/*! 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 35 Use the vertex and intercepts to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 35

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)=2 x^{2}+4 x-3$$

Short Answer

Expert verified
The axis of symmetry is \(x=-1\). The x-intercepts are at \(x=-1-\sqrt{\frac{5}{2}}\) and \(x=-1+\sqrt{\frac{5}{2}}\). The domain of \(f(x)\) is \(\mathbb{R}\) and the range of \(f(x)\) is \([-5, +\infty)\).

Step by step solution

01

Write the Quadratic Function in Vertex Form

The standard form of a quadratic function is \(y=a(x-h)^2+k\), where \((h, k)\) is the vertex of the parabola. The given function is \(f(x)=2x^2+4x-3\). Using the method of completing the square, \(f(x)\) can be written in vertex form. \[f(x)=2(x^2+2x)-3=2(x^2+2x+1-1)-3=2[(x+1)^2-1]-3=2(x+1)^2-5\]. So, the vertex of the parabola is at (-1,-5) which is \((h,k)\).
02

Finding the x-intercepts

The x-intercepts are determined by solving for \(x\) when \(f(x)=0\). Doing so gives: \[0=2(x+1)^2-5 \Rightarrow (x+1)^2=\frac{5}{2}\] The solutions are \(x=-1-\sqrt{\frac{5}{2}}\) and \(x=-1+\sqrt{\frac{5}{2}}\). Therefore, the x-intercepts are \(-1-\sqrt{\frac{5}{2}}\) and \(-1+\sqrt{\frac{5}{2}}\)
03

Finding the Axis of Symmetry

The axis of symmetry is the vertical line \(x=h\), where \(h\) is the x-coordinate of the vertex. In this case, the axis of symmetry is \(x=-1\)
04

Identifying the Domain and the Range of the Function

The domain of any quadratic function is all real numbers, so the domain of \(f(x)\) is \(\mathbb{R}\). Since the coefficient of \(x^2\) is positive, the parabola opens upwards, which means the lowest point is the vertex. Therefore, the range is \([-5, +\infty)\)

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!

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm solving a polynomial inequality that has a value for which the polynomial function is undefined.

Write an equation that expresses each relationship. Then solve the equation for \(y .\) \(x\) varies jointly as \(y\) and the square of \(z\).

The rational than \(f(x)=\frac{27,725(x-14)}{x^{2}+9}-5 x\) models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\). a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the \(200 \mathrm{M}\) and \(\overline{\mathrm{TRACE}}\), features or the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?

In this exercise, we lead you through the steps involved in the proof of the Rational Zero Theorem. Consider the polynomial equation $$a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0}=0$$ and let \(\frac{P}{q}\) be a rational root reduced to lowest terms. a. Substitute \(\frac{p}{q}\) for \(x\) in the equation and show that the equation can be written as $$a_{n} p^{n}+a_{n-1} p^{n-1} q+a_{n-2} p^{n-2} q^{2}+\cdots+a_{1} p q^{n-1}=-a_{0} q^{n}$$ b. Why is \(p\) a factor of the left side of the equation? c. Because \(p\) divides the left side, it must also divide the right side. However, because \(\frac{P}{q}\) is reduced to lowest terms, \(p\) and \(q\) have no common factors other than \(-1\) and 1 Because \(p\) does divide the right side and has no factors in common with \(q^{n},\) what can you conclude? d. Rewrite the equation from part (a) with all terms containing \(q\) on the left and the term that does not have a factor of \(q\) on the right. Use an argument that parallels parts (b) and (c) to conclude that \(q\) is a factor of \(a_{n}\).

The illumination provided by a car's headlight varies inversely as the square of the distance from the headlight. A car's headlight produces an illumination of 3.75 footcandles at a distance of 40 feet. What is the illumination when the distance is 50 feet?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.