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Chapter 0: Problem 23

Find the point \((s)\) of intersection of the graphs of the functions. Express your answers accurate to five decimal places. $$ f(x)=0.3 x^{2}-1.7 x-3.2 ; \quad g(x)=-0.4 x^{2}+0.9 x+6.7 $$

Short Answer

Expert verified
The points of intersection are approximately \(s_1 鈮� (2.40018, -4.78686)\) and \(s_2 鈮� (-1.97155, 7.38055)\).

Step by step solution

01

Set f(x) equal to g(x)

To find the x-coordinate(s), set the two functions equal to each other: \(0.3x^2 - 1.7x - 3.2 = -0.4x^2 + 0.9x + 6.7\)
02

Simplify the equation

Combine like terms to simplify the equation: \(0.7x^2 - 2.6x - 9.9 = 0\)
03

Solve for x-coordinates using the Quadratic Formula

Use the Quadratic Formula to solve for x: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) In the equation \(0.7x^2 - 2.6x - 9.9 = 0\), we have \(a = 0.7, b = -2.6\), and \(c = -9.9\). Plug these values into the Quadratic Formula: \(x =\frac{2.6 \pm \sqrt{(-2.6)^2 - 4(0.7)(-9.9)}}{2(0.7)}\)
04

Calculate the x-coordinates

Evaluate the expression to find the x-coordinates: \(x =\frac{2.6 \pm \sqrt{6.76 + 27.72}}{1.4}\) \(x =\frac{2.6 \pm \sqrt{34.48}}{1.4}\) Approximating the x-coordinates up to five decimal places, we get: \(x_1 鈮� 2.40018\) \(x_2 鈮� -1.97155\)
05

Calculate the y-coordinates

Use either function to find the corresponding y-coordinates. We will use the function f(x): \(y_1 = f(x_1) = 0.3(2.40018)^2 -1.7(2.40018) -3.2\) \(y_1 鈮� -4.78686\) \(y_2 = f(x_2) = 0.3(-1.97155)^2 -1.7(-1.97155) -3.2\) \(y_2 鈮� 7.38055\)
06

Write the points of intersection

The points of intersection are: \(s_1 鈮� (2.40018, -4.78686)\) \(s_2 鈮� (-1.97155, 7.38055)\)

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Key Concepts

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

Functions
Functions in mathematics represent an important concept where each input is associated with a single output. Think of a function as a machine that takes an input, performs a specific task, and produces an output. In this exercise, we have two functions:
  • The function \( f(x) = 0.3x^2 - 1.7x - 3.2 \) represents a parabola that opens upwards due to the positive coefficient of \( x^2 \).
  • The function \( g(x) = -0.4x^2 + 0.9x + 6.7 \) is another parabola, but it opens downwards because the coefficient of \( x^2 \) is negative.
Functions can be used to model real-world processes, and understanding their behavior is crucial for problem-solving in mathematics and various fields.
Intersection Points
Intersection points occur where two graphs meet. These points represent the values of \( x \) and \( y \) that satisfy both equations simultaneously. In our problem, we are looking for these intersection points by setting \( f(x) = g(x) \). This creates an equation that we need to solve to find the \( x \)-coordinates. Once we find the \( x \)-coordinates, we plug them back into one of the original functions to find the corresponding \( y \)-coordinates. These coordinates together form the intersection points.
  • The intersection point \( s_1 鈮� (2.40018, -4.78686) \) shows where both functions have the same value.
  • The point \( s_2 鈮� (-1.97155, 7.38055) \) represents another intersection point where the two equations are equal.
Understanding intersection points is key to solving many mathematical problems, as they often represent solutions to equations.
Quadratic Formula
The Quadratic Formula is a powerful tool used to solve equations of the form \( ax^2 + bx + c = 0 \). It provides an exact solution for any quadratic equation, making it essential for solving intersection problems involving parabolas. The formula is given by:\[{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}\]In this problem, the equation \( 0.7x^2 - 2.6x - 9.9 = 0 \) needs to be solved using the Quadratic Formula.
  • Identify the coefficients: \( a = 0.7 \), \( b = -2.6 \), \( c = -9.9 \).
  • Calculate the discriminant \( b^2 - 4ac \) to check the nature of the roots, which in this case is positive, indicating two real solutions.
  • Plug the values into the formula to find \( x_1 \) and \( x_2 \), which lead to the intersection points.
Mastering the Quadratic Formula helps with understanding and solving a wide variety of mathematical and real-life problems.
Graphs
Graphs visually represent functions, making them a great tool for analyzing and understanding the behavior of equations. Each function can be plotted on a coordinate plane, with the \( x \)-axis representing the input and the \( y \)-axis representing the output.
  • The graph of \( f(x) = 0.3x^2 - 1.7x - 3.2 \) will appear as a U-shaped curve due to its upward-opening parabola.
  • The graph of \( g(x) = -0.4x^2 + 0.9x + 6.7 \) forms an arch or downward opening parabola.
By plotting these functions on the same graph, we can quickly visualize where they intersect, offering a solution that is both intuitive and mathematically accurate. Graphs are essential in fields such as science, engineering, and economics, where complex relationships need to be understood and communicated effectively.

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Most popular questions from this chapter

Find \(f \circ g \circ h .\) $$f(x)=\sqrt{x}, g(x)=2 x+1, \quad h(x)=x^{2}-1$$

Find the point \((s)\) of intersection of the graphs of the functions. Express your answers accurate to five decimal places. $$ f(x)=-0.3 x^{2}+0.6 x+3.2 ; \quad g(x)=0.2 x^{2}-1.2 x-4.8 $$

Fighting Crime Suppose that the reported serious crimes (crimes that include homicide, rape, robbery, aggravated assault, burglary, and car theft) that end in arrests or in the identification of suspects is \(g(x)\) percent, where \(x\) denotes the total number of detectives. Next, suppose that the total number of detectives in year \(t\) is \(f(t)\), where \(t=0\) corresponds to 2001 . a. If \(f(1)=406\) and \(g(406)=23\), find \((g \circ f)(1)\), and interpret your result. b. If \(f(6)=326\) and \(g(326)=18\), find \((g \circ f)(6)\), and interpret your result. c. Comment on the results of parts (a) and (b). Source: Boston Police Department.

Find the point \((s)\) of intersection of the graphs of the functions. Express your answers accurate to five decimal places. $$ f(x)=\sin ^{2} x ; \quad g(x)=\sqrt{x^{2}-x^{4}} $$

Hotel Occupancy Rate The occupancy rate of the all-suite Wonderland Hotel, located near an amusement park, is given by the function $$ r(t)=\frac{10}{81} t^{3}-\frac{10}{3} t^{2}+\frac{200}{9} t+55 \quad 0 \leq t \leq 11 $$ where \(t\) is measured in months and \(t=0\) corresponds to the beginning of January. Management has estimated that the monthly revenue (in thousands of dollars) is approximated by the function $$ R(r)=-\frac{3}{5000} r^{3}+\frac{9}{50} r^{2} \quad 0 \leq r \leq 100 $$ where \(r\) (percent) is the occupancy rate. a. What is the hotel's occupancy rate at the beginning of January? At the beginning of July? b. What is the hotel's monthly revenue at the beginning of January? At the beginning of July?
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