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Chapter 3: Problem 1

Plot the Curves : $$ y=1+x^{2}-\frac{1}{2} x^{4} $$

Short Answer

Expert verified
Plot points for \( x \) from -3 to 3 and connect smoothly for the curve of \( y = 1 + x^2 - \frac{1}{2}x^4 \).

Step by step solution

01

Understanding the Function

The function given is a polynomial: \( y = 1 + x^2 - \frac{1}{2}x^4 \). This equation describes a curve where \( y \) is expressed in terms of \( x \). To plot this curve, we will compute \( y \) for a range of \( x \) values.
02

Selecting Range of x-values

Choose a range of \( x \) values to evaluate the function. Typically, for simple plots, we can choose \( x \) values from -3 to 3. This range gives a clear view of the behavior of the polynomial without extending to extreme values where it could behave unpredictably.
03

Calculating y-values

For each selected \( x \)-value, substitute this into the function to compute \( y \). For example, for \( x = -3 \), the \( y \)-value would be \( y = 1 + (-3)^2 - \frac{1}{2}(-3)^4 = 1 + 9 - 40.5 = -30.5 \). Repeat for other values such as \( x = -2, -1, 0, 1, 2, 3 \).
04

Plotting the Points

Using the calculated \( y \)-values, plot the corresponding points on a graph with \( x \) on the horizontal axis and \( y \) on the vertical axis. Mark each point clearly.
05

Drawing the Curve

Connect the plotted points smoothly to form the curve of the polynomial. Since the polynomial is a continuous function, the resulting curve should also be smooth, without any breaks.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plotting Graphs
Plotting graphs is a fundamental skill in understanding mathematical functions, especially polynomial functions like the one provided, which is expressed as \( y = 1 + x^2 - \frac{1}{2}x^4 \). To plot the graph of a polynomial function, follow these steps:
  • Choose a range of \( x \) values, typically centered around zero, to adequately capture the key features of the function.
  • Calculate the corresponding \( y \) values by substituting each \( x \) into the polynomial equation.
  • Plot these \( (x, y) \) points on a coordinate grid.
Once the points are plotted, the final step is to connect the dots smoothly. This visual representation will help in further analyses such as identifying key features like intercepts, turning points, and symmetry.
Curve Sketching
Curve sketching involves going beyond just plotting points; it requires understanding the shape and important characteristics of a curve based on its function. For polynomial functions, here's what to consider:
  • Look for symmetry: The given polynomial \( y = 1 + x^2 - \frac{1}{2}x^4 \) is an even function, since all exponents of \( x \) are even. This suggests it is symmetric around the y-axis.
  • Identify intercepts: Find where the curve intersects the axes by setting \( x = 0 \) to find the y-intercept, and \( y = 0 \) to solve for x-intercepts.
  • Determine end behavior: Analyze the leading term \( -\frac{1}{2}x^4 \) to see how the function behaves as \( x \) approaches infinity or negative infinity.
  • Find critical points: Derive the function, set the derivative to zero, and solve for \( x \) to locate turning points and analyze concavity.
Using these techniques, you can sketch more detailed, insightful curves that show a deeper understanding of the polynomial function.
Polynomial Equations
Polynomial equations form the backbone of many algebraic and calculus problems. A polynomial like \( y = 1 + x^2 - \frac{1}{2}x^4 \) showcases important characteristics:
  • Order: This is a fourth-degree polynomial because the highest power of \( x \) is four.
  • Coefficients: The numbers in front of \( x \)'s powers, such as 1, 1, and \(-\frac{1}{2}\), influence the shape and orientation of the graph.
  • Zeros: Solving the equation \( 1 + x^2 - \frac{1}{2}x^4 = 0 \) can help identify the points where the curve intersects the x-axis, providing key insights into the nature of the polynomial.
Understanding polynomial equations helps in predicting the behavior of complex systems not just mathematically, but also in practical, real-world scenarios. This comprehension extends to recognizing patterns in data and modeling phenomena, making polynomial equations a versatile mathematical tool.

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