Sec. 2.5:  Demonstrate the transformations of graphs for functions in the form of

(need dpgraph for all links)

Vertical shifts of y=x² : Turn on scrollbar with variable D to move graph up or down.

Horizontal shifts of y=x³ : Turn on scrollbar with variable C to move graph left or right.

Horizontal reflections with y=√x : Turn on scrollbar with variable B to reflect graph.

Vertical stretch and reflections of 

   y=x(x+2)(x-1) : Turn on scrollbar with variable A to stretch graph or reflect it.

All 4 with y=³√x : Turn on scrollbar with variable A, B, C or D.

Sec. 2.5:  Demonstrate the transformations of graphs for functions in the form of

y=a*(x-h)²+k.

(need Winplot for this link)

Graph: Use scrollbars to change a, h, and k.

Sec. 2.5:  Demonstrate the transformations of graphs for functions in the form of

y=a*f(x-h)+k.

(need Winplot for this link)

Graph: The graph will start with f(x)=x², but you can change this. Use scrollbars to change a, h, and k.

Sec. 2.5:  Demonstrate the transformations of graphs for functions in the form of

y=a*f(b(x-h))+k.

(need Winplot for this link)

Graph: The graph will start with f(x)=x², but you can change this under Equa:User Functions... . Use scrollbars to change a, b, h, and k.

(need dpgraph for all links)

1. Find the basic graph.

  Since this one deals with a square root the basic graph is y=√x.

   The basic graph.

2. If there is a negative on x, factor out -1 inside the function.

  So now we have y=-4√-(x-3) +7.

3. Do the horizontal and vertical shifts.

  The horizontal shift is 3 right.

  The vertical shift is 7 up.

  Move the basic graph. Use scrollbar with c for horizontal shift and d for vertical shift.

4. Plot the basic point.

  The graph of the square root function has an endpoint at the origin. So the endpoint will now be at (3,7).

  The graph so far.

5. If there is a horizontal reflection, reflect relative to this point.

  Reflect graph. Use scrollbar with b.

6. Note if there is a vertical reflection.

  Since a=-4<0, there is a vertical reflection.

7. Perform vertical reflection and stretch/shrink.

  Reflect and stretch graph. Use scrollbar with a.

8. Plot other points as required.

  The square root function needs the endpoint and one other point plotted. In the basic graph it goes through the point (1,1) which is 1 to the right and 1 up from the endpoint.

  This graph will have that same point, but now it's 1 to left because of the horizontal reflection. Also, it's now 4 down because of the vertical reflection and stretch. 

  So that's from (3,7), 1 left and 4 down.

9. Draw curve.

Sec. 3.2:  Graph

f(x) =

 (1/40)(3x-5)²(x+4)(x+1).

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1. Find x-intercepts and multiplicity. Factor if needed.

2. Find y-intercept.

     y-int.=f(0)

            =(1/40)(-5)²(4)(-1)

            =2.5

3. Graph intercepts.

     The graph so far.

4. Make sure you plot at least one point between each consecutive pair of x-intercepts.

     Since the y-int. is between -1 and 5/3, we just need to plot another one between -4 and -1. So let's use -3.

      f(-3)

        = (1/40)(-14)²(1)(-2)

        = -9.8

     The graph so far.

5. Determine end behavior.

     Since as x goes to infinity, all of the factors go to infinity, the right side of the graph will go up. The graph so far.

6. Use multiplicity to determine how the graph behaves at the x-intercepts. (even=bounces, odd=goes through)

7. Finish graph.

     Final graph.

Sec. 3.3-3.4:  Find all rational zeros of f(x) =

25x⁴-45x³+49x²-24x+4.

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1. Find the factors of the constant term.

    Factors of 4=1,2,4

2. Find the factors of the leading coefficient.

    Factors of 25=1,5,25

3. Find all possible rational zeros by dividing every factor from step 1 by all the factors from step 2. Remember to include the ± since we don't know about the signs of the zeros.

    PRZ=±1,±1/5,±1/25,

            ±2,±2/5,±2/25,

            ±4,±4/5,±4/25

4. Graph the function using a graphing utility.

      The graph of f(x) .

5. Use graph to limit the list.

    Since the only visible x-int. is between 0 and 1, the list narrows to

         1/5,1/25,2/5,

          2/25,4/5,4/25

    Also, looking closer, the x-int. is near 1/2, so the list becomes just 2/5.

6. Use synthetic division to reduce the polynomial.

7. Use the graph to determine possible multiplicity.

    Since the graph seems to bounce off the x-axis, 2/5's multiplicity is probably 2, and thus we should try 2/5 again.

8. Keep trying possible zeros until you reduce the polynomial to 2nd degree. Then, solve the resulting equation.

    We are now down to 

9. Answer question.

    Zeros=2/5,

Sec. 3.6: Find the solution set in interval notation for

.

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1. Solve the related equation.

2. Find the domain restrictions on the inequality.

3. Set up the intervals using the solutions from step #1 and the restrictions from step#2.

4. Find out which intervals are included using test points or the graph of y=(left side) - (right side).

Graph of y=.

Since after moving the 1 to the left side you would get "<0", we are looking for where the graph is below the x-axis. Thus, from the graph only the center interval would be included.

5. Determine which endpoints are included and which aren't.

We can't include where the inequality is undefined. So -4 isn't included.

Since we have a non-strict inequality we include the rest of the endpoints.

6. State solutions set.

Thus the solution set is (-4,9].