Problem 60
Question
Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-0.01 x$$
Step-by-Step Solution
Verified Answer
X-intercepts: \((0,0), (0.1,0), (-0.1,0)\); Y-intercept: \((0,0)\); End behavior: left down, right up.
1Step 1: Understanding the Polynomial Function
The function given is a cubic polynomial: \( f(x) = x^3 - 0.01x \). This function is in the standard form of a cubic polynomial where the leading term is \( x^3 \). Cubic functions typically have a characteristic 'S' shaped curve.
2Step 2: Graph the Polynomial Using a Calculator
To graph the function \( f(x) = x^3 - 0.01x \), enter the equation into a graphing calculator. Set an appropriate window to observe the full behavior of the graph. You might want to start with \([-10, 10]\) for both \(x\) and \(y\) values.
3Step 3: Determine the X-Intercepts
To find the x-intercepts, set \( f(x) = 0 \). Solve \( x^3 - 0.01x = 0 \) for \( x \). Factor the equation to get \( x(x^2 - 0.01) = 0 \), yielding roots at \( x = 0, x = \sqrt{0.01}, x = -\sqrt{0.01} \). This means the x-intercepts are at \( (0,0), (0.1,0), (-0.1,0) \).
4Step 4: Determine the Y-Intercept
The y-intercept occurs where \( x = 0 \). Substituting into \( f(x) \), we find \( f(0) = 0^3 - 0.01(0) = 0 \). Therefore, the y-intercept is at \( (0, 0) \).
5Step 5: Analyze the End Behavior of the Polynomial
The end behavior of the polynomial is determined by the leading term \( x^3 \). As \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \). This indicates that the left end of the graph goes downward, and the right end goes upward.
Key Concepts
Graphing PolynomialsX-InterceptsY-InterceptsEnd Behavior of Polynomials
Graphing Polynomials
Graphing a polynomial helps us understand its overall behavior, such as its intercepts and end behavior. For cubic functions like \( f(x) = x^3 - 0.01x \), this typically results in an `S` shaped curve. This occurs because of the highest power of \( x \), which in this case is 3. The graph of this cubic polynomial will show how the function behaves over a range of \( x \) values by crossing the axes at certain points called intercepts. To visually explore this function, you can use a graphing calculator which allows you to see this behavior clearly. Make sure you set a window that shows the entire important parts of the graph, a starting point could be \( [-10, 10] \) for both \( x \) and \( y \) values. This gives you a complete and detailed perspective of how the polynomial looks.
X-Intercepts
X-intercepts are the points where the graph crosses the x-axis. For our function \( f(x) = x^3 - 0.01x \), these intercepts can be found by solving the equation \( f(x) = 0 \). This involves setting the polynomial equal to zero and solving for \( x \).
Looking at the polynomial \( x^3 - 0.01x = 0 \), you can factor it to \( x(x^2 - 0.01) = 0 \). This shows us that the x-intercepts are found when \( x = 0 \), \( x = \sqrt{0.01} \), and \( x = -\sqrt{0.01} \). Simplified, we have intercepts at the points \( (0, 0), (0.1, 0), \text{and} (-0.1, 0) \).
These intercepts indicate where the polynomial will touch or cross the x-axis, offering insights into the roots of the equation.
Looking at the polynomial \( x^3 - 0.01x = 0 \), you can factor it to \( x(x^2 - 0.01) = 0 \). This shows us that the x-intercepts are found when \( x = 0 \), \( x = \sqrt{0.01} \), and \( x = -\sqrt{0.01} \). Simplified, we have intercepts at the points \( (0, 0), (0.1, 0), \text{and} (-0.1, 0) \).
These intercepts indicate where the polynomial will touch or cross the x-axis, offering insights into the roots of the equation.
Y-Intercepts
The y-intercept is a specific point where the graph crosses the y-axis. It's important because it tells you the function's value when \( x = 0 \). For the function \( f(x) = x^3 - 0.01x \), determining the y-intercept involves substituting \( x = 0 \) directly into the function.
Performing this substitution, we find that \( f(0) = 0^3 - 0.01(0) = 0 \). Therefore, the y-intercept is at the point \( (0, 0) \).
In other words, the point where the graph meets the y-axis is at the origin, providing a foundational reference for the graph's position.
Performing this substitution, we find that \( f(0) = 0^3 - 0.01(0) = 0 \). Therefore, the y-intercept is at the point \( (0, 0) \).
In other words, the point where the graph meets the y-axis is at the origin, providing a foundational reference for the graph's position.
End Behavior of Polynomials
Understanding the end behavior of a polynomial is essential for grasping its long-term behavior as you move towards positive or negative infinity along the x-axis. For the polynomial \( f(x) = x^3 - 0.01x \), the leading term \( x^3 \) dictates this behavior.
The term \( x^3 \) causes a distinct cubic characteristic. As \( x \to \infty \) (moving to the right infinitely), \( f(x) \to \infty \), meaning the graph goes upwards. Conversely, as \( x \to -\infty \) (moving to the left infinitely), \( f(x) \to -\infty \), thus the graph tends downwards.
These observations define the end behavior: the graph ascends to the right and descends to the left, giving it the signature shape of most cubic polynomials.
The term \( x^3 \) causes a distinct cubic characteristic. As \( x \to \infty \) (moving to the right infinitely), \( f(x) \to \infty \), meaning the graph goes upwards. Conversely, as \( x \to -\infty \) (moving to the left infinitely), \( f(x) \to -\infty \), thus the graph tends downwards.
These observations define the end behavior: the graph ascends to the right and descends to the left, giving it the signature shape of most cubic polynomials.
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