Summarize the Rational Function Task Rational Functions Summarize the Rational Function Task Holt McDougal Algebra 2 Holt Algebra 2

Let’s make sure you got the conclusions form the task. Given functions f(x) = (x – a)(x + b)(x – c) and g(x) = (x – d)(x + e) where a, b, c, d, and e are positive real numbers such that a ≠ b ≠ c ≠ d ≠ e, Where are the roots of the function d and -e

Given functions f(x) = (x – a)(x + b)(x – c) and g(x) = (x – d)(x + e) where a, b, c, d, and e are positive real numbers such that a ≠ b ≠ c ≠ d ≠ e, b) Where are the vertical asymptotes of the function a, -b and c

Given functions f(x) = (x – a)(x + b)(x – c) and g(x) = (x – d)(x + e) where a, b, c, d, and e are positive real numbers such that a ≠ b ≠ c ≠ d ≠ e, c) Where are the roots of the function d and -e

Given functions f(x) = (x – a)(x + b)(x – c) and g(x) = (x – d)(x + e) where a, b, c, d, and e are positive real numbers such that a ≠ b ≠ c ≠ d ≠ e, d) Where are the vertical asymptotes of the function a, -b and c

d) Where are the vertical asymptotes of the function Given functions f(x) = (x – a)(x + b)(x – c) and g(x) = (x – d)(x + e) where a, b, c, d, and e are positive real numbers such that a ≠ b ≠ c ≠ d ≠ e, d) Where are the vertical asymptotes of the function d and -e

What type of functions were f(x) and g(x)? Polynomial Functions

3. Give a definition of a rational function? A rational function is a function whose rule can be written as a ratio of two polynomials.

For any simplified rational function, what information can you obtain from the numerator? If you set the numerator = to 0, you determine the x-intercepts (zeros, solutions, roots).

5. For any simplified rational function, what information can you obtain from the denominator? If you set the denominator = to 0, you determine the vertical asymptotes.

Summarize one more time….. Notice it says with no common factors!! Factoring is ALWAYS the first step.

Example 3: Graphing Rational Functions with Vertical Asymptotes Identify the zeros and vertical asymptotes of f(x) = . (x2 + 3x – 4) x + 3 Step 1 Factor!! Then find the zeros & vertical asymptotes. Factor the numerator. (x + 4)(x – 1) x + 3 f(x) = The numerator is 0 when x = –4 or x = 1. Zeros: –4 and 1 The denominator is 0 when x = –3. Vertical asymptote: x = –3

Vertical asymptote: x = –3 Example 3 Continued Identify the zeros and vertical asymptotes of f(x) = . (x2 + 3x – 4) x + 3 Vertical asymptote: x = –3 Step 2 Graph the function. Plot the zeros and draw the asymptote. Then make a table of values to fill in missing points. x –8 –4 –3.5 –2.5 1 4 y –7.2 4.5 –10.5 –1.3 3.4

Check It Out! Example 3 Identify the zeros and vertical asymptotes of f(x) = . (x2 + 7x + 6) x + 3 Step 1 Facotr!! Find the zeros & vertical asymptotes. (x + 6)(x + 1) x + 3 f(x) = Factor the numerator. The numerator is 0 when x = –6 or x = –1 . Zeros: –6 and –1 The denominator is 0 when x = –3. Vertical asymptote: x = –3

Check It Out! Example 3 Continued Identify the zeros and vertical asymptotes of f(x) = . (x2 + 7x + 6) x + 3 Step 2 Graph the function. Plot the zeros and draw the asymptote. Then make a table of values to fill in missing points. Vertical asymptote: x = –3 x –7 –5 –2 –1 2 3 7 y –1.5 –4 4.8 6 10.4

Cwk/Hwk Worksheet 1-9