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Answer :
To solve the question related to the polynomial function [tex]\( f(x) = 10x^6 + 7x - 7 \)[/tex] being graphed, we need to consider a few possible aspects:
1. Identifying the Degree and Leading Term:
- The given polynomial [tex]\( f(x) = 10x^6 + 7x - 7 \)[/tex] is a degree 6 polynomial.
- The leading term is [tex]\( 10x^6 \)[/tex], which indicates the end behavior of the graph. Since the coefficient of the highest degree term is positive, as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex] and as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
2. Roots and Intercepts:
- To determine the roots or intercepts, you would typically set the function equal to zero: [tex]\( 10x^6 + 7x - 7 = 0 \)[/tex].
- Solving for [tex]\( x \)[/tex] would require numerical methods or graphing technology, as this is not easily factorable by hand. In a graph, intercepts are points where the curve crosses the x-axis.
3. Y-Intercept:
- The y-intercept of a polynomial function is found by evaluating the function at [tex]\( x = 0 \)[/tex].
- [tex]\( f(0) = 10(0)^6 + 7(0) - 7 = -7 \)[/tex]. So, the y-intercept is at the point (0, -7).
4. Behavior Near the Y-Intercept:
- Around the y-intercept, you might consider how the function behaves by looking at smaller or larger values of x. However, for this polynomial, what's significant is how it approaches y = -7 at x = 0.
5. Symmetry:
- This function is not symmetric, as the terms do not allow it to exhibit even or odd functions' symmetry properties.
6. End Behavior and Turning Points:
- With a degree 6 polynomial, the graph could potentially have up to 5 turning points. The specific nature and location of these points would need graphical or numerical analysis.
The graph would show how the polynomial behaves as it moves through these intercepts, turning points, and tails off to infinity in the specified directions. If additional context or specific instructions were needed, you might consider exploring graphing calculators or software to visualize these aspects more thoroughly.
1. Identifying the Degree and Leading Term:
- The given polynomial [tex]\( f(x) = 10x^6 + 7x - 7 \)[/tex] is a degree 6 polynomial.
- The leading term is [tex]\( 10x^6 \)[/tex], which indicates the end behavior of the graph. Since the coefficient of the highest degree term is positive, as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex] and as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
2. Roots and Intercepts:
- To determine the roots or intercepts, you would typically set the function equal to zero: [tex]\( 10x^6 + 7x - 7 = 0 \)[/tex].
- Solving for [tex]\( x \)[/tex] would require numerical methods or graphing technology, as this is not easily factorable by hand. In a graph, intercepts are points where the curve crosses the x-axis.
3. Y-Intercept:
- The y-intercept of a polynomial function is found by evaluating the function at [tex]\( x = 0 \)[/tex].
- [tex]\( f(0) = 10(0)^6 + 7(0) - 7 = -7 \)[/tex]. So, the y-intercept is at the point (0, -7).
4. Behavior Near the Y-Intercept:
- Around the y-intercept, you might consider how the function behaves by looking at smaller or larger values of x. However, for this polynomial, what's significant is how it approaches y = -7 at x = 0.
5. Symmetry:
- This function is not symmetric, as the terms do not allow it to exhibit even or odd functions' symmetry properties.
6. End Behavior and Turning Points:
- With a degree 6 polynomial, the graph could potentially have up to 5 turning points. The specific nature and location of these points would need graphical or numerical analysis.
The graph would show how the polynomial behaves as it moves through these intercepts, turning points, and tails off to infinity in the specified directions. If additional context or specific instructions were needed, you might consider exploring graphing calculators or software to visualize these aspects more thoroughly.
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