Solve the system of equations. Type in all points of intersection for the two functions and round to the nearest tenth if necessary.
[tex]
\begin{array}{l}
f(x)=-0.5 x+2 \\
g(x)=x^3-5 x^2+3
\end{array}
[/tex]
Answer (2)
Set the two functions equal to each other: − 0.5 x + 2 = x 3 − 5 x 2 + 3 .
Rearrange the equation: x 3 − 5 x 2 + 0.5 x + 1 = 0 .
Find the roots of the cubic equation: x ≈ − 0.4 , 0.5 , 4.9 .
Calculate the corresponding y values and round to the nearest tenth: ( − 0.4 , 2.2 ) , ( 0.5 , 1.7 ) , ( 4.9 , − 0.4 ) .
The points of intersection are: ( − 0.4 , 2.2 ) , ( 0.5 , 1.7 ) , ( 4.9 , − 0.4 ) .
Explanation
Problem Analysis We are given a system of equations with two functions:
f ( x ) = − 0.5 x + 2 g ( x ) = x 3 − 5 x 2 + 3
We need to find the points of intersection of these two functions. These points occur where f ( x ) = g ( x ) .
Setting the Equations Equal To find the intersection points, we set the two functions equal to each other:
− 0.5 x + 2 = x 3 − 5 x 2 + 3
Rearranging the equation, we get:
x 3 − 5 x 2 + 0.5 x + 1 = 0
Finding the Roots We need to solve the cubic equation x 3 − 5 x 2 + 0.5 x + 1 = 0 for x . We can use a numerical method or a calculator to find the approximate real roots of this equation. The roots are approximately:
x 1 ≈ − 0.3869 x 2 ≈ 0.5324 x 3 ≈ 4.8546
Calculating the Y Values Now, we need to find the corresponding y values for each x value. We can use either f ( x ) or g ( x ) to find the y values. Let's use f ( x ) = − 0.5 x + 2 :
For x 1 ≈ − 0.3869 :
y 1 = − 0.5 ( − 0.3869 ) + 2 ≈ 0.19345 + 2 ≈ 2.19345
For x 2 ≈ 0.5324 :
y 2 = − 0.5 ( 0.5324 ) + 2 ≈ − 0.2662 + 2 ≈ 1.7338
For x 3 ≈ 4.8546 :
y 3 = − 0.5 ( 4.8546 ) + 2 ≈ − 2.4273 + 2 ≈ − 0.4273
Rounding to the Nearest Tenth We need to round the x and y values to the nearest tenth:
For x 1 ≈ − 0.3869 , y 1 ≈ 2.19345 :
Rounded values: x 1 ≈ − 0.4 , y 1 ≈ 2.2 Point: ( − 0.4 , 2.2 )
For x 2 ≈ 0.5324 , y 2 ≈ 1.7338 :
Rounded values: x 2 ≈ 0.5 , y 2 ≈ 1.7 Point: ( 0.5 , 1.7 )
For x 3 ≈ 4.8546 , y 3 ≈ − 0.4273 :
Rounded values: x 3 ≈ 4.9 , y 3 ≈ − 0.4 Point: ( 4.9 , − 0.4 )
Final Answer The points of intersection, rounded to the nearest tenth, are:
( − 0.4 , 2.2 ) ( 0.5 , 1.7 ) ( 4.9 , − 0.4 )
Examples
Understanding systems of equations is crucial in various real-world applications. For instance, in economics, it helps determine market equilibrium where supply and demand curves intersect. Similarly, in physics, it can be used to find the point where the trajectories of two objects meet. By solving systems of equations, we can model and analyze complex relationships between different variables, leading to informed decision-making and problem-solving in diverse fields.
The points of intersection for the functions f ( x ) and g ( x ) are approximately ( − 0.4 , 2.2 ) , ( 0.5 , 1.7 ) , and ( 4.9 , − 0.4 ) . These points indicate where the two functions have the same output. Each coordinate pair shows an intersection point on the graph of the functions.
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