Graph the polynomial function $f(x)=-2(x-1)^2\left(x^2-16\right)$ using parts (a) through (e).
(a) Determine the end behavior of the graph of the function.
The graph of $f$ behaves like $y=-2 x^4$ for large values of $|x|$.
b) Find the $x$ - and $y$-intercepts of the graph of the function.
he $x$-intercept(s) is/are $-2,-1,2$.
implify your answer. Type an integer or a fraction. Use a comma to separate answers as
Answer (1)
The x-intercepts are found by solving f ( x ) = 0 , resulting in x = − 4 , 1 , 4 .
The y-intercept is found by evaluating f ( 0 ) , resulting in y = 32 .
The end behavior is determined by the leading term, behaving like y = − 2 x 4 .
The graph touches the x-axis at x = 1 (multiplicity 2) and crosses at x = − 4 and x = 4 (multiplicity 1), with the final answer being a description of these features to aid in graphing.
Explanation
Problem Analysis We are asked to graph the polynomial function f ( x ) = − 2 ( x − 1 ) 2 ( x 2 − 16 ) . To do this, we need to find the end behavior, x- and y-intercepts, and analyze the multiplicity of the roots.
Finding x-intercepts First, let's find the x-intercepts by setting f ( x ) = 0 :
− 2 ( x − 1 ) 2 ( x 2 − 16 ) = 0 This gives us ( x − 1 ) 2 = 0 or x 2 − 16 = 0 . Thus, x = 1 (with multiplicity 2) and x = { − 4 , 4 } (each with multiplicity 1). So the x-intercepts are -4, 1, and 4.
Finding y-intercept Next, let's find the y-intercept by setting x = 0 :
f ( 0 ) = − 2 ( 0 − 1 ) 2 ( 0 2 − 16 ) = − 2 ( 1 ) ( − 16 ) = 32 So the y-intercept is 32.
Analyzing End Behavior The end behavior is given as y = − 2 x 4 . This means that as x approaches positive or negative infinity, f ( x ) approaches negative infinity. In other words, the graph opens downwards.
Analyzing Multiplicity of Roots Now, let's analyze the multiplicity of the roots. The root x = 1 has a multiplicity of 2, which means the graph touches the x-axis at x = 1 but does not cross it. The roots x = − 4 and x = 4 each have a multiplicity of 1, which means the graph crosses the x-axis at these points.
Summary of Key Features Putting it all together, we have:
x-intercepts: -4, 1, 4
y-intercept: 32
End behavior: behaves like y = − 2 x 4
Multiplicity: x = 1 (multiplicity 2), x = -4 (multiplicity 1), x = 4 (multiplicity 1)
Examples
Understanding the behavior of polynomial functions is crucial in many real-world applications. For example, engineers use polynomial functions to model the trajectory of a projectile, such as a ball thrown in the air. By knowing the roots (x-intercepts) and the end behavior of the polynomial, they can predict where the projectile will land and how high it will go. Similarly, economists use polynomial functions to model economic growth or decline, and biologists use them to model population growth or decay. The ability to analyze and graph polynomial functions is therefore a valuable skill in many fields.
