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Chapter 2: Problem 60

Use a graphing utility to find all real solutions. You may need to adjust the window size manually or use the ZOOMFIT feature to get a clear graph. Graphically solve \(\sqrt{x-k}=x\) for \(k=-2,0,\) and 2 How many solutions does the equation have for each value of \(k ?\)

Short Answer

Expert verified
The number of real solutions depends on the value of \(k\). The exact number of solutions should be found by graphing the equation with a graphing utility for each given \(k\) value.

Step by step solution

01

Graphing for k=-2

Let's substitute \(k=-2\) into the equation. That will give us \(\sqrt{x+2}=x\). Using a graphing utility, plot the function \(\sqrt{x+2}\) and the function \(x\). The points at which the two graphs intersect are the real solutions.
02

Counting solutions for k=-2

The number of solutions are the points where the line \(x\) and the graph \(\sqrt{x+2}\) intersect. Count these points.
03

Graphing for k=0

Follow the same strategy, but this time with \(k=0\), the equation becomes \(\sqrt{x}=x\). Plot the two functions \(\sqrt{x}\) and \(x\) and find the points of intersection.
04

Counting solutions for k=0

Again, the points of intersection on the graph are the real solutions. Count these points.
05

Graphing for k=2

In this case, with \(k=2\), the equation becomes \(\sqrt{x-2}=x\). Use the graphing utility to plot \(\sqrt{x-2}\) and \(x\) and find the intersection points.
06

Counting solutions for k=2

The solutions to the equation are the intersection points on the graph. Count these points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Solutions
Real solutions are the points where the graph of an equation and the x-axis intersect. When dealing with the equation \( \sqrt{x-k} = x \), we are looking for the values of \( x \) that satisfy the equation for different values of \( k \). These solutions are real numbers as opposed to imaginary numbers.
To find real solutions graphically, we can plot both sides of the equation separately and observe where they intersect.
This way we identify where the values of \( x \) are equal for both expressions. Each intersection point represents a real solution to the equation.
Intersection Points
The intersection points are crucial for identifying the real solutions of an equation. When you plot two functions, the intersection points show where the functions have the same value for the same \( x \).
In our example \( \sqrt{x-k} \) and \( x \) were both plotted. Depending on the value of \( k \), these graphs intersect at different points.
  • For \( k = -2 \), the equation transforms into \( \sqrt{x+2} = x \) leading to the graph potentially intersecting at different points than other \( k \) values.
  • For \( k = 0 \), the graphs are \( \sqrt{x} \) and \( x \) which might intersect differently.
  • For \( k = 2 \), intersection points need to be found graphing \( \sqrt{x-2} \) and \( x \).
By counting these intersection points, we determine how many real solutions exist for each value of \( k \).
Graphing Utility
Using a graphing utility is an invaluable tool for visualizing mathematical equations and understanding their solutions. It allows us to plot complex functions easily and visually identify intersections and real solutions.
For the exercise, utilizing a graphing utility can make the task of finding intersection points straightforward - you can often rely on zooming features such as ZOOMFIT to adjust the view.
This helps to gain a clearer, detailed view of how the functions \( \sqrt{x-k} \) and \( x \) interact. Adjust the window size if necessary, so you can easily spot where the graphs meet.
Equation Solving
Solving equations graphically is a strategic method in finding solutions to mathematical problems. In this context, it involves interpreting where two graphs intersect.
  • By substituting different values of \( k \) in \( \sqrt{x-k} = x \), each transformation of the equation provides a unique graph to analyze.
  • The graphical approach gives immediate visual confirmation of solution existence, making it easier to understand where equations meet.
As each value of \( k \) alters the equation, the number of solutions may change. This graphical method complements algebraic solving and helps to visualize potential multiple solutions or in some cases, no solutions at all.

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Most popular questions from this chapter

Attendance at Broadway shows in New York can be modeled by the quadratic function \(p(t)=0.0489 t^{2}-0.7815 t+10.31,\) where \(t\) is the number of years since 1981 and \(p(t)\) is the attendance in millions. The model is based on data for the years \(1981-2000 .\) (Source: The League of American Theaters and Producers, Inc.) (a) Use this model to estimate the attendance in the year \(1995 .\) Compare it to the actual value of 9 million. (b) Use this model to predict the attendance for the year 2006 (c) What is the vertex of the parabola associated with the function \(p\), and what does it signify in relation to this problem? (d) Would this model be suitable for predicting the attendance at Broadway shows for the year \(2025 ?\) Why or why not? (e) \(\quad\) Use a graphing utility to graph the function \(p\) What is an appropriate range of values for \(t ?\)

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Let \(f(x)=a x+b\) and \(g(x)=c x+d,\) where \(a, b, c,\) and \(d\) are constants. Show that \((f+g)(x)\) and \((f-g)(x)\) also represent linear functions.

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\)-intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=-x^{2}+4 x-4$$
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