/*! 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 13 Sketch the graph of each quadrat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 13

Sketch the graph of each quadratic function and compare it with the graph of \(y=x^{2}\). (a) \(f(x)=\frac{1}{2} x^{2}\) (b) \(g(x)=-\frac{1}{8} x^{2}\) (c) \(h(x)=\frac{3}{2} x^{2}\) (d) \(k(x)=-3 x^{2}\)

Short Answer

Expert verified
The graphs of \(f(x) = 0.5x^2\) and \(h(x) = 1.5x^2\) are wider and narrower, respectively, than the graph of \(y = x^2\), and open upwards. The graphs of \(g(x) = -0.125x^2\) and \(k(x)=-3x^2\) are wider and narrower, respectively, than the graph of \(y = x^2\), and open downwards.

Step by step solution

01

Sketch the graphs

First, sketch the graph of each given function, and \(y = x^2\). Pay attention to the vertex (in each case the vertex is at the origin), the shape (whether it opens upwards or downwards), and the width of the parabola (narrower for absolute values of coefficients greater than 1, broader for absolute values of coefficients less than 1).
02

Analyze the graphs

Second, analyze the graphs. For \(f(x) = 0.5x^2\), the graph is wider than \(y = x^2\) and opens upwards. For \(g(x) = -0.125x^2\), the graph is even wider, but it opens downwards. The graph of \(h(x) = 1.5x^2\) is narrower than \(y = x^2\) and opens upwards. Finally, for \(k(x)= -3x^2\), the graph is narrower, but opens downwards.
03

Describe comparing the graphs

In summary, the main difference between these functions and \(y = x^2\) is the 'stretch' or 'compression' along the y-axis, which is determined by the coefficient. If the coefficient is greater than 1 or less than -1, the graph is 'stretched' (it is narrower). If the coefficient is between -1 and 1, the graph is 'compressed' (it is wider). Moreover, a negative coefficient means the graph opens downwards; a positive coefficient means the graph opens upwards.

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 the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{3}+9 x^{2}+27 x+35$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x+4=0$$

Think About It \(\quad\) A cubic polynomial function \(f\) has real zeros \(-2, \frac{1}{2},\) and \(3,\) and its leading coefficient is negative. Write an equation for \(f\) and sketch its graph. How many different polynomial functions are possible for \(f ?\)

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{2}+10 x+17$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Decision 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.