Question
Analyze the polynomial function \( f(x)=x^{2}(x-3) \) using parts (a) through (e). (a) Determine the end behavior of the graph of the function. The graph of \( f \) behaves like \( y=x^{3} \) for large values of \( |x| \). (b) Find the \( x \) - and \( y \)-intercepts of the graph of the function. The \( x \)-intercept(s) is/are (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.)
Ask by Munoz George. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The x-intercepts are x = 0 and x = 3.
Solution
We are given the function f(x) = x²(x – 3), which can also be written as f(x) = x³ – 3x².
(a) End Behavior:
Since this is a cubic polynomial with a positive leading coefficient, as x → ∞, f(x) behaves like x³ and goes to +∞; and as x → –∞, f(x) behaves like x³ and goes to –∞.
(b) Intercepts:
• x-intercepts: Set f(x) = 0.
x²(x – 3) = 0
This gives x² = 0 or x – 3 = 0,
so x = 0 or x = 3.
Thus, the x-intercepts are x = 0 and x = 3.
• y-intercept: Evaluate f(0).
f(0) = (0)²(0 – 3) = 0.
So, the y-intercept is 0.
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Simplify this solution Bonus Knowledge
The \( x \)-intercept occurs when \( f(x) = 0 \). Setting the polynomial equal to zero gives: \[ x^{2}(x - 3) = 0 \] This leads to two equations: \( x^{2} = 0 \) and \( x - 3 = 0 \). Solving \( x^{2} = 0 \) gives \( x = 0 \), and solving \( x - 3 = 0 \) gives \( x = 3 \). Therefore, the \( x \)-intercepts are \( 0, 3 \). To find the \( y \)-intercept, evaluate \( f(0) \): \[ f(0) = 0^{2}(0 - 3) = 0 \] Thus, the \( y \)-intercept is also \( 0 \). The \( x \)-intercepts are \( 0, 3 \) and the \( y \)-intercept is \( 0 \).
