/*! 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 59 Graphical Analysis In Exercises ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 59

Graphical Analysis In Exercises \(59-64,\) use a graphing utility to graph the quadratic function. Find the \(x\) -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when \(f(x)=0 .\) $$ f(x)=x^{2}-4 x $$

Short Answer

Expert verified
The \(x\)-intercepts are at \(x = 0\) or \(x = 4\).

Step by step solution

01

Plot the Function

Using a graphing utility, the function \(f(x)=x^{2}-4 x\) is plotted. Observe the graph and where it intersects or touches the \(x\)-axis, because these points are known as the x-intercepts.
02

Find the x-intercepts

The solutions to the equation are also the x-intercepts (where \(y = 0\)). To find them algebraically, set the function \(f(x) = 0\) and solve for \(x\). You will get \(0 = x^{2} - 4x\). From this equation, factor out an \(x\) to get \(0 = x(x - 4)\). This implies \(x = 0\) or \(x = 4\). These are the x-intercepts and can be cross verified from the graph.
03

Compare and Conclude

Compare the x-intercepts found through the algebraic solution and those identified on the graph. They should match. Therefore, the solutions of the quadratic equation when \(f(x) = 0\) are \(x = 0\) and \(x = 4\).

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.

x-intercepts
The concept of x-intercepts is crucial when analyzing quadratic functions. These are the points where the graph of the function intersects the x-axis. For any given function, the value of \( y \) is zero at the x-intercepts since these points lie directly on the x-axis. In the case of the quadratic function \( f(x) = x^2 - 4x \), the process to find the x-intercepts involves setting the equation to zero, \( f(x) = 0 \).To calculate this, we set up the equation \( x^2 - 4x = 0 \) and factor it: \( x(x - 4) = 0 \). This indicates that the x-intercepts are at \( x = 0 \) and \( x = 4 \). Graphically, these points can be seen where the parabola crosses the x-axis. It's interesting to check the accuracy of these solutions by plotting the function on a graph, as they should perfectly coincide.
Quadratic Equation Solutions
Quadratic equations often have two solutions, which correspond to the x-intercepts of their graphs. The quadratic in question, \( f(x) = x^2 - 4x \), can be solved by factoring. We start by setting the equation to zero: \( x^2 - 4x = 0 \). Factoring this yields \( x(x - 4) = 0 \), giving the solutions \( x = 0 \) and \( x = 4 \).These solutions are also where the graph of the quadratic function intersects the x-axis, hence they are the x-intercepts. The method of solving quadratic equations by factoring is efficient and directly gives the roots, or solutions, of the equation. This way, we can also verify these roots graphically to ensure they match the x-intercepts we identify on a graphing utility.
Graphing Utility
A graphing utility is a powerful tool to visualize functions and easily find key characteristics such as x-intercepts. For the quadratic function \( f(x) = x^2 - 4x \), the use of a graphing utility can help us see the parabola's shape and where it intersects the x-axis.By inputting the function into a graphing device or software, the graph line is generated and evident x-intercepts become visible. These visual x-intercepts can reinforce our algebraic solutions, which in this case are \( x = 0 \) and \( x = 4 \). Such utilities are helpful not only for confirming manually calculated intercepts but also to foster a deeper understanding of the behavior of quadratic functions. They allow students to explore the symmetries and vertex of parabolas, making learning interactive and dynamic.

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

Forensics At \(8 : 30\) A.M., a coroner went to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person's temperature twice. At \(9 : 00\) A.M. the temperature was \(85.7^{\circ} \mathrm{F},\) and at \(11 : 00\) A.M. thetemperature was \(82.8^{\circ} \mathrm{F}\) . From these two temperatures, the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula $$t=-10 \ln \frac{T-70}{98.6-70}$$ where \(t\) is the time in hours elapsed since the person died and \(T\) is the temperature (in degrees Fahrenheit) of the person's body. (This formula comes from a general cooling principle called Newton's Law of Cooling.It uses the assumptions that the person had a normal body temperature of \(98.6^{\circ} \mathrm{F}\) at death and that the room temperature was a constant \(70^{\circ} \mathrm{F} .\) ) Use the formula to estimate the time of death of the person.

Cost, Revenue, and Profit The revenue and cost equations for a product are \(R=x(75-0.0005 x)\) and \(C=30 x+250,000,\) where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold. How many units must be sold to obtain a profit of at least \(\$ 750,000 ?\) What is the price per unit?

In Exercises 73 and \(74,\) use the position equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ where s represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet?

Think About It Sketch the graph of a fifth-degree polynomial function whose leading coefficient is positive and that has a zero at \(x=3\) of multiplicity 2 .

Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x .\) Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56 .\) Which of these values is a physical impossibility in the construction of the box? Explain.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.