/*! 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 38 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 38

Sketch the graph of the function. $$f(x)=\left\\{\begin{array}{ll}x^{2}+5, & x \leq 1 \\\\-x^{2}+4 x+3, & x>1\end{array}\right.$$

Short Answer

Expert verified
To sketch the graph, draw the first parabola \(X^{2}+5\) for \(x \leq 1\), it opens upward with the vertex (0,5). Draw the second parabola \( -x^{2}+4x+3\) for \(x > 1\), it opens downward with the vertex (2,5). These two parabolas meet at \(x=1\), creating a piecewise function graph.

Step by step solution

01

Examination of first half

Consider the first half of the function \(f(x)=x^{2}+5\), which applies to all \(x\) where \(x \leq 1\). This is a parabolic function shifted 5 units upwards. The vertex of this parabola is at (0, 5), and as \(x\) decreases, it opens upwards.
02

Examination of second half

Now, look at the second half of the function \(f(x)=-x^{2}+4x+3\), valid when \(x > 1\). This is also a parabolic function. The vertex of the parabola can be found using the formula \(\frac{-b}{2a}\), which gives \(x=2\). Substituting 2 into the equation gives the y-coordinate as 5. So, the vertex of this parabola is (2,5), and it opens downwards.
03

Plotting the functions

Using the aforementioned information, draw the first half (part 1) of the function for \(x \leq 1\) and the second half (part 2) for \(x > 1\). Now you can get the union of these 2 segments in a single plot.
04

Joining at juncture

At \(x = 1\), the value from both sides is checked to see if it coincides to form a continuous function. If substituting \(x = 1\) into both functions yields the same result, that means the function is continuous and can be smoothly joined at this point.

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

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=\frac{5}{6}-\frac{2}{3} x$$

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=x^{3}-1$$

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=4+(1 / x)$$

Sketch the graph of the function. $$k(x)=\left\\{\begin{array}{ll}2 x+1, & x \leq-1 \\\2 x^{2}-1, & -11\end{array}\right.$$

During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of 0.5 inch per hour for the final hour. Write and graph a piecewise- defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.