/*! 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 4 Write an equation of a quadratic... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 4

Write an equation of a quadratic function that corresponds to each pair of \(x\)-intercepts. Assume there is no vertical stretch or shrink. a. \(2.5\) and \(-1\) (a) b. \(-4\) and \(-4\) c. \(-2\) and 2 d. \(r_{1}\) and \(r_{2}\)

Short Answer

Expert verified
(a) \(f(x) = x^2 - 1.5x - 2.5\); (b) \(f(x) = x^2 + 8x + 16\); (c) \(f(x) = x^2 - 4\); (d) \(f(x) = (x - r_1)(x - r_2)\).

Step by step solution

01

Understand the Quadratic Function

A quadratic function with given \(x\)-intercepts \((q_1, 0)\) and \((q_2, 0)\) can be expressed as \(f(x) = a(x - q_1)(x - q_2)\). If there is no vertical stretch or shrink, \(a = 1\).
02

Determine the Equation for Pair (a)

The \(x\)-intercepts given are \(2.5\) and \(-1\), meaning \(q_1 = 2.5\) and \(q_2 = -1\). Substitute these into the equation: \(f(x) = (x - 2.5)(x + 1)\).
03

Simplify the Equation for Pair (a)

Expand \((x - 2.5)(x + 1)\) to find \(f(x) = x^2 - 1.5x - 2.5\).
04

Determine the Equation for Pair (b)

For \(x\)-intercepts \(-4\) and \(-4\), \(q_1 = q_2 = -4\). The function is \(f(x) = (x + 4)^2\).
05

Simplify the Equation for Pair (b)

Expand \((x + 4)^2\) to find \(f(x) = x^2 + 8x + 16\).
06

Determine the Equation for Pair (c)

For \(x\)-intercepts \(-2\) and \(2\), substitute \(q_1 = -2\) and \(q_2 = 2\): \(f(x) = (x + 2)(x - 2)\).
07

Simplify the Equation for Pair (c)

Expand \((x + 2)(x - 2)\) to get \(f(x) = x^2 - 4\).
08

Determine the Equation for Pair (d)

For general \(x\)-intercepts \(r_1\) and \(r_2\), the function is \(f(x) = (x - r_1)(x - r_2)\).
09

Solutions

The solutions to the exercise are: (a) \(f(x) = x^2 - 1.5x - 2.5\) (b) \(f(x) = x^2 + 8x + 16\) (c) \(f(x) = x^2 - 4\) (d) \(f(x) = (x - r_1)(x - r_2)\).

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.

x-intercepts
The term "x-intercepts" refers to the points where a graph crosses the x-axis. These points are special because the y-value at these intersections is zero. Understanding x-intercepts is crucial when working with quadratic functions as they provide vital information about the solutions (or roots) of the equation.

To find the x-intercepts of a quadratic function like \[ f(x) = a(x - q_1)(x - q_2) \] set the equation equal to zero: \[ 0 = a(x - q_1)(x - q_2) \]Here,
  • \(q_1\) and \(q_2\) are the x-intercepts.
  • These values of \(x\) make \(f(x) = 0\).
In this context, x-intercepts are also called roots or solutions of the quadratic equation. Identifying these helps to graph the function and understand its behavior.
factoring quadratics
Factoring quadratics is a method used to express a quadratic polynomial as a product of its linear factors. This is incredibly helpful when seeking to find the x-intercepts (roots) of the function.

Consider a quadratic function given by:\[ f(x) = ax^2 + bx + c \]To factor this, we aim to rewrite it in terms of two binomials:\[ f(x) = a(x - q_1)(x - q_2) \]Here are key steps to factor a quadratic:
  • If possible, find two numbers that multiply to \(c\) and add up to \(b\).
  • Use these numbers to express \(bx\) as the sum of two terms, then factor by grouping.
  • If the quadratic is in the form \((x - q_1)(x - q_2)\), this reveals the x-intercepts \(q_1\) and \(q_2\).
In instances where factoring is challenging, other methods like the quadratic formula or completing the square might be more appropriate.
polynomials
A polynomial is a mathematical expression made up of variables (also called indeterminates) and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are foundational in algebra, and they come in many forms.

Quadratic functions are a special type of polynomial and have the general form:\[ f(x) = ax^2 + bx + c \]Here:
  • \(a\), \(b\), and \(c\) are constants where \(a eq 0\).
  • The highest power of the variable \(x\) is 2, hence the name quadratic.
Polynomials can also be factored or expanded just like quadratic functions. Understanding how to manipulate these expressions is crucial in algebra for solving equations and analyzing their graphs. Polynomials can model various real-world situations, making them an essential element of mathematical language.

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 each product. Check your answers by using calculator tables or graphs. a. \((x+1)\left(2 x^{2}+3 x+1\right)\) b. \((2 x-5)\left(3 x^{2}+2 x-4\right)\)

Convert each equation from vertex form to general form. Check your answers by entering both expressions into the \(\mathrm{Y}=\) screen on your calculator, then graphing. a. \(y=(x+5)^{2}+4\) b. \(y=2(x-7)^{2}-8\) c. \(y=-3(x+4)^{2}+1\) (a) d. \(y=0.5(x-3)^{2}-4.5\)

Graph the equation \(y=x^{2}+3 x+5\). Use the graph and the quadratic formula to answer these questions. a. How many \(x\)-intercepts does the graph have? b. Use the quadratic formula to try to find the roots of \(0=x^{2}+3 x+5\). What happens when you take the square root? c. Without looking at a graph, how can you use the quadratic formula to tell if a quadratic equation has any real roots?

Perform the operations, then combine like terms. Check your answers by using tables or graphs. a. \(\left(8 x^{3}-5 x\right)+\left(3 x^{3}+2 x^{2}+7 x+12\right)\) b. \(\left(8 x^{3}-5 x\right)-\left(3 x^{3}+2 x^{2}+7 x+12\right)\) c. \(\left(2 x^{2}-6 x+11\right)+\left(-8 x^{2}-7 x+9\right)\) d. \(\left(2 x^{2}-6 x+11\right)\left(-8 x^{2}-7 x+9\right)\)

APPLICATION The school ecology club wants to fence in an area along the riverbank to protect endangered wildflowers that grow there. The club has enough money to buy 200 feet of fencing. It decides to enclose a rectangular space. The fence will form three sides of the rectangle, and the riverbank will Tkice form the fourth side. a. If the width of the enclosure is 30 feet, how much fencing material is available for the length? Sketch this situation. What is the area? (a) b. If the width is \(w\) feet, how much fencing material remains for the length, \(l\) ? c. Lse your answer from \(11 \mathrm{~b}\) to write an equation for the area of the rectangle in factored form. Check your equation with your width and area from 11 a. d. Which two different widths would give an area equal to 0? (a) e. Which width will give the maximum area? What is that area?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.