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Given the function P(x) = x³ + x² x² - 42x. The y-intercept is The x-intercepts is/are When →∞, y → ? v When →-∞, y → ? ✓

Given the function P(x) = x³ + x² x² - 42x. The y-intercept is The x-intercepts is/are When →∞, y → ? v When →-∞, y → ? ✓

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 35E
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### Polynomial Function Analysis Given the function \( P(x) = x^3 + x^2 - 42x \). 1. **Finding the y-intercept:** The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when \( x = 0 \). The y-intercept is: \[ P(0) = 0^3 + 0^2 - 42 \cdot 0 = 0 \] 2. **Finding the x-intercepts:** The x-intercepts are the points where the graph of the function intersects the x-axis. This occurs when \( P(x) = 0 \). We need to solve the equation: \[ x^3 + x^2 - 42x = 0 \] Factor out \( x \): \[ x(x^2 + x - 42) = 0 \] Next, factor the quadratic equation \( x^2 + x - 42 \): \[ x(x + 7)(x - 6) = 0 \] Thus, the x-intercepts are: \[ x = 0, x = -7, x = 6 \] 3. **End Behavior Analysis:** To determine the behavior of the function as \( x \) approaches positive or negative infinity: - When \( x \to \infty \), \( y \to \infty \) - When \( x \to -\infty \), \( y \to -\infty \) Therefore, the detailed questions and answers should appear as follows on the educational website: 1. **The y-intercept is:** \[ 0 \] 2. **The x-intercepts is/are:** \[ 0, -7, 6 \] 3. **End Behavior:** - When \( x \to \infty \), \( y \to \infty \) - When \( x \to -\infty \), \( y \to -\infty \) This completes the analysis of the given polynomial function \( P(x) = x^3 + x^2 - 42x \).
Transcribed Image Text:### Polynomial Function Analysis Given the function \( P(x) = x^3 + x^2 - 42x \). 1. **Finding the y-intercept:** The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when \( x = 0 \). The y-intercept is: \[ P(0) = 0^3 + 0^2 - 42 \cdot 0 = 0 \] 2. **Finding the x-intercepts:** The x-intercepts are the points where the graph of the function intersects the x-axis. This occurs when \( P(x) = 0 \). We need to solve the equation: \[ x^3 + x^2 - 42x = 0 \] Factor out \( x \): \[ x(x^2 + x - 42) = 0 \] Next, factor the quadratic equation \( x^2 + x - 42 \): \[ x(x + 7)(x - 6) = 0 \] Thus, the x-intercepts are: \[ x = 0, x = -7, x = 6 \] 3. **End Behavior Analysis:** To determine the behavior of the function as \( x \) approaches positive or negative infinity: - When \( x \to \infty \), \( y \to \infty \) - When \( x \to -\infty \), \( y \to -\infty \) Therefore, the detailed questions and answers should appear as follows on the educational website: 1. **The y-intercept is:** \[ 0 \] 2. **The x-intercepts is/are:** \[ 0, -7, 6 \] 3. **End Behavior:** - When \( x \to \infty \), \( y \to \infty \) - When \( x \to -\infty \), \( y \to -\infty \) This completes the analysis of the given polynomial function \( P(x) = x^3 + x^2 - 42x \).
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