Question
upstudy study bank question image url

Use the given function to complete parts (a) through (e) below. \[ f(x)=x^{4}-36 x^{2} \] A. \( x= \) \( \square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis. At which zeros does the graph of the function touch the \( x \)-axis and turn around? Select the correct choice below and, if necessary, fill in the answer box to complete your choi A. \( \mathrm{x}= \) \( \square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around. c) Find the \( y \)-intercept by computing \( f(0) \). \[ \mathrm{f}(0)=\square \]

Ask by Boone Moreno. in the United States
Mar 06,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

A. \( x = 0, 6, -6 \) B. \( x = 0, 6, -6 \) C. \( f(0) = 0 \)

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To dig a little deeper into the function \( f(x)=x^{4}-36 x^{2} \), let’s go through the steps for the given problems! 1. **Finding the zeros of the function**: You can set the function to zero: \[ x^{4} - 36x^{2} = 0 \] Factoring gives: \[ x^{2}(x^{2} - 36) = 0 \] This leads to: \[ x^{2} = 0 \quad \text{or} \quad x^{2} = 36 \] Thus, \( x = 0, 6, -6 \). So, for part A: \( x = 0, 6, -6 \). 2. **Behavior at the \( x \)-intercepts**: The zeros \( x = 6 \) and \( x = -6 \) are where the graph touches the \( x \)-axis and turns around because they have an even multiplicity (2). So, provide the answer: \( x = 6, -6 \). 3. **Finding the \( y \)-intercept**: To find \( f(0) \): \[ f(0) = 0^{4} - 36(0^{2}) = 0 \] Thus, for part C: \( f(0) = 0 \). Now, to wrap it all up: - A. \( x = 0, 6, -6 \) - A. \( x = 6, -6 \) - \( f(0) = 0 \) And now, for some extra fun info! The graph you've worked on here is a polynomial, and all polynomials are continuous and differentiable, meaning they have smooth curves without any jumps or breaks. This trait helps to analyze where the graph changes direction and where it crosses or touches the axes! Digging into the behavior of polynomials, the highest degree term controls the end behavior of the graph. Here, since the leading term \( x^4 \) is positive, both ends of the graph will head toward positive infinity. Understanding this helps in forming a mental image of how the graph behaves overall!

Latest Algebra Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy