1.4 Polynomial functions and rates of change
State the degree of the following terms:
8
- 3 y^{3}
- 4^{5} y^{4}
State the numerical coefficient of:
Consider the polynomial P \left( x \right) = 9.
State the degree.
State the constant term.
For the following polynomials:
State the degree.
State the leading coefficient.
State the constant term.
Describe the end behavior.
Describe the rise and fall of the graph.
P \left( x \right) = 3 x^{4} + 5 x^{2} + 4 x + 3
P \left( x \right) = 2 x^{5} - 3 x^{4} + 6 x^3 - x^2 + 20x-12
P \left( x \right) = 3 - \dfrac{6}{7} x^{6}
P \left( x \right) = \dfrac{x^{6}}{2} + \dfrac{x^{5}}{10} + 8
Given the polynomial function: g(x) = 3x^4 - 8x^3 + 4x^2 + 2x - 1
State the type of function each term in the polynomial function represents.
Identify the leading term, leading coefficient, and the degree of the polynomial. Justify your answer.
Explain the implications of the leading term, leading coefficient, and the degree of the polynomial to the other terms of the function.
A construction company is planning to build a road. The cost of constructing the road is modeled by the polynomial function C(x) = 2x^3 - 5x^2 + 3x + 10, where x represents the length of the road in kilometers.
Identify the degree and leading coefficient of this polynomial function.
Discuss implications of the degree and leading coefficient in terms of the cost of construction.
Consider the polynomial function f(x) = 2x^{4} - 3x^{2} + 5x - 1.
Determine the degree of the polynomial function.
Discuss the implications of the degree of the polynomial function in terms of its behavior.
State the number of solutions it may have.
Explain how the degree of a polynomial function can affect the end behavior of its graph.
Consider the function y = x^{6} - x^{2}.
Determine the leading coefficient of the polynomial function.
Is the degree of the polynomial odd or even?
Does y = x^{6} - x^{2} rise or fall to the left?
Does y = x^{6} - x^{2} rise or fall to the right?
Find the zeros of the function.
Sketch the graph of y = x^{6} - x^{2}.
Identify the approximate coordinates of the local and global maxima and minima of this polynomial function.
Consider the polynomial function x^4 - 8x^3 + 18x^2 - 16x.
Sketch the graph of the function.
Determine the approximate zeros of the polynomial function.
Determine the approximate local and global maxima and minima of the polynomial function.
Using this function, prove why there must be a local maximum or minimum between every two distinct real zeros of a nonconstant polynomial function.
Identify the intervals on which the function has local maxima or minima, and provide a brief explanation for your answer.
Consider the following polynomial function f(x)=-2x^3+x^2+2x-1.
Determine the degree and leading coefficient of the polynomial function and explain their significance to the behavior of the graph.
Sketch the graph of the function.
Identify the x-intercepts (zeros) of the function and interpret their meaning in the context of the problem.
Find any approximate maxima or minima of the function and explain their significance in relation to the graph.
Determine if there are any points of inflection in the graph of the function and discuss their implications.
Describe the end behavior of the graph as x approaches positive and negative infinity.
For the following graphs of polynomial functions:
Estimate the local maxima and minima of the function and identify their approximate x-values.
Estimate the zeros of the function and provide their approximate x-values.
Identify any points of inflection on the graph and explain their significance in terms of the function's behavior.
Determine if there is a global maximum or minimum on the graph and justify your answer.
Describe the end behavior of the graph as x approaches positive and negative infinity.
Consider the following graph:
Determine the points of inflection for the given polynomial function.
Interpret the points of inflection in terms of the behavior of this function.
Describe how the concavity of the function changes at the points of inflection.
Discuss any potential changes in the rate of change of the function at the points of inflection.
A farmer is growing crops and has modeled the yield of a particular crop using the polynomial function Y(x) = -2x^3 + 5x^2 + 10x - 20, where x represents the amount of fertilizer applied.
Identify the coordinates of the local maxima and minima of this polynomial function.
Identify the global maxima and minima of this polynomial function.
Explain the significance of the local maxima and minima in terms of the crop yield.
Explain the significance of the global maxima and minima in terms of the crop yield.
Consider the polynomial function f(x) = -2x^{4} + 5x^{2} - 3.
Determine the degree of the polynomial function.
Based on the degree of the polynomial function, predict whether there exists a global maximum or minimum. Justify your prediction.
Explain the relationship between the degree of a polynomial function and the existence of a global maximum or minimum.
Provide a brief explanation of why polynomial functions of an even degree may have a global maximum or minimum.
Given the polynomial function: f(x) = -x^4+6 x^2-8
Identify the leading term, leading coefficient, and the degree of the polynomial. Justify your answer.
Explain the implications of the leading term, leading coefficient, and the degree of the polynomial to the behavior of the function.
Explain the implications of the leading term, leading coefficient, and the degree of the polynomial to the behavior of the function and the graph.
Manipulate the function to find the possible zeros of the function.
Sketch a graph of the polynomial function, identifying key features such as maxima, minima, and points of inflection.
A construction company is planning to build a roller coaster. The shape of the roller coaster can be modeled by a polynomial function f(x) = -2x^3 + 12x^2 - 16x + 8, where x represents the distance from the starting point of the roller coaster (in meters) and f(x) represents the height of the roller coaster at that distance (in meters).
Determine the degree of the polynomial function and explain what it represents in the context of the roller coaster.
Sketch the graph of the function.
Identify the potential locations of maxima and minima on the roller coaster and interpret their significance.
Find the points of inflection, if any, on the roller coaster and discuss their implications for its shape.
Based on the polynomial function, analyze the overall shape of the roller coaster and discuss any key features or characteristics.
The height of a projectile launched into the air can be modeled by the polynomial function h(t) = -5t^2 + 20t + 10, where h(t) represents the height in meters and t represents the time in seconds.
Sketch the graph of the function.
Determine the maximum height reached by the projectile and find the corresponding time of flight.
Estimate the positive zeros of the function and interpret them in the context of the projectile's motion.
Determine if there are any points of inflection in the trajectory of the projectile and explain their significance.
Manipulate the polynomial function to find the vertex form of the function, and interpret the vertex in terms of the maximum or minimum point of the projectile's trajectory.
Consider the polynomial function: f(x) = -2(x - 3)^2(x + 2)^3
Explain what each term in the polynomial function represents, and discuss the leading term, leading coefficient, and the degree of the polynomial.
Identify the degree of the polynomial function and discuss its implications.
Determine the zeros of the polynomial function and explain the relationship between the zeros and the existence of local maxima and minima.
Predict the existence of a global maximum or minimum for the polynomial function and explain your reasoning.
Describe the possible graph of the polynomial function, identifying key features such as maxima, minima, and points of inflection.
A polynomial function models the total amount of carbon dioxide emissions in a city over a 10-year period. The function is given by C(t) = 3t^3 - 12t^2 -9t +54, where t represents the number of years since the start of the period and C(t) is the total amount of carbon emissions in thousand tons.
Determine the degree of the polynomial function and discuss its implications for the carbon emissions.
Sketch a graph of the function, identifying key features such as approximate maxima, minima, and points of inflection.
Discuss the significance of any local maxima or minima and points of inflection in the context of the carbon emissions.
Estimate the number of years after which the carbon emissions will be at its lowest and explain your reasoning.
Predict the carbon emission after 10 years.
The viewers of an online streamer during its first five months in the platform can be modeled by the polynomial function: P(x) = 0.75x^3-5x^2+9x+1, where x represents the number of months since the streaming was started and P(x) represents the number of viewers in thousands.
Determine the degree of the polynomial function and discuss its implications for the company's profit.
Sketch a graph of the profit function, identifying key features such as the approximate maxima, minima, and points of inflection.
Discuss the significance of any local maxima or minima and points of inflection in the context of the streamer's number of viewers.
Given the polynomial function, estimate when the streamer will reach its highest viewers within the first five months and explain your reasoning.