Chapter 5

What is true of the appearance of graphs that refl ct a direct variation between two variables?

What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

Describe a use for the Remainder Theorem.

Explain why we cannot find inverse functions for all polynomial functions.

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

Explain the difference between the coeffici\(\mathrm{t}\) of a power function and its degree.

What is the difference between an \(x\) -intercept and zero of a polynomial function \(f\) ?

Explain the advantage of writing a quadratic function in standard form.

Explain the difference between the coefficient of a power function and its degree.

If two variables vary inversely, what will an equation representing their relationship look like?

What is the fundamental difference in the graphs of polynomial functions and rational functions?

Why must we restrict the domain of a quadratic function when fi ding its inverse?

Explain why the Rational Zero Theorem does not guarantee fi ding zeros of a polynomial function.

If a polynomial of degree \(n\) is divided by a binomial of degree 1 , what is the degree of the quotient?

If a polynomial function is in factored form, what would be a good fi st step in order to determine the degree of the function?

If a polynomial function of degree \(n\) has \(n\) distinct zeros, what do you know about the graph of the function?

Is there a limit to the number of variables that can jointly vary? Explain.

If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?

Explain why we cannot fi d inverse functions for all polynomial functions.

When finding the inverse of a radical function, what restriction will we need to make?

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(x^{2}+5 x-1\right) \div(x-1) $$

In general, explain the end behavior of a power function with odd degree if the leading coeffici\(t\) is positive.

Explain how the Intermediate Value Theorem can assist us in fi ding a zero of a function.

What is the difference between rational and real zeros?

Explain why the condition of \(a \neq 0\) is imposed in the defin tion of the quadratic function.

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as \(x\) and when \(x=6, y=12\).

Can a graph of a rational function have no vertical asymptote? If so, how?

The inverse of a quadratic function will always take what form?

If Descartes' Rule of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn?

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(2 x^{2}-9 x-5\right) \div(x-5) $$

What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

Explain how the factored form of the polynomial helps us in graphing it.

What is another name for the standard form of a quadratic function?

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the square of \(x\) and when \(x=4, y=80\).

Can a graph of a rational function have no \(x\) -intercepts? If so, how?

For the following exercises, find the inverse of the function on the given domain. $$ f(x)=(x-4)^{2},[4, \infty) $$

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(3 x^{2}+23 x+14\right) \div(x+7) $$

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As \(x \rightarrow-\infty, f(x) \rightarrow-\infty\) and as \(x \rightarrow \infty\), \(f(x) \rightarrow-\infty\)

If the graph of a polynomial just touches the \(x\) -axis and then changes direction, what can we conclude about the factored form of the polynomial?

If synthetic division reveals a zero, why should we try that value again as a possible solution?

What two algebraic methods can be used to fi d the horizontal intercepts of a quadratic function?

What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the square root of \(x\) and when \(x=36, y=24\).

For the following exercises, find the domain of the rational functions. $$ f(x)=\frac{x-1}{x+2} $$

For the following exercises, find the inverse of the function on the given domain. $$ f(x)=(x+2)^{2},[-2, \infty) $$

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(x^{4}-9 x^{2}+14\right) \div(x-2) $$

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(4 x^{2}-10 x+6\right) \div(4 x+2) $$

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$ f(x)=x^{5} $$

For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ C(t)=2(t-4)(t+1)(t-6) $$

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. $$ f(x)=x^{2}-12 x+32 $$