Rational Functions & Solving NonLinear Inequalities Chapter 2 part 2

Definition The defining characteristic of a rational function is that there is a variable in the denomimator. Features of the graph of a rational function: ◦ y-intercept ◦ zeros (aka x-intercepts) ◦ vertical asymptotes (related to domain) ◦ horizontal or slant asymptote (aka end behavior) ◦ holes (also related to domain)

Preliminary Work Rewrite as a single fraction in both nonfactored and factored forms.

y-intercept let x = 0 every term in the nonfactored form becomes zero except the constant terms y-int = ratio of constant terms ◦ use nonfactored form

x-intercepts (aka zeros) let y or f(x) = 0 If N(x) / D(x) = 0 then N(x) = 0 and D(x) ≠ 0 therefore zeros of f(x) = zeros of N(x) find zeros of N(x) ◦ use factored form

Vertical Asymptotes occur when you attempt to divide by zero find zeros of D(x) ◦ use factored form

Horizontal Asymptote (aka end behavior) recall from polynomial functions that the leading term determined end behavior. ratio of leading coefficients ◦ use nonfactored form ◦ when degree of N(x) = degree of D(x)  May need to add a zero term to N(x) or D(x)

Summary of Features Holes: will learn later y-int: ratio of constant terms x-int: zeros of N(x) V asy: zeros of D(x) ◦ Domain: x ≠ V asy H asy: ratio of leading coefficients ◦ when degree of N(x) = degree of D(x) S asy: will learn later

Slant Asymptote occurs when degree of N(x) = 1 + degree of D(x) use polynomial division or synthetic division to find the slant asymptote. ◦ ignore any remainders

Holes occur when you attempt 0 ÷ 0 zeros common to both N(x) and D(x) ◦ use factored form ◦ need to adjust notes for x-intercepts and vertical asymptotes

Summary of Features Holes: zeros common to N(x) and D(x) y-int: ratio of constant terms x-int: zeros of N(x) not common to D(x) V asy: zeros of D(x) not common to N(x) ◦ Domain: x ≠ holes or V asy H asy: ratio of leading coefficients ◦ when degree of N(x) = degree of D(x) S asy: occurs when degree of N(x) = 1 + degree of D(x) ◦ use polynomial or synthetic division to find

Solving NonLinear Inequalities more complex than solving linear inequalities first find all critical values which are x-values where the function has the potential to switch from pos to neg or vice versa. then test the regions created by the critical values to determine if they make the inequality true or false. (there will always be one more region than critical value). solution set includes all regions and critical values that make the inequality true

Solving Polynomial Inequalities the only critical values are zeros of f(x). find zeros by p/q test, synthetic division, factoring and/or quadratic formula. when testing regions, you only need to determine the sign of the output. include critical values for ≤ or ≥. Do not include for.

Solving Rational Inequalities critical values include both zeros and vertical asymptotes of f(x) find zeros of both N(x) and D(x) when testing regions, you only need to determine the sign of the output Include critical values which are ◦ zeros for ≤ or ≥ Do not include critical values which are ◦ zeros for ◦ vertical asymptotes