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Chapter 8: Problem 43

In Exercises \(41-48,\) solve the system using a graphing utility. Round all values to three decimal places. $$\left\\{\begin{array}{l} y=-x^{2}+3 \\ y=3^{x} \end{array}\right.$$

Short Answer

Expert verified
The intersection point of the two functions (which is the solution to this system of equations) will have its x-coordinate rounded to three decimal places. The exact values depend on the accuracy of the used graphing utility.

Step by step solution

01

Plot the functions

Start by plotting the two functions, \(y=-x^{2}+3\) and \(y=3^{x}\), on a graphing utility. Observe the shape and position of each graph.
02

Identify Intersection Points

Next, identify the point(s) where the two graphs intersect. This point represents the solution to the system of equations, as it's the x-value where both equations are true at the same time.
03

Compute Intersection Points

Using the graphing utility, zoom in on the point of intersection and find its x-coordinate. Make sure to round the values to three decimal places.

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Key Concepts

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

Systems of Equations
A system of equations is a set of two or more equations with the same variables. Solving such a system means finding the values of the variables that satisfy all equations simultaneously. In our example, we have two equations:
  • The quadratic equation: \( y = -x^2 + 3 \)
  • The exponential equation: \( y = 3^x \)
In a graphing utility, these are plotted as separate curves. The solutions to the system are the x-values where the graphs intersect.

Understanding how different types of functions behave can make it easier to predict where they might intersect. For example, quadratic functions typically form a parabola, while exponential functions show rapid growth or decay.
Intersection Points
Intersection points are critical in solving systems of equations graphically. They show us where the solutions to both equations lie.

When two curves meet at one or more points on a graph, those points are intersection points. In the context of our problem, these are the locations where both the quadratic function \( y = -x^2 + 3 \) and the exponential function \( y = 3^x \) have the same y-value:
  • The x-coordinate of these points indicates the solution(s) to the system.
To find these exact x-values using a graphing utility, we often use tools like zooming in for greater precision. Once identified, these x-values can be rounded to three decimal places to get an accurate estimate.
Quadratic Functions
Quadratic functions are polynomial functions of degree 2. They have the general form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

In our exercise, the quadratic function is \( y = -x^2 + 3 \). Here:
  • The term \( -x^2 \) indicates the parabola opens downward.
  • The constant \( +3 \) moves the vertex of the parabola up to the point (0, 3) on the y-axis.
Graphically, quadratic functions create a distinctive U-shape called a parabola. The nature of the quadratic might imply whether it has real, imaginary, multiple, or no intersection points with other graphs. Understanding this can help anticipate whether a solution exists when checking against other functions.

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Most popular questions from this chapter

The following table lists the caloric content of a typical fast-food meal. Food (single serving) Calories Cheeseburger Medium order of fries Medium cola (21 oz) \(\begin{array}{lc}\text { Food (single serving) } & \text { Calories } \\\ \text { Cheeseburger } & 330 \\ \text { Medium order of fries } & 450 \\\ \text { Medium cola }(210 z) & 220\end{array}\) (a) After a lunch that consists of a cheeseburger, a medium order of fries, and a medium cola, you decide to burn off a quarter of the total calories in the meal by some combination of running and walking. You know that running burns 8 calories per minute and walking burns 3 calories per minute. If you exercise for a total of 40 minutes, how many minutes should you spend on each activity? (b) Rework part (a) for the case in which you exercise for a total of only 20 minutes. Do you get a realistic solution? Explain your answer.

For the given matrices \(A, B,\) and \(C,\) evaluate the indicated expression. $$\begin{aligned}&A=\left[\begin{array}{ll}4 & 1 \\\0 & 2 \\\5 & 1\end{array}\right] ; \quad B=\left[\begin{array}{rr}4 & 3 \\\\-6 & 2 \\\3 & -1\end{array}\right]\\\&C=\left[\begin{array}{rrr}1 & 2 & 3 \\\\-2 & -3 & -1 \\\3 & 1 & 2\end{array}\right] ; \quad C(B-A)\end{aligned}$$

If \(A=\left[\begin{array}{cc}a^{2}-3 a+3 & 1 \\ 0 & 2 b+5\end{array}\right]\) and \(B=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right],\) for what values of \(a\) and \(b\) does \(A B=\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right] ?\)

Minimize \(P=16 x+10 y\) subject to the following constraints. $$\left\\{\begin{array}{l} y \geq 2 x \\ x \geq 5 \\ x \geq 0 \\ y \geq 0 \end{array}\right.$$

The total revenue generated by a film comes from two sources: box-office ticket sales and the sale of merchandise associated with the film. It is estimated that for a very popular film such as Spiderman or Harry Potter, the revenue from the sale of merchandise is four times the revenue from ticket sales. Assume this is true for the film Spiderman, which grossed a total of \(\$ 3\) billion. Find the revenue from ticket sales and the revenue from the sale of merchandise.
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