Problem 28 Find the $x$ - and $y$ -inte... [FREE SOLUTION]

91影视

Precalculus with Calculus Previews

Dennis G. Zill, Jacqueline M. Dewar

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 6: Problem 28

Find the $x$ - and $y$ -intercepts of the graph of the given function. Do not graph. $$ f(x)=4^{x} x^{4}-4^{x+1} $$

Short Answer

Expert verified

The y-intercept is (0, -4), and the x-intercepts are ($\sqrt{2}$, 0) and (-$\sqrt{2}$, 0).

Step by step solution

Finding the y-intercept

The y-intercept of a function occurs where the graph intersects the y-axis. This happens when $x = 0$. To find the y-intercept for $f(x) = 4^x x^4 - 4^{x + 1}$, substitute $x = 0$ into the function:\[f(0) = 4^0 \, 0^4 - 4^{0 + 1}\]Calculate:\[f(0) = 1 \, \cdot 0 - 4 = -4\]So, the y-intercept is $(0, -4)$.

Finding the x-intercepts

The x-intercepts of a function occur where the graph intersects the x-axis. This is where $f(x) = 0$. Thus, set the function equal to zero and solve for $x$:\[4^x x^4 - 4^{x+1} = 0\]Factor the expression:\[4^x (x^4 - 4) = 0\]Here, either $4^x = 0$ or $x^4 - 4 = 0$:- $4^x$ can never be zero as any positive number raised to any power remains positive.- Solve $x^4 - 4 = 0$ using the difference of squares:\[(x^2 - 2)(x^2 + 2) = 0\]Solve each factor:- $x^2 - 2 = 0$ gives $x = \pm \sqrt{2}$ - $x^2 + 2 = 0$ gives no real solutions since $x^2 = -2$ is not possible for real numbers.Thus, the x-intercepts are $(\sqrt{2}, 0)$ and $(-\sqrt{2}, 0)$.

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.

Understanding Function Graphs

A function graph visually represents the relationship between the input values, often denoted as $ x $, and the output values, represented as $ f(x) $ or $ y $. To effectively understand and analyze these graphs, one must be familiar with key features such as intercepts, asymptotes, and the overall shape. The intercepts, both x- and y-intercepts, are points where the graph crosses the axes.

Y-intercept: This is the point where the graph crosses the y-axis. It occurs when $ x = 0 $. To find it, substitute $ x = 0 $ in the function. In our problem, this yields the y-intercept as $(0, -4)$.
X-intercepts: These are the points where the graph crosses the x-axis, indicating where the function value, $ f(x) $, equals zero. Solving this involves setting the function to zero and solving for $ x $.

Understanding function graphs helps in visualizing algebraic solutions and conveys intuitive understanding of numerical relationships. They are essential tools in solving real-world problems, providing insights into how functions behave across different values of $ x $.

Solving Equations for Intercepts

Solving equations is a fundamental skill in mathematics, especially when finding intercepts of a function graph. For any function $ f(x) $, intercepts are determined by solving specific equations:

Finding the Y-intercept: Substitute $ x = 0 $ into $ f(x) $ to solve for $ y $. The solution gives the y-intercept value.
Finding the X-intercepts: Set $ f(x) = 0 $ and solve this equation to find all possible $ x $ values where the function crosses the x-axis.

In our specific problem, solving $ 4^x x^4 - 4^{x + 1} = 0 $ involves isolating terms and simplifying using algebraic identities like the difference of squares. By simplifying and factoring, the equation $ x^4 - 4 = 0 $ is analyzed using well-known methods to find the x-intercepts $(\sqrt{2}, 0)$ and $(-\sqrt{2}, 0)$.

Application of the Difference of Squares

The difference of squares is a helpful algebraic identity used to simplify and solve polynomial equations. It applies to expressions of the form $ a^2 - b^2 $, and it is factored as $(a - b)(a + b)$.
In our problem, $ x^4 - 4 $ can be seen as a difference of squares, $ (x^2)^2 - 2^2 $, allowing us to factor it as $(x^2 - 2)(x^2 + 2)$.

This identity aids in quickly identifying solutions for $ x $, simplifying the task of finding intercepts.
It also highlights the importance of recognizing patterns and applying algebraic techniques to reduce complex problems into simpler ones.

Once factored, the equation $ (x^2 - 2)(x^2 + 2) = 0 $, when set to zero, simplifies the process of solving for $ x $. This is instrumental in finding the real solutions $ x = \pm \sqrt{2} $, while acknowledging $ x^2 + 2 = 0 $ yields no real solutions, since no real number squared equals a negative value.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the \(x\) - and \(y\) -intercepts of the graph of the given function. Do not graph. $$ f(x)=4^{x} x^{4}-4^{x+1} $$

Short Answer

Step by step solution

Finding the y-intercept

Finding the x-intercepts

Key Concepts

Understanding Function Graphs

Solving Equations for Intercepts

Application of the Difference of Squares

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Logic and Functions

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.