/*! 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 36 Use the vertex and intercepts to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 36

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. $$f(x)=3 x^{2}-2 x-4$$

Short Answer

Expert verified
The vertex is at \((1/3, -13/3)\), the intercepts are at \((0, -4)\), \((-1, 0)\), and \((4/3, 0)\), the axis of symmetry is at \(x=1/3\), the domain is \((-\infty, +\infty)\), and the range is \([-13/3, +\infty)\).

Step by step solution

01

Identify the vertex

The vertex of a parabola, \(y=ax^{2}+bx+c\), is given by the formula \((-b/2a , f(-b/2a))\). For the function \(f(x)=3x^2-2x-4\), substitution of \(a=3\) and \(b=-2\) into the vertex formula gives a vertex at \((1/3, -13/3)\).
02

Find the intercepts

The y-intercept is found by evaluating the function at \(x=0\). Thus, the y-intercept is at \((0, -4)\). The x-intercepts are found by setting \(f(x)=0\) and solving. For \(f(x)=3x^2-2x-4\), this yields x-intercepts at \((-1, 0)\) and \((4/3, 0)\).
03

Determine the axis of symmetry

The axis of symmetry for a parabola \(y=ax^{2}+bx+c\) is the vertical line \(x=-b/2a\). For the function \(f(x)=3x^2-2x-4\), substitution of \(a=3\) and \(b=-2\) into the formula gives the axis of symmetry at \(x=1/3\).
04

Determine the domain and range

The domain of a quadratic function is all real numbers, so the domain of \(f(x)=3x^2-2x-4\) is \((-\infty, +\infty)\). The range of a parabola that opens upwards (which is the case for \(f(x)=3x^2-2x-4\), since the coefficient of \(x^{2}\) is positive) is \([-D, +\infty)\), where -D is the y-coordinate of the vertex. So the range of \(f(x)=3x^2-2x-4\) is \([-13/3, +\infty)\).

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!

Key Concepts

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

Vertex of a Parabola
Understanding the vertex of a parabola is crucial for graphing quadratic functions such as \(f(x)=3x^2-2x-4\). The vertex is the highest or lowest point on the parabola, acting as a pivotal point from which the parabola either opens upwards or downwards. For the equation \(y=ax^2+bx+c\), the vertex can be found using the formula \((-\frac{b}{2a}, f(-\frac{b}{2a}))\).

For our example, the coefficients are \(a=3\) and \(b=-2\). Plugging these into the formula, the vertex is at \((\frac{1}{3}, -\frac{13}{3})\). This point is essential for sketching the curve, as it marks the precise turn of the graph and helps in identifying the direction in which the parabola opens.
X-intercepts and Y-intercepts
To graph a quadratic function, knowing where it crosses the x-axis and y-axis, also known as the x-intercepts and y-intercept, is essential. The y-intercept is simply the function's value at \(x=0\). Hence, for our function \(f(x)=3x^2-2x-4\), we find the y-intercept at \((0, -4)\).

To find x-intercepts where the graph crosses the x-axis, we set the function equal to zero, \(0=3x^2-2x-4\), and solve for \(x\). This gives us the points \((-1, 0)\) and \((\frac{4}{3}, 0)\), which are crucial for plotting the function as they represent the points at which the graph intersects the x-axis.
Axis of Symmetry
The axis of symmetry in a parabola is a vertical line that passes through the vertex, dividing the graph into two mirror images. For quadratic functions like \(y=ax^2+bx+c\), the axis of symmetry formula is \(x=-\frac{b}{2a}\).

For our function, by substituting \(a=3\) and \(b=-2\), we find that the axis of symmetry is at \(x=\frac{1}{3}\). This axis of symmetry not only helps in creating a well-balanced graph but also identifies properties related to the vertex and intercepts, contributing to understanding the function's behavior.
Domain and Range of a Quadratic Function
The domain of a quadratic function is the set of all possible input values. For any quadratic function, this is all real numbers, represented as \((-fty, +fty)\).

The range is the set of possible output values, which depends on the direction the parabola opens. For a parabola that opens upwards, like our function \(f(x)=3x^2-2x-4\), the range starts at the y-coordinate of the vertex and goes to positive infinity, denoted as \([-\frac{13}{3}, +fty)\). These concepts serve not just to sketch the graph but to predict and control the outputs for various inputs within the context of real-world problems.

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. Use a graphing utility to graph \(y=2 x^{2}-82 x+720\) in a standard viewing rectangle. What do you observe? b. Find the coordinates of the vertex for the given quadratic function. c. The answer to part (b) is \((20.5,-120.5) .\) Because the leading coefficient, \(2,\) of the given function is positive, the vertex is a minimum point on the graph. Use this fact to help find a viewing rectangle that will give a relatively complete picture of the parabola. With an axis of symmetry at \(x=20.5,\) the setting for \(x\) should extend past this, so try \(\mathrm{Xmin}=0\) and \(\mathrm{Xmax}=30 .\) The setting for \(y\) should include (and probably go below) the \(y\) -coordinate of the graph's minimum \(y\) -value, so try \(\mathrm{Ymin}=-130\) Experiment with Ymax until your utility shows the parabola's major features. d. In general, explain how knowing the coordinates of a parabola's vertex can help determine a reasonable viewing rectangle on a graphing utility for obtaining a complete picture of the parabola.

Write a polynomial inequality whose solution set is \([-3,5]\)

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=1,\) a slant asymptote whose equation is \(y=x, y\) -intercept at \(2,\) and \(x\) -intercepts at \(-1\) and 2

Describe in words the variation shown by the given equation. \(z=k x^{2} \sqrt{y}\)

If you have difficulty obtaining the functions to be maximized in Exercises \(73-76,\) read Example 2 in Section \(1.10 .\) On a certain route, an airline carries 8000 passengers per month, each paying \(\$ 50 .\) A market survey indicates that for each \(\$ 1\) increase in the ticket price, the airline will lose 100 passengers. Find the ticket price that will maximize the airline's monthly revenue for the route. What is the maximum monthly revenue?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

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.