Piecewise functions 9-20 class notes

Homework for 9-21 is page 25, items a, b, 7 and 8.

Yesterday and today we used easy, integer x's as inputs for a table of values for a piecewise function.  When an x makes a condition true, you use the associated rule to obtain the function or y- value.

If the condition has a "<" or ">" inequality without an equal sign, the graph of that piece will have an open dot at the end of the line, meaning you can't use the x-value for that position to create a y-value.  That piece of the function has no value, no x, no y, no orederd pair at that position.

If the condition has a "<" or ">" inequality including an equal sign, the graph of that piece will have a closed dot at the end of the line, meaning you use the x-value for that position to create a y-value.  That piece of the function has a definite value and ordered pair at that position.

When we make a table of values, we use integers so it's easier to plot the points on a grid.  When we connect the points with a line, it means there are many points besides our easy integers which satisfy the expression and the condition.  The domain and range of a function piece that is a line can have fraction and rational values, numbers in between the integers.  The domain and range are not limited to integers or easy points.  In item 5, the domain is "x is all real numbers", and the range is "y is all real numbers > 0."

We discussed why the graphs of piecewise functions look strange.  We are familiar with functions having a single rule, or expression, and one or no restrictions (conditions).  Each piece of a piecewise function can have a different shape: Two straight lines with different slopes, or a curve and a straight line.

Homework for 9-21 is page 25, items a, b, 7 and 8.

Quiz preparation:

Given the graph of a piecewise function, state the domain and range.

Given the expressions for a piecewise function, make a table of values and draw a graph.

Write a description of a piecewise function, describing where it changes from one piece to another, what the shapes of the pieces are, and what the domain and range are.

Cornell notes 9-19 functions and relations

EA 1 graphing project more hints 9-19

Solve for the coordinates of the vertex where red crosses green
Use the red equation in a system with the green equation
4x + 8y = 2400 and
9x + 3y = 1800                         
Hint - make the equations simpler; divide the 1st equation by 4

x + 2y = 600

and divide the second equation by 3.

3x + y = 600

Then use an "x =" or a "y =" to substitute, for example

x = 600 - 2y

make x tmporarily go away

3(600 - 2y) + y = 600     by substituting for x next to the 3

1800 - 6y + y = 600     by distribution

1800 - 5y = 600     by combining "y" terms

-5y = -1200     by subtracting 1800 from each side

y = 240    by dividing by -5 on each side

bring back x

x = 600 - 2(240)     by substituting for y in the earlier equation

x = 600 - 480

x = 120

the solution (x,y)  where these graphs intersect is (120, 240)

(120, 240) is a vertex for the feasible region

Solve for the coordinates of the vertex where green crosses blue
Use the green equation in a system with the blue equation
9x + 3y = 1800 and
9x + y = 1700                           
Hint - use a "y = " equation to substitute

y = 1700 - 9x     by subtracting 9x from each side

make y go away temporarily

9x + 3(1700 - 9x) = 1800     by substituting for y next to the 3

9x + 5100 - 27x = 1800     by distributing

-18x + 5100 = 1800     by combining "x" terms

-18x = -3300     by subtracting 5100 from each side

x = 183     by dividing each side by -18

bring back y

9(183) + y = 1700     by substituting 183 for x next to the 9 in the earlier equation

1647 + y = 1700     by multiplying

y = 53     by subtracting 1647 from both sides

        

the solution (x,y)  where these graphs intersect is (183, 53)

(183, 53) is a vertex for the feasible region

The calculated vertices for the feasible region are (0,0), (0, 300), (120, 240), (183, 53) and (200,)

These are the combinations of the numbers of games made at plant 1 and plant 2 that you will use to test in the objective function P = 90x + 70y.

Plug in all five ordered pairs and see which one gives you a maximum value for P.

EA 1 graphing project solve pairs of equations for two vertices

EA 1 graphing project solve pairs of equations for two vertices.

Directions for 5., revised.  You only need two ordered pairs, each from solving a system of 2 equations in 2 variables.

The dashed line is the boundary of the feasible region.  A solution for the red line and the blue line is not necessary because the intersection is outside the feasible region.

Solve for the coordinates of the vertex where red crosses green
Use the red equation in a system with the green equation
4x + 8y = 2400 and
9x + 3y = 1800                         
Hint - make the equations simpler; divide the 1st equation by 4
and divide the second equation by 3. Then use an "x =" or a "y =" to substitute
You will get (x,y) for this vertex

Solve for the coordinates of the vertex where green crosses blue
Use the green equation in a system with the blue equation
9x + 3y = 1800 and
9x + y = 1700                           
Hint - use a "y = " equation to substitute
You will get (x,y) for theis vertex

EA 1 graphing project 9-16 sample of student work

Partial work on the project by Miss Stacey

EA 1 Graphing Project class notes

Continue 2. by finding easy points for the two other constraints.

3.  Graph the three constraints using two easy points each.

4.   Make estimates for the coordinates of the intersections.  Miss Ledcke asked why do we read a graph and make estimates if the estimate is inexact and we won't use it.  The answer is that someone will give you a graph (picture information) and not give you the equations (math information), so you have to make an educated guess based on the information you have.  If you get more information, or if you can create more information by writing and solving a system of equations, you can get a more precise answer.

5.  In class we looked at where the red dashed line intersects the green dashed line.  That point is interesting because it's the solution of two equations, and it's the vertex of a feasible region.  We used substitution to find the coordinates of the point.

6. Compare this result with the estimate you made in 4.

7, 8, 9, 10, 11 build on the board diagram shown above.

Homework for Monday 9-19 is to finish the Embedded Assessment 1 graphing project.

EA 1 graphing project and class notes

Math 11    Embedded Assessment 1    Graphing Project

Page 21.  We have determined that the profit objective function is P = 90x + 70y, where x is the number of games made at plant 1 and y is the number of games made at plant 2.  The table of hours used at each plant activity has given us three constraints, 9x + y LTE 1700, 9x + 3y LTE 1800, and

4x + 8y LTE 2400.

1.        On grid paper create axes in quadrant I with an x-scale of 600 and a y-scale of 1700.

2.        Find easy points (0,y) and (x,0) for the three constraint boundaries. 

9x + y = 1700, 9x + 3y = 1800, and 4x + 8y = 2400

You will get six easy points, two for each of the three lines.

3.       Graph the three constraint boundaries using different colors.

4.       Make estimates for the coordinates of the three intersections of the three lines.  You will get three ordered pairs.

5.        Take the constraint equations two at a time and use substitution or elimination to solve for x and y.  You will get three ordered pairs.

6.       Compare the results of 4. And 5.

7.       On grid paper create axes in quadrant I with an x-scale of 400 and a y-scale of 400.

8.       Look at the graph from 3. and look closely at the area where x < 400 and y < 400

9.       Use the intercepts from 2. and the points from 5. to draw an enlarged graph on grid 7.

10.   Shade the feasible region.

11.   Identify the vertices.

12.   Test the coordinates of the vertices in the profit function.  This means substitute the coordinates of a vertex into P = 90x + 70y to get a value for P.

13.   Determine which vertices or points give the greatest profit.

14.   Explain what this means for manufacturing game systems.

The homework for Friday 9-16 is to work ahead and finish this project.