SUMMARY OF PROPERTY 1 PROPERTY 5

Transcription

1 SUMMARY OF PROPERTY 1 PROPERTY 5 There comes a time when putting a puzzle together that we start to see the final image. The same is true when we represent the first five (5) FUNction Summary Properties (FSP) geometrically on the same coordinate system. Although the properties found by using f do NOT tell us everything about its graph, they get us off to a good start. Since this is the case, we can obtain reasonable graphs for many functions studied in precalculus. These graphs can be fine tuned with calculus techniques. To illustrate what f tells us and does not tell us about its own graph, we use Master Example 1 and Master Example. Master Example 1: Use FUNction Summary Properties 1 5 to obtain an approximate graph 3 of f( x) = 4x + 13x + 4x 1. Solution: [[P]] We summerize FSP 1 5 below: 1. Dom f =R. The intercept points are ( 0, 1), (,0) 3. Cont f =R 3 a. Pos f =, b. Neg f = (, ), 4 4. Behavior at/toward ± : and 3,0 4. a. Lim f( x) x b. Lim f( x) x + = =+ 5. Symmetry : Neither even nor odd Putting the geometrical interpretation of these properties on the same coordinate system, we obtain FSP 1 5 Summary FSP1-5:1 004 TherapyUSA.biz

2 Using the positive/negative nature of the graph, we see that the graph must have a high point (relative maximum point) at (,0) and a low point (relative minimum point) somewhere on 3 the interval,. At this point, if you draw the best graph you can using the FSP information, you will obtain something like where your dashed portion may look slightly different. The graph above is accurate since <<GRAPHING Tool>> was used to generate it. However, even as accurate of a graph as this graph is, it does NOT accurately show the low point. We will fine-tune this graph further by finding the relative minimum point in FUNction Summary Property 7. In FUNction Summary Property 8 we will discuss its bends and in FUNction Summary Property 9, we will determine the changes in its bends. <<FUNctions R FUN!>> FSP 1 5 Summary FSP1-5: 004 TherapyUSA.biz

3 Master Example : Use FUNction Summary Properties 1 5 to obtain an approximate graph of f( x) = ( x 4) ( x+ 3) Solution: [[P]] Recall that 1. Dom f = R \{ 3} The intercept points are 0, ( 0, 0.59) 3. Cont f = R \{ 3} a. Pos f = ( 3, ) (, ) (, + ) b. Neg f = (, 3) c. The line x= 3 is a vertical asymptote. 4. Behavior at/toward ± : Lim f x = a. ( ) x b. Lim f( x) x + =+ c. The slant asymptote is y= x Symmetry : Neither even nor odd Using the information above, we construct a partial graph: , (,0) and (,0 ) Note that the portion of the graph showing the behavior at/toward infinity must be as shown due to the continuity information and the fact the this graph and its slant asymptote do NOT intersect. The graph you draw using the one above should approximate the one drawn below: FSP 1 5 Summary FSP1-5:3 004 TherapyUSA.biz

4 Another graph showing its detail around the origin is shown below: Fine-tuning this and other graphs is the purpose of FUNction Summary Properties 6 9. <<FUNctions R FUN!>> We have just seen that frequently there is more to a graph than the first five (5) FUNction Summary Properties and that calculus is required to find many of the remaining properties. However, there are times when we can find all of the FSP - even the ones we have not formally studied for certain functions considered in Precalculus. We now consider several of these functions. FSP 1 5 Summary FSP1-5:4 004 TherapyUSA.biz

5 Example 1: Analyze and sketch the graph of f( x) 3 = x+ 4. Solution: [[P]] We know from algebra that the graph of f is a straight line with slope y-intercept ( 0,4 ) : y-axis 3 and x-axis From the graph it is easy to obtain most of the FSP 1 5 information: 1. Dom f =R. Intercepts: a. y: ( 0,4 ) 8 b. x:, x+ 4= 0 x= 4 x= 3 3. Cont f =R 8 a. Pos f =, 3 8 b. Neg f =, Behavior at/toward ± : Lim f x =+ a. ( ) x b. Lim f( x) x + ; this follows from setting f ( x ) = 0 : = 5. Symmetry : Neither even nor odd FSP 1 5 Summary FSP1-5:5 004 TherapyUSA.biz

6 At this point, the reader should read the Concept section of FUNction Summary Properties 6 13, paying particular attention to the basic graphs that illustrate the various properties. This will provide the reader with an informal introduction to each property. We now discuss FUNction Summary Properties 6 13 for the function given above: 6. Increasing/Decreasing: Here we consider the graph from left to right ( L R ) a. The answer to increasing is the x-axis intervals/regions where the graph of f is going up : Inc f = φ (the empty set ; f is NEVER going up ) b. The answer to decreasing is the x-axis intervals/regions where the graph of f is going down : Dec f =R 7. Relative Maximun/Minimum Points: These are the points where the graph has high/low points, but NOT necessarily highest/lowest points. The answer to relative maximum/minimum points is just that: Points. For this function, the answer is none! 8. Concavity: Here we consider the graph from left to right ( L R ) a. The answer to concave upward is the x-axis intervals/regions where the graph of f is bent upward : CU f = (the graph does NOT bend at all!). b. The answer to concave downward is the x-axis intervals/regions where the graph of f is bent downward : CD f = (the graph does NOT bend at all!). 9. Inflection Points: These are the points where the graph changes its bend. The answer to inflection points is just that: Points. Since f doesn t bend, it does not have inflection points. 10. Graph: The graph was so easy to obtain that we essentially started our analysis with it. 11. Absolute Maximum/Minimum Points: These are the points on the graph that have the largest/smallest y-values. Since there are no relative maximum/minimum points, there cannot be any absolute ones. 1. Range: The answer to range is the projection of the graph onto the y-axis (vertical axis): Range f =R 13. Additional Information: None Example : Analyze and sketch the graph of ( ) f x = x + x 15. <<FUNctions R FUN!>> Solution: [[P]] The function f is quadratic with a= 1, b=, c= 15. We know from algebra that we can obtain an accurate graph by connecting the skeleton points : y-intercept, x-intercepts, and vertex. Let s find them so we can obtain the graph: ( 0, 0 ) = ( 0, 15) 1. y: f ( ). x: ( 5,0) ; ( 3,0 ) This is true since f( x ) = 0 implies FSP 1 5 Summary FSP1-5:6 004 TherapyUSA.biz

7 x + x 15= 0 ( x )( x ) = 0 x= 5;3 3. Vertex: We borrow the formula for the vertex from algebra: b b, f = ( 1, 16) a a Using these points, we construct the graph: y-axis x-axis From this graph, we obtain the FSP: 1. Dom f =R. Intercepts: 0, 15 a. y: ( ) b. x: ( 5,0) ; ( 3,0 ) 3. Cont f =R a. Pos f = (, 5) ( 3, + ) b. Neg f = ( 5,3) 4. Behavior at/toward ± : Lim f x =+ a. ( ) x b. Lim f( x) x + =+ 5. Symmetry: Neither even nor odd 6. Increasing/Decreasing: Inc f = 1, + a. [ ) b. Dec f = (, 1] FSP 1 5 Summary FSP1-5:7 004 TherapyUSA.biz

8 7. Relative Maximun/Minimum Points: ( 1, 16) is a relative minimum point. 8. Concavity: a. CU f =R b. CD f = 9. Inflection Points: None! 10. Graph: See above. 11. Absolute Maximum/Minimum Points: ( 1, 16) is an absolute maximum point. 1. Range: Range f = [ 16, + ) 13. Additional Information: The graph is symmetric with respect to the vertical line x= 1. <<FUNctions R FUN!>> One of the types of functions studied in precalculus involves taking a basic functionf and h x = A f Bx+ C + D where A, B, C, and D are real creating a new one h of the form ( ) ( ) numbers. Since we know all the FUNction Summary Properties of f, we can find all the FUNction Summary Properties of h. Functions of this type are the subject of another manual. FSP 1 5 Summary FSP1-5:8 004 TherapyUSA.biz

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