2 - 4x + 2\).

Step by step solution

01

Set up equations using given points

The function is given by the equation \( y = a x^2 + b x + c \). Plug the points \((1, -1)\), \((3, -1)\), and \((-2, 14)\) into this equation to form a system of equations. For point \((1, -1)\): \( -1 = a(1)^2 + b(1) + c \) which simplifies to \( a + b + c = -1 \). For point \((3, -1)\): \( -1 = a(3)^2 + b(3) + c \) which simplifies to \( 9a + 3b + c = -1 \).For point \((-2, 14)\): \( 14 = a(-2)^2 + b(-2) + c \) which simplifies to \( 4a - 2b + c = 14 \).
02

Write the system of equations

The system of equations derived from step 1 is as follows: 1. \( a + b + c = -1 \) 2. \( 9a + 3b + c = -1 \) 3. \( 4a - 2b + c = 14 \)
03

Subtract equations to eliminate c

To eliminate \(c\), subtract equation 1 from equation 2:\( (9a + 3b + c) - (a + b + c) \) which simplifies to \( 8a + 2b = 0 \) or \( 4a + b = 0 \). Next, subtract equation 1 from equation 3: \( (4a - 2b + c) - (a + b + c) \) which simplifies to \( 3a - 3b = 15 \) then \( a - b = 5 \).
04

Solve for a and b

We now have two simpler equations: \(4a + b = 0\) and \(a - b = 5\). Solve the first equation for \(b\): \( b = -4a \). Substitute \(b\) into the second equation: \( a - (-4a) = 5 \) which simplifies to \( 5a = 5\) thus \( a = 1\). Now substitute \(a\) back into equation \(4a + b = 0\): \( 4(1) + b = 0 \) which simplifies to \( b = -4 \).
05

Solve for c

Substitute \(a = 1\) and \(b = -4\) into p\(( a + b + c = -1 )\) which becomes \( 1 - 4 + c = -1 \). This simplifies to \( c = 2 \).
06

Write the final function

Combine the values for \(a\), \(b\), and \(c\): \[ y = 1x^2 - 4x + 2 \].

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
When dealing with multiple unknowns, a system of equations can be utilized to find consistent solutions for each variable. In this case, the quadratic function passes through specific points:
  • (1, -1)
  • (3, -1)
  • (-2, 14)
Let's create equations from these points, remember from the standard quadratic equation: \( y = ax^2 + bx + c \). The main goal is to form and solve equations to find the unknowns \(a\), \(b\), and \(c\). This system lets us work with predictable relationships and seek out consistent variable values that satisfy all the given conditions.
Solving for Variables
Once the system of equations is set up, the next step is to solve for variables \(a\), \(b\), and \(c\). Here, we eliminate one variable at a time. For this specific exercise: We subtract equations to eliminate the variable \(c\): \( (9a + 3b + c) - (a + b + c) = 8a + 2b = 0 \) that further simplifies to \( 4a + b = 0 \). Likewise, we subtract another set to get \( a - b = 5 \). It鈥檚 clear now, the task is simplified to solving these new equations:
  • \( 4a + b = 0 \)
  • \( a - b = 5 \)
Solving these, we start with one equation to find a variable and substitute back to find others. This chain-solving strategy helps in reducing complex equations into more manageable forms.
Graphing Quadratic Equations
Quadratic functions are frequently represented by parabolas in a graph. Graphing these helps in visualizing the function's roots, vertex, and overall shape. Let鈥檚 consider our obtained solution: \( y = x^2 - 4x + 2 \). To graph this:
  • Find the vertex
  • Locate the axis of symmetry
  • Identify x-intercepts (roots)
  • Determine the y-intercept
The vertex can be derived using \( x = -\frac{b}{2a} \). Substituting \( a=1 \) and \( b=-4 \), we get \( x = 2 \). By inserting \( x=2 \) back into the equation, we calculate the vertex as \( y = -2 \). The parabola graph will curve upwards since the coefficient of \( x^2 \) is positive.
Parabola
A parabola is the graphical representation of a quadratic function. It鈥檚 crucial to understand its properties:
  • The vertex is the highest or lowest point depending on if it opens upwards or downwards.
  • The axis of symmetry is a vertical line passing through the vertex, splitting the parabola into two symmetrical parts.
  • The direction of the opening is determined by the sign of the coefficient of \( x^2 \): positive opens upwards while negative opens downwards.
For our function \( y = x^2 - 4x + 2 \),
  • The vertex is at (2, -2).
  • The axis of symmetry is the line \( x = 2 \).
  • The parabola opens upwards as the coefficient of \( x^2 \) is 1 (positive).
By graphing the parabola, you can analyze key features and present a clear visual understanding of quadratic functions.

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

A young couple has \(\$ 25,000\) to invest. As their financial consultant, you recommend that they invest some money in Treasury bills that yield \(7 \%,\) some money in corporate bonds that yield \(9 \%,\) and some money in junk bonds that yield \(11 \% .\) Prepare a table showing the various ways that this couple can achieve the following goals: (a) \(\$ 1500\) per year in income (b) \(\$ 2000\) per year in income (c) \(\$ 2500\) per year in income (d) What advice would you give this couple regarding the income that they require and the choices available?

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} 2 x-2 y+3 z=6 \\ 4 x-3 y+2 z=0 \\ -2 x+3 y-7 z=1 \end{array}\right. $$

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} x+y-z=6 \\ 3 x-2 y+z=-5 \\ x+3 y-2 z=14 \end{array}\right. $$

If \(f(x)=\frac{\sqrt{25 x^{2}-4}}{x}\) and \(g(x)=\frac{2}{5} \sec x, 0

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} 3 x-4 y=4 \\ \frac{1}{2} x-3 y=-\frac{1}{2} \end{array}\right. \\ x=2, y=\frac{1}{2} ;\left(2, \frac{1}{2}\right) \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.