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Gr10 step function ppt

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• 1. Step functions
• The graph of a step function is not linear – it consists of horizontal line segments, usually having a closed circle on one end and an open circle on the other end
• The graph generally resembles a series of steps, hence the name
• 2. GREATEST INTEGER FUNCTION
• One prominent example of a step function is the greatest integer function
• This function is written f(x) =
• The symbol represents the greatest integer LESS THAN OR EQUAL to x
• For example, [ 5.2 ] = 5, [2.9] = 2, [1] = 1
• Note that integers stay the same, but numbers with a decimal portion are rounded down
• [-3.4] = -4
• 3. Greatest Integer Function – the most notable step function Step Functions – functions whose graphs resemble sets of stair steps. Notation of Greatest Integer Function : Meaning of Greatest Integer Function : the greatest integer less than or equal to x. x
• 4. Let’s evaluate some greatest integers… 2 ? 9 ? -3 ? the greatest integer less than or equal to x. 2 = 9 = -3 =
• 5. Let’s evaluate some greatest integers… 2 ? 0 ? -5 ? the greatest integer less than or equal to x. 2.2 = 1 / 2 = -4.1 = 0 1 2 3 -1 -2 -3 -4 -5
• 6. Let’s evaluate some greatest integers… 9 ? 5 ? -3 ? the greatest integer less than or equal to x. 9.1 = 5 1 / 3 = -2 2 / 9 = 0 1 2 3 -1 -2 -3 -4 -5
• 7. F(x) = [x]
• Note that we have an open-circle on the right of each step, where the function “jumps up” to the next step
• Thus this graph does represent a function – a vertical line will not pass thru more than one point!
• Note also that while the domain is all real numbers, the range is limited to integer values!
• 8. Now that you know how to evaluate greatest integer functions, you can graph them. x y 0 0 .25 0 .5 0 .75 0 1 1 1.25 1 1.5 1 1.75 1 2 2 y = x
• 9. Now that you know how to evaluate greatest integer functions, you can graph them. y = x - 4 x y 0 .25 .5 .75 1 1.25 1.5 1.75 2
• 10. Least Integer functions
• We can also have step functions where you basically round up the number
• These are defined as least integer functions – the function’s output is the least integer that is greater than or equal to the input
• Some word problems involve this type of function, such as the ones on the following slides – let’s take a look!
• 11. Example 6-1a Psychology One psychologist charges for counseling sessions at the rate of \$85 per hour or any fraction thereof. Draw a graph that represents this solution. Explore The total charge must be a multiple of \$85, so the graph will be the graph of a step function. Plan If the session is greater than 0 hours, but less than or equal to 1 hour, the cost is \$85. If the time is greater than 1 hour, but less than or equal to 2 hours, then the cost is \$170, and so on .
• 12. Example 6-1b Solve Use the pattern of times and costs to make a table, where x is the number of hours of the session and C ( x ) is the total cost. Then draw the graph. 425 340 255 170 85 C(x) x
• 13. Example 6-1c Examine Since the psychologist rounds any fraction of an hour up to the next whole number, each segment on the graph has a circle at the left endpoint and a dot at the right endpoint. Answer:
• 14. Example 6-1d Sales The Daily Grind charges \$1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation. Answer: