Please Complete

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• Please Complete
• 1) Find f(-100), f(-10), f(0), f (1/2), f(10),
  and f(100) when f(x) 1/x.
• (hint substitute each number for x)
• 2) What happens to f(x) as x gets bigger?

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Lets check our work
1) f(-100) 1 / -100 -0.01 f(-10) 1 /
-10 -0.1 f(0) 1 / 0 Error, no such
number. f(1/2) 1 / (1/2) 1 (2/1) 2
f(10) 1 / 10 0.1 f(100) 1 / 100
0.01 2) What happens to f(x) as x gets bigger?
As x gets bigger, f(x) is getting smaller.
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Lesson 1 Characteristics of Functions

• Exploring the graphs of basic functions Rational

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• Name
• Date
• Math 1A/ Block?
• Unit 1 Lesson 1 Characteristics of Functions
• Essential Question- What is a function?
• Key Question- What are the characteristics of a
  rational function?

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Parent Functions? Linear f(x)x Quadratic f(x)x2 Absolute Value f(x) x Cubic f(x)x3 Square Root Rational
1. What does the graph look like? 2. What does the equation look like? 3. What are the x and y-intercepts? 4. What is the domain and range? 5. What is the end behavior? 6. Intervals of Increase or Decrease
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Parent Functions? Linear f(x)x Quadratic f(x)x2 Absolute Value f(x) x Cubic f(x)x3 Square Root Rational
1. What does the graph look like? 2. What does the equation look like? 3. What are the x and y-intercepts? 4. What is the domain and range? 5. What is the end behavior? 6. Intervals of Increase or Decrease Line f(x) x The highest exponent (degree) 1 (0,0) Domain (-8, 8) Range (- 8, 8) Left - 8 Right 8 Increasing (- 8, 8) U-shaped Graph called a parabola f(x)x2 The highest exponent (degree) 2 (0,0) Domain (-8, 8) Range 0, 8) Left 8 Right 8 Decreasing (- 8, 0) Increasing (0, 8) V-shaped Graph f(x) x The highest exponent (degree) 1 but it has an absolute value bar (0,0) Domain (-8, 8) Range 0, 8) Left 8 Right 8 Decreasing (- 8, 0) Increasing (0, 8) A Water-slide f(x)x3 The highest exponent (degree) 3 (0,0) Domain (-8, 8) Range (- 8, 8) Left - 8 Right 8 Increasing (- 8, 8) Side-Ways Half a U The highest exponent (degree) 1/2 or a radical sign (0,0) Domain 0, 8) Range 0, 8) Left 0 Right 8 Increasing (0, 8)
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What are the Characteristics of the Rational
Function?

• What does the graph look like?
• What does the equation look like?
• What are the x and y-intercepts?
• What is the domain and range?
• What is the end behavior?

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Rational Functions

• Parent equation

Complete the table of values on your notes and
then plot the points. Do not attempt to connect
the dots yet
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Rational Functionsf(x) 1/x
x f(x)
-4 -1/4
-3 -1/3
-2 -1/2
-1 -1
-3/4 -4/3
-1/2 -2
-1/3 -3
-1/4 -4
x f(x)
0 undefined
¼ 4
1/3 3
½ 2
¾ 4/3
1 1
2 ½
3 1/3
4 1/4
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Rational Functionsf(x) 1/x

• Youve plotted your points, now lets see what
  this function looks like

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What is the domain?
Domain (-8,0) and (0,8)
Why is there a parenthesis () by the zero and not
a bracket ?
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Lets take a closer look

• We are going to look at the same function, but
  instead of showing only 6 points on the graph,
  were showing 20.
• Make a prediction Will this graph ever touch
  the y-axis? In other words, can x ever be zero???

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It looks as if it touches zero, but it actually
does not Lets magnify this graph and take a
closer look?
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• We can see that the line does not actually touch
  the x-axis. X will never be zero.
• This is called an asymptote.
• We would say that the function has a vertical
  asymptote at x0.

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What is the range?
Range (-8,0) and (0,8)
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Rational Functions

• Again, if we magnify the graph and look at it
  when x 20, we can see that the function never
  touches the x-axis.

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• Y will never be zero. Look at your parent
  equation
• Why can 1/x ever equal zero???
• We would say that the function has a horizontal
  asymptote at y0.

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What are the x-intercepts (zeros)?
There are no x-intercepts. The function never
crosses the x- axis.
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What is the y-intercept?
There are no y-intercepts because the graph never
touches the y-axis.
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What is the maximum and minimum?
There is no maximum or minimum. The graph
continues on to infinity in both the positive and
negative y directions.
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Is this function increasing or decreasing or both?
Remember how to tell Trace it with your finger.
This function is decreasing.
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Another example of a rational function

• What does the graph of look like?

What is the domain and range?
Is it increasing or decreasing or both?
Does is have any asymptotes?
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Review
What function family is this?
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Review
What function family is this?
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Review
What function family is this?
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Review
What function family is this?
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Review
What function family is this?
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Review
What function family is this?
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Review
What function family is this?
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Review
What function family is this?