Warm Up Graph the function

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Presentation transcript:

Warm Up Graph the function 𝑓 𝑥 =− 𝑥+3 2 +4

Graphing Rational Functions We have graphed several functions, now we are adding one more to the list! Graphing Rational Functions

Parent Function: 𝒇 𝒙 = 𝟏 𝒙

Pay attention to the transformation clues! f(x) = + k a x – h (-a indicates a reflection in the x-axis) vertical translation (-k = down, +k = up) horizontal translation (+h = left, -h = right) Watch the negative sign!! If h = -2 it will appear as x + 2.

Asymptotes Places on the graph the function will approach, but will never touch.

Vertical Asymptote: x = 0 Horizontal Asymptote: y = 0 f(x) = 1 x Graph: Vertical Asymptote: x = 0 Horizontal Asymptote: y = 0 No horizontal shift. No vertical shift. A HYPERBOLA!!

W𝐡𝐚𝐭 𝐝𝐨𝐞𝐬 𝒇 𝒙 =− 𝟏 𝒙 look like?

Vertical Asymptote: x = -4 Horizontal Asymptote: y = 0 Graph: f(x) = 1 x + 4 x + 4 indicates a shift 4 units left Vertical Asymptote: x = -4 No vertical shift Horizontal Asymptote: y = 0

1 Graph: f(x) = – 3 x + 4 x + 4 indicates a shift 4 units left Vertical Asymptote: x = -4 –3 indicates a shift 3 units down which becomes the new horizontal asymptote y = -3. Horizontal Asymptote: y = 0

x Graph: f(x) = + 6 x – 1 x – 1 indicates a shift 1 unit right Vertical Asymptote: x = 1 +6 indicates a shift 6 units up moving the horizontal asymptote to y = 6 Horizontal Asymptote: y = 1

You try!! 1. 𝑦= 1 𝑥 +2 2. 𝑦= 1 𝑥+3 −4

How do we find asymptotes based on an equation only?

Vertical Asymptotes (easy one) Set the denominator equal to zero and solve for x. Example: 𝑦= 6 𝑥−3 x-3=0 x=3 So: 3 is a vertical asymptote.

Horizontal Asymptotes (H.A) In order to have a horizontal asymptote, the degree of the denominator must be the same, or greater than the degree in the numerator. Examples: 𝑦= 𝑥 2 −3 𝑥+7 No H.A because 2>1 𝑦= 𝑥 3 −2 𝑥 3 −2 Has a H.A because 3=3. 𝑦= 𝑥+1 𝑥 2 Has a H.A because 1<2

3 cases

If the degree of the denominator is GREATER than the numerator. The Asymptote is y=0 ( the x-axis)

If the degree of the denominator and numerator are the same: Divide the leading coefficient of the numerator by the leading coefficient of the denominator in order to find the horizontal asymptote. Example: 𝑦= 6𝑥 3 3𝑥 3 −2 Asymptote is 6/3 =2.

If there is a Vertical Shift The asymptote will be the same number as the vertical shift. (think about why this is based on the examples we did with graphs) Example: 5 𝑥−3 +7 Vertical shift is 7, so H.A is at 7.

Practice