Download Calculator Workshop

Lesson One: Solving Equations
To solve equations using TI TECHNOLOGY:
1.
2.
3.
4.
Enter the left side of the equation into Y1 - Use ( ) where necessary
Enter the right side of the equation into Y2 - Use ( ) where necessary
Make sure PLOT is off
Graph using zoom 6 - adjust the "window" if necessary so that the point(s) of
intersection are in the graphing window
5. 2nd Trace (Calc) – 5 (for Intersect) - Enter (for 1st curve) - Enter (for 2nd curve) –
Enter (for guess)
6. The x-coordinate of the point of intersection is a solution to the equation
7. If there is more than one point of intersection (more than one solution), press
2nd Trace (Calc) – 5 – Enter – Enter – but before pressing enter for guess,
type a guess that is close to the second point of intersection. Repeat as needed
to determine all solutions to the equation.
Solve the following equations using the Point of Intersection Method.
1. 2.7( x  4) 
4.
3
x 1
5
4x  2  3  7
7. 4(x-1) = 3
2.
2x  6
5
7
3.
4x  2  3  1
5. 3.4x2 – 4.1x – 2.6 = 0
6. –2.3x3 = 4x – 7
8. log x = 7
9. 2sin(x) + 3 = 4
[0<x<2]
10. 2sin(3x) + 3 = 4
[0<x<2]
APPLICATION: Trig - Page 261 – Problem 64
Lesson Two: Solving Inequalities
To Solve inequalities using TI TECHNOLOGY:
1.
2.
3.
4.
"Move" everything to the left, leaving zero on the right
Enter the expression into y1 and 0 into y2
Graph in an appropriate window
Determine the x-intercepts using the "equation solving" steps
(adjust the window as needed)
Use your graph to determine the solution to the following:
3
1. 2.7( x  4)  x  1
5
3.
2.
3.4x2 – 4.1x – 2.6 < 0
5. –2.3x3 > 4x – 7
4.
2x  6
5
7
4x  2  3  1
6. 2sin(3x) + 3 > 4
{0 < x < 2}
1.06
2.27
.87
5.06
4.36
.17
Lesson Three – Solving Systems of Equations
To solve Systems of Equations with two variables using TI Technology:
1.
2.
3.
4.
Get the y by itself in each equation
Enter one equation into y1
Enter the other equation into y2 – set line style to thick
Graph and find the point(s) of intersection
Solve these systems of equations:
1. y = x2
y = -x2 + 4x + 5
2. y = x  3
y + 8 = x2
3. xy = 7
y=x-3
To solve Systems of Equations with two variables using TI Technology:
1. Matrix, ,  (for edit)
2. type the number that corresponds to an empty matrix
3. type the row and column size of the augmented matrix (including a column for answers)
4. type the i,j coefficients pressing enter after each entry
5. 2nd, quit
6. Matrix,  (for math)
7. down arrow to option B (or type alpha matrix for B)
This selects rref (reduced row echelon form)
8. Matrix, type the number of the matrix that contains the coefficients, and close the parentheses
9. Use the right arrow to scroll over –
The entries in the last column are the solution to the system of equations.
1. Solve the system of equations:
3a + 4b – 2c = 16
5a + 2b + c = 9
7a – 4b + c = 7
Application:
Write the equation of a parabola that passes through the points:
(2,5) (-3,10) (-1,-4)
Lesson Four: Functions
To evaluate a function with TI Technology use:
TRACE to evaluate a function from the graph as follows:
Enter the function f(x) = 2.37x2 – 3.1x – 5 into Y1=
Graph using ZOOM 6
From the graph screen press:
TRACE
Type the value for "x" and press ENTER. The corresponding "y" value will be displayed
Note: The value you type for x must be between xmin and xmax – you may
adjust your window as needed
Use this method to determine f(5.32) = _____ f(-1.4) = ______ f(13.2) = ________
Or use Tables as Follows:
2nd WINDOW (for table set)
Type the x value (56) into Tblstart and a value of 1 for increment in Tbl
Press 2nd Graph for Table
The Y1 value is f(x).
So f(56) = _______
Use this method to determine f(-22) = ______ and f(173) = ________
Or use the "Ask" function as follows:
2nd Window – for Table set
to highlight Ask
Enter
2nd Table
Type the value for x
The Y1 value is f(x).
Use this method to determine f(-23) _______
Application: PreCalc Page 173 Problem 94
f( 2 ) = ________
Lesson Four - Continued
Functions and Special Points – Intercepts, maximum, minimum
To determine the turning points and intercepts of a function using TI Technology, use:
Maximum or Minimum to find the turning points as follows:
Press Y=
If plot1 is highlighted turn it off and clear any equations
Type the function f(x) = 2x3 + 3x2 – 2x – 5 into Y1
ZOOM 6
(If the turning point is not visible on the screen press WINDOW and adjust as needed)
2nd TRACE (for CALC)
3 (if the vertex is a minimum) or 4 (if maximum)
use the left arrow to move the cursor to the left of the turning point and press ENTER
(or you may type an x value less than the x value of the turning point then enter)
use the right arrow to move the cursor to the right of the turning point and press ENTER
(or you may type an x value greater than the x value of the turning point then enter)
ENTER (one more time)
Record the coordinates of the turning point: ___________
Repeat as needed to determine additional turning points and record your results: __________
f(0) TO DETERMINE THE Y INTERCEPT AS FOLLOWS:
With the graph of the function still on the screen press
TRACE
type 0 (for x=)
ENTER
Record the y-intercept: ______________
2nd TRACE 5 TO DETERMINE THE X INTERCEPTS (if they exist) AS FOLLOWS:
y=
type 0 into Y2
GRAPH
(If the intercepts are not visible on the screen press WINDOW and adjust as needed)
2nd TRACE (for CALC)
5
ENTER, ENTER, ENTER
Record the first x-intercept: ______________
If there is a second x-intercept press repeat these steps, but BEFORE pressing the third enter for guess,
type a guess close to the second intercept, then press ENTER.
Record additional x-intercepts: ______________
Use your graphing calculator to sketch the graph and label the vertex, y-intercept, and xintercepts:
1. y = -2(x + 3)2 – 7
2. 4x + y – 2x2 = –9
Lesson Four - Continued
Functions and Special Points – Intercepts, maximum, minimum
Application:
Precalc - Page 182 Problem 80
LESSON 5: Evaluating Expressions Involving Functions
To evaluate a function using the TI Technology VARS:
1. If f(x) = 40x – 2x2 Evaluate f(5).
Instructions for evaluating f(5) = Y 1 (5):
Y=, type function
2nd Mode (for Quit)
VARS  Y-VARS
ENTER (for 1:Function)
ENTER (for 1:Y 1 )
Y 1 should now be on your home screen.
To evaluate the function at x = 5, type (5) next to Y 1 to get Y 1 (5), ENTER.
2. Find the average rate of change over [4,5] of f(x) = 40x – 2x2
To find the average rate of change you enter the difference quotient below following the instructions in
1 above to enter Y 1 (5) and Y 1 (4).
(Y (5)  Y1 (4))
f (5)  f ( 4)
= 1
54
(5  4)
3. Using the same function in (1), Fill in the table for the average rate of change over the intervals
[a,a+h], where a=4 and h = 1, 0.1, 0.01, 0.001, 0.0001
Round your answers to four decimal places (MODE, FLOAT, 4).
2nd ENTER (for recall the last entry) allows you to recall the difference quotient and edit it. This feature
is helpful in repetitive calculations. Remember to use 2nd Del (for insert).
h
average rate
of change
Over the
Interval
1
f ( 4  h )  f ( 4)
h
[4, 4+h]
[4,4+1]=
[4,5]
0.1
0.01
0.001
0.0001
[4,4+.1]=
[4,4.1]
The visual image of the difference quotient on the calculator screen helps to reinforce this important
quotient that is used in the definition of derivative.
Lesson Six: Write the equation of…….
To write the equation that models a given set of data using TI Technology:
ENTER THE DATA INTO A LIST:
STAT
1 ( for Edit)
To clear existing data:
use the arrow keys to put the cursor on L1
CLEAR
ENTER
Type the x values (column one) into L1 pressing ENTER after each number
Repeat this process to enter the y values into L2
***After pressing ENTER for the last data value, press
2nd MODE (for QUIT) to exit the data entry screen
SET THE WINDOW (Viewing Rectangle)
Before graphing the data, you must tell the calculator what values to use on the x and y axis. These values are
based upon the values you entered into L1 (x) and L2 (y). When entering minimum and maximum values, use
rounded values. For example, if your data has a minimum of 23 and a maximum of 47, you might use min=20
and max=50.
To set the window manually use the following steps:
WINDOW
Type a rounded number for Xmin that is slightly lower than the lowest x value entered
ENTER
Type a rounded number for Xmax that is slightly higher than the highest x value entered
ENTER
Type a rounded number for Xscl that determines the interval between the marks on
the x axis - for example, Xscl=5 will put marks every 5th number, starting at Xmin
ENTER
repeat for Ymin, Ymax and Yscl
Xres - leave this set at 1
GRAPH
Y= (to display the graph screen)
up arrow - select PLOT 1 (highlight PLOT 1 – this turns on the plot function)
GRAPH
Note: You can let the calculator set the window automatically by using ZOOM 9, but the min and max numbers
will not be “round” numbers. To use this option, press ZOOM 9 in place of “GRAPH”.
Generate a "line of best fit" or a "least squares regression line"
MODE, FLOAT (floating decimals)
STAT,  (CALC), 4 (Linreg), Enter
Your calculator displays values for a and b
Record your equation as y = ax + b, rounding a and b to the nearest hundredth.
Note: Some calculators also display a value for r. This is known as the correlation
coefficient – the closer r is to a value of 1, the closer the line is to an EXACT fit.
To check the results, "paste" your equation into Y1 and graph using:
y=, VARS, 5 (statistics),  (eq), 1 (RegEQ), GRAPH
Your line should go APPROXIMATELY through your data points
Use these steps to solve PreCalc Page 172 Problem 92
If the data is not linear:
You can use the same steps to enter the data, or you can Link to get the data:
Connect the graph link cord to your calculator and your instructor's
2nd X,T,  ,n (for LINK)
 (for RECEIVE)
ENTER
Then follow the steps for a linear regression, substituting option 5 (quadreg)
Use these procedures to solve PreCacl Page 184 Problem 96 -
Or when given just two or three points:
In 1998 worldwide sales of hand-held PCs reached 4 million, and in 2000 this number
was 7.8 million.
a. Write a linear function to model the data.
b. Graph this function in an appropriate window.
c. To the nearest year, when will the number of PCs reach 12 million?
d. Predict to the nearest million the sales in 2007.
Or:
Write the equation of a parabola that passes through the points:
(2,5) (-3,10) (-1,-4)
Lesson Seven: Logistics Growth
Precalculus and Calculus Investigations
To determine logistic functions using TI Technology:
Give a function in the form
f ( x) 
c
1  aebx
1. Enter the above formula for a logistics function into your y-list, using the letters A, B, and C in your
formula. (Use the Alpha key to enter A, B and C.) Then 2nd quit to return to the home screen.
2. Then on your home screen, assign the values A =35, B = 0.15, and C = 300.
Use the store key. 35 sto A enter will assign A the value of 35 .
3. Graph the function in the window [0,50,10] by [0,350,50].
Notice the S – Shaped appearance
Next - Investigations of how the values A, B, and C affect the graph by storing new values into A, B
and C and graphing the function again.
Investigations with exponential versus logistics growth
A simulated Flu epidemic: [Spreading Gossip]
Activity needs 20 to 25 people to work well
For N participants
0
1. Assign each participant a number
2. Generate a random number from 1 to N
on TI 82 ipart(rand*N) + 1
in TI 83 randInt(1,N)
3. Ask that person to stand
4. For each day generate a sequence of k random numbers, where k is the
number of students standing from previous day
TI 82 ipart(rand*N) + 1 k times
TI 83 randInt(1,N,k)
5. Have participants with those numbers stand. Record the number standing for
each day (trial) Compete this process until all are standing
Spread of a Virus
Day
1
2
3
4
5
6
7
8
9
10
11
12
# newly infected
total # infected
Enter into lists and graph. Generate a logistics equation to model data and graph.
Lesson Eight – Another Fun Activity with Regression
The M & M Model or
Can You Generate an Exponential Function
1. Pour out the 40 M & Ms from your snack bag onto the paper plate.
2. Remove the ones with the M & M logo showing.
3. Count the ones that are left and record this number in the table.
4. Do this for five trials.
Table of Trials
Number of Trial
_____0_______
_____1_______
_____ 2_______
_____ 3_______
_____ 4_______
_____ 5_______
M & M s left
____40___
_________
__________
__________
__________
__________
Enter this data into your calculator in the following steps:
1. Press STAT.
2. Press ENTER.
3. Type in the number of trails in L1 and the M&M count in L2.
4. Press 2nd Mode (quit)
5. Press STAT.
6. Highlight CALC
7. Press down arrow until you highlight ExpReg
8. Press ENTER.
9. Press ENTER (again)
Write your Exponential Function below ________________.
Answer Sheet
Lesson One – Equations
1. –5.62
2. –14.5, 20.5
6. 1.06
7. 1.79
10. .17, .87, 2.27, 2.97, 4.36, 5.06
Application: See text
3. no solution
8. 10,000,000
Lesson Two – Inequalities
1. [-5.62, )
2. [–14.5, 20.5]
4. all real numbers 5. (-, 1.06)
4. 4.5
5. -.46, 1.67
9. 2.62, .52
3. (-.46, 1.67)
6. (.17, .87)U(2.27, 2.97)U(4.36, 5.06)
Lesson Three – Systems
Two variables
1. (-.87, 2.87)
2. (–2.89, 3.24)
3. (4.54, -1.54)
Three variables
1. [2, 1, -3]
2. Parabola: 2x2 + x – 5 = 0
Lesson Four:
f(5.32) = 45.58
f(56) = 7253.7
f(-23) = 1320
f(-1.4) = 3.99
f(-22) = 1210.3
f( 2 ) = -4.64
f(13.2) = 367.03
f(173) = 70390
Application: See text
Special points: Minimum = (-1.26, -1.72) Maximum = (.26,-5.28)
Y int (-5,0) X int (1.17, 0)
1. Max = (-3,-7) x = n/a y = -25
2. Min = (1, -11)
x: (-1.35, 0) (3.35, 0) y = -9
Application: See text
Lesson Five: 1. 150
2. 22
3. 22, 23.80, 23.98, 23.998, 23.9998
Lesson Six: See text for first two problems
Third: y = 1.92x – 3792
Fourth: 2x2 + x – 5 = 0
Lesson Seven: Answers may vary
Lesson Eight: Answers may vary