📝 Chapter 1: Polynomial Functions

State the degree, leading coefficient, and constant term of \( f(x) = -4x^5 + 7x^3 - 2x + 9 \).

Describe the end behaviour of \( g(x) = 3x^6 - x^4 + 2 \).

A table has constant 3rd differences of 12. The degree is:

The y-intercept of \( f(x) = -(x+3)(x-1)^2(x-5) \) is:

At \( x = 1 \) in \( f(x) = -(x+3)(x-1)^2(x-5) \), the graph:

Is \( f(x) = 2x^4 - x^2 \) even, odd, or neither?

The average rate of change of \( f(x) = x^2 - 4x + 1 \) on \([1, 3]\) is:

A degree 5 polynomial has: negative leading coefficient, zeros at \( x = -2 \) (order 2), \( x = 0 \) (order 1), \( x = 3 \) (order 2), and \( f(1) = 12 \). Find the equation.

Without graphing technology, factor \( g(x) = x^4 - 8x^2 + 12 \) completely and determine the number of x-intercepts. Show all reasoning.

A quartic function has turning points at \( x = -2, 1, 4 \) and is positive for all \( x \). Is this possible? Explain your reasoning with a sketch if applicable.

A polynomial passes through \( (0,0), (2,0), (5,0) \) with degree 4 and positive leading coefficient. Between which pairs of zeros must there be a turning point? Explain.

A student claims: "A polynomial of degree 4 always has exactly 3 turning points." Is this correct? Provide a clear explanation with an example and counterexample. Write the correct statement.

Explain the difference between the graph at a zero of order 1, order 2, and order 3. Include a sketch for each and describe the behaviour.

Describe a step-by-step method for sketching a polynomial given in factored form. Cover: intercepts, end behaviour, multiplicity, and additional points.

A student calculated the average rate of change of \( f(x)=x^2 \) on \([1,4]\) as 5 and said the instantaneous rate at \( x=2.5 \) is also 5. Is the ARoC correct? Is the reasoning about IRoC correct? Estimate the actual IRoC at \( x=2.5 \).

A box is made by cutting squares of side \( x \) from each corner of a 20cm × 30cm cardboard. Write \( V(x) \), state the domain in context, and determine the degree.

Bacteria population: \( P(t) = -0.5t^4 + 8t^3 - 36t^2 + 200 \). Find: a) initial population, b) ARoC from \( t=0 \) to \( t=4 \), c) interpret in context.

Roller coaster: \( h(x) = -0.001(x+10)^2(x-20)(x-50)^2 \). Where is it at ground level? Where does it touch and return vs pass through?

Profit: \( P(x) = -2x^3 + 15x^2 - 24x \). For what production levels is profit positive? Estimate the ARoC on \([1,3]\). Should the company produce 4000 items?