8-2: Reciprocal Function

What does domain mean? Are there any other numbers in the domain? in the domain? Are we missing any other numbers any other numbers that are in the domain? that are in the domain? Domain the numbers on the x-axis that are covered the x-axis that are covered by a “shadow” when you by a “shadow” when you shine a flashlight from the shine a flashlight from the top or the bottom. top or the bottom. Domain: all real numbers. real numbers.

Finding the domain and range from a graph (half circle). (half circle). What is the maximum y-value? What is the minimum x-value ? What is the minimum y-value?  range: from ~3 to 6 ~2.5 ~ What is the maximum x-value ?  domain: -2 to ≤ x ≤ 2 -2 ≤ x ≤ 2 3 ≤ y ≤ 6 3 ≤ y ≤ 6

Your turn: 1. What the domain of this relation

Will this result in a real number? If you are trying to find the time when the ball reaches its maximum height after throwing it, does negative time make any sense?

What will “restrict the domain”? Your turn: Fill in the blanks to list the conditions that will restrict the domain division by zero. even root of a negative #. even root of a negative #. Doesn’t make sense in the real world (negative time, negative length, putting (negative time, negative length, putting more gas in the gas tank that it can hold, etc.) more gas in the gas tank that it can hold, etc.) If the input value results in ____, _____, or _______

“Up against the wall (please)” Get up. Walk to the nearest wall. Stand about 3 feet away from the wall. Face the wall. Follow my directions.

Vocabulary Asymptote: A vertical or horizontal line that the graph approaches but NEVER reaches. Asymptotes are not part of the graph but you can see them easily.

x = 0 x = 0 Parent function: the simplest function in a family of functions. y = 0 y = 0 What value is not part of the domain? What value is not part of the range?

x = 0 x = 0 Your turn: 5. What will happen to the reciprocal function if you add 3 to it? y = 3 y = 3 The new horizontal asymptote is: y = 3 asymptote is: y = 3 What value is not part of the domain? What value is not part of the range? x = 0 x = 0 y = 3 y = 3

x = 0 x = 0 Your turn: 6. What will happen to the reciprocal function if you subtract 2 from it? y = -2 y = What is the new horizontal asymptote? 8. What is the domain and range of the new function? of the new function?

x = 4 x = 4 Your turn: 9. What will happen to the reciprocal function if you replace ‘x’ with ‘x – 4’ ? y = 3 y = 3 The new vertical asymptote is: x = 4

The “Reciprocal” function Parent Function

The “Reciprocal” function Parent Function

Your turn: 10. What is the effect of multiplying the “reciprocal” function with a number that is greater than 1? function with a number that is greater than 1? (Parent Function)

Your turn: How is the graph of the parent function is transformed by each of the following equations? is transformed by each of the following equations? (hint describe how it changes by how the horizontal (hint describe how it changes by how the horizontal and vertical asymptotes are changed) and vertical asymptotes are changed)

x = 1 x = 1 General Form: y = 3 y = 3 Vertical asymptote: x = h Horizontal asymptote: y = k Vertical asymptote: x = 1 Horizontal asymptote: y = 3 Vertical asymptote: the value of ‘x’ that makes the denominator = zero.

Your turn: What are the horizontal and vertical asymptotes for each of the following reciprocal functions? of the following reciprocal functions? Asymptotes are lines. I want the equation of the line. y = 0.5 y = 0.5 y = -7 y = -7 x = 0 x = 0 x = 2 x = 2 y = 0 y = 0 x = -6 x = -6

x = 0, y = 1 x = 0, y = 1 Your turn: What are the horizontal and vertical asymptotes? x = -1, y = 0 x = -1, y = 0 x = 4, y = 3 x = 4, y = 3

End here.

Vertical Asymptotes We graph these as dotted lines (not part of the actual graph) (not part of the actual graph) f(x) will approach but will not cross these vertical dotted lines. cross these vertical dotted lines. Vertical asymptote: the value of ‘x’ that makes the denominator = zero.

What is a rational Number? A ratio of integers.

What is a rational Function? A ratio of functions. 17. From the given functions f(x) and g(x), build the function f(x). Your turn: 18. From the given functions f(x) and g(x), build the function k(x).

Rational Functions Graph this on your calculator. Can you tell what the vertical asymptote is? asymptote is? Any value of x that makes the denominator = zero is a vertical asymptote. is a vertical asymptote. What is the vertical asymptote?

Your turn: Write the equation of each asymptote

“Zeros” of a function y = 0 x = -1 x = -1 What does a “zero of a function” mean? x = +2 x = +2 Vocabulary: Zero of a function: the input values (x) that makes the output values (y) equal to zero. output values (y) equal to zero. -1 and 2 are the “zeroes” of the function above -1 and 2 are the “zeroes” of the function above

Zeroes of Rational Funtions ‘x’ = ? so that f(x) = 0 ? ‘x’ = ? so that f(x) = 0 ? What number in the denominator can make a fraction equal zero? can make a fraction equal zero? ‘x’ = ? so that f(x) = 0 ? ‘x’ = ? so that f(x) = 0 ? NO number!

Zeroes of Rational Funtions Only the numerator can cause f(x) to = “0” ‘x’ = ? so that f(x) = 0 ? ‘x’ = ? so that f(x) = 0 ? x = 0 x = 0

Your Turn: 24. What is the “zero” of this function? 25. What are the vertical asymptotes?

Review for Rational Functions 1. Vertical asymptotes: values of ‘x’ that make the denominator = 0. that make the denominator = Vertical asymptotes: are “zeroes” of the denominator. the denominator. 3. x-intercepts: are values of ‘x’ that make the numerator = 0. that make the numerator = x-intercepts: are “zeroes” of the numerator. the numerator.

Rational Function: General Form Remember Polynomials ?: degree and lead coefficient determined “end behavior” coefficient determined “end behavior” Rational Functions: Degree and lead coefficients of both numerator and denominator functions determine “end behavior.” They also determine horizontal asymptotes.

Horizontal Asymptotes. m < n y = 0 (horizontal line) m = n y = m > n No horizontal asymptotes. “End Behavior” same as: y = (horizontal line) m: Numerator degree, n: denominator degree.

Finding vertical asymptotes m < n : y = 0 m = n : y = 1. x-intercepts: zeroes of numerator 2. vertical asymptotes: zeroes of denominator 3. horizontal asymptotes: m > n : y = 4. End behavior:

Steps for graphing Rational Functions 1. Find the x-intercepts. 2. Find the vertical asymptotes. What are the “zeroes” of the denominator? 3. Find the horizontal asymptotes (if any). Use the table that compares ‘m’ to ‘n’. What are the “zeroes” of the numerator? 4. Identify the end behavior. Use the table that compares ‘m’ to ‘n’ OR Graph it.

Your Turn: Graph 9. What are the vertical asymptotes (if any) ? 10. What are the horizontal asymptotes (if any) ? 11. What are the “zeroes” of the function (if any) ? 12. If a horizontal asymptote exists, what is the end behavior?

Graph the Function 2. Vertical asymptote:  denominator cannot equal zero. Vertical asymptote:  x = 1 3. Horizontal asymptote:  compare top/bottom degrees Horizontal asymptote:  m = 2, n = 1  m > n  no horizontal asymptote 1. x-intercepts:  Factor numerator x = -2, End behavior:  m > n End behavior same as: End behavior same as:  +,odd

X intercepts: x = -2, -3 Vertical asymptotes: y = 1 Horizontal Asymptotes: none End Behavior: right—rises, left—falls

Remember “End Behavior” Odd Power Positive leading coefficient Odd Power Negative leading coefficient x x y y

Remember “End Behavior” Even Power Positive leading coefficient Even Power Negative leading coefficient x y x y

What happens to f(x) as x  0 (approaches ‘0’) from the “left” ? xf(x) As x  0 from left

What happens to f(x) as 0  x (approaches ‘0’) from the “right” ? xf(x) As x  0 from right