Review for Algebra 1 Test 7

Functions

100

How do you know  a relation is a function?

A relation is a function if EVERY input is assigned EXACTLY ONE output. 

200

Is the relation a function {(4,7),(5,8),(-7,6),(7,-2)}? Explain. (HINT: Show ALL 3 steps)

300

What is the error? Redo it without the error.

Step 1 should be D={6,7,8} and NOT D={6,7,8,6}, because we CAN'T repeat the same numbers in each set. The error led to an error in the map, which led to an error in the conclusion.

400

_{Marcel claims that the graph below represents a function.State whether Marcel is correct. Justify your answer.}

Marcel is NOT correct, because the graph fails the Vertical Line test at x=2. 

500

_{A relation is graphed on the set of axes below. Is it a function? Explain.}

_{Yes, it's a function because the graph passes the vertical line test.}

Solve System of Equation (Go in order)

100

Read the problem below. What key words or phrases should you annotate?

EX: The cost of three notebooks and four pencils is $8.50. The cost of five notebooks and eight pencils is $14.50. Determine the cost of one notebook and the cost of one pencil.

3 notebooks, 4 pencils, is $8.50

then 5 notebooks, 8 pencils, is 14.50 

"Determine the cost of one notebook and the cost of one pencil."

200

STEP 1: What two things are you asked to find? Label them x and y.

EX: The cost of three notebooks and four pencils is $8.50. The cost of five notebooks and eight pencils is $14.50. Determine the cost of one notebook and the cost of one pencil.

In the last sentence I am told to find:

x=the cost of one notebook

y=the cost of one pencil

300

Step 1 Continued: Write the system of equations that represents both shopping trips.

The cost of three notebooks and four pencils is $8.50. The cost of five notebooks and eight pencils is $14.50. Determine the cost of one notebook and the cost of one pencil.

3x+4y=8.50

5x+8y=14.50

400

Step 2: Find the cost of one variable using elimination

The cost of three notebooks and four pencils is $8.50. The cost of five notebooks and eight pencils is $14.50. Determine the cost of one notebook and the cost of one pencil.

Multiply by opposite lead coefficients. Make one negative. 

  -5(3x+4y=8.50) ->-15x-20y=-42.5 

 3(5x+8y=14.50)-> 15x+24y=43.50 

The x terms eliminate and the following comes down when you add the rest: 

4y=1

(4y)/4=1/4

y=0.25

500

Step 3: Find the cost of the other variable using substitution

The cost of three notebooks and four pencils is $8.50. The cost of five notebooks and eight pencils is $14.50. Determine the cost of one notebook and the cost of one pencil.

Pick one of the original equations: 

3x+4x=8.50

Then substitute the y=0.25 into the equation.

3x+4(0.25)=8.50

3x+1=8.50

3x+1-1=8.50-1

3x=7.50

(3x)/3=7.50/3

x=2.50

The cost of one pencil is $0.25 and the cost of one notebook is $2.50

Solve quadratics

100

Joey thinks the equation below CANNOT be solved by FACTORING and can ONLY be solved by COMPLETING THE SQUARE, because there are NO two numbers that multiply to "-4" and add up to "5." Do you agree or disagree? What are the solutions?

x^2+5x=-4

Joey is not correct, because FIRST STEP for solving by FACTORING is to set the equation EQUAL TO ZERO. If you do that you get  x^2+5x+4=0  an an equation that CAN be factored as (x+4)(x+1)=0. Then set each factor equal to zero and solve for x to get  x={-4,-1} 

200

Explain what you are doing in ALL the steps for solving quadratic equations by COMPLETING THE SQUARE

STEP 1:  Isolate the constant. (get the number by itself alone one side of the =)

Step 2: find "b" the coefficient of the variable. Then evaluate  (b/2)^2 

Step 3: Complete the square by starting with (x+b/2)^2= step 1 constant + step 2 last number

and then solve for x

300

Solve the equation: 

x^2-5x-50=0

Step 1: Set equal to zero is done!

x^2-5x-50=0

Step 2: Find two numbers that multiply to the constant "-50" and add up to the coefficient of x, "-5" 

(-10)(5) and (-10)+(5)

Step 3: Factor:  

(x-10)(x+5)=0

Step 4: Apply the Zero Product Property and solve

x-10=0 and x+5=0

+10 +10    -5 -5

x={10,-5}

400

Solve the equation:  x^2-4x=15 

Are the solutions irrational or rational?

x^2-4x=15

Step 1: done!! the constant "15" is isolated (alone on one side)

Step 2: b=-4 so 

(-4/2)^2=(-2)^2=4

Step 3: Complete the Square and Solve for x

(x-2)^2=15+4

sqrt((x-2)^2)=sqrt(19)

x-2=+-sqrt(19)

+2      +2 

 x={2+-sqrt(19)}

*solutions are irrational because when you plug into calculator you don't get integers*

500

Solve by completing the square:  x^2+4x=2 

Round to nearest hundredth.

Step 1: Constant is already isolated

Step 2: b=4 

(4/2)^2=(2)^2=4

Step 3: 

(x+2)^2=2+4

sqrt((x+2)^2)=sqrt(6)

x+2=+-sqrt(6)

-2      -2

x={-2+-sqrt(6)}

x={0.45,-4.45}

Linear Regression

100

What are the calculator steps that you must for Linear Regression problems?

[2nd] [0]  Go to "Diagnostic On" and click [enter] twice so it says DONE

[STAT] [Enter] and input x values in L1 and y values in L2

[STAT]  go to "CALC" then "4:LinReg" 

then Click on  CALCULATE

200

_{To the nearest hundredth (1) Create the linear regression equation & (2) state the Correlation Coefficient and explain the strength}

1) y = 25.33x+353.16

2) r=0.93 suggests a strong positive correlation because r is close to 1.

300

To the nearest hundredth, (1) State the linear regression equation and (2) State the correlation coefficient & explain what the data suggest.

1)  y=-13.64x+81 

2)  r=-0.98 this suggests there is a strong negative correlation because r is close to  -1 

400

Emma recently purchased a new car. She decided to keep track of how many gallons of gas she used on five of her business trips.  (1) Write the linear regression equation for these data where miles driven is the independent variable. (Round all values to the nearest hundredth.) (2) If Emma drives 2,000 miles, how many gallons would you use (round to nearest integer)?

1)  y=0.05x-92 

2)  y=0.05(2000)-92=99.08 

So, if Emma drives 2,000 she would expect to use 99 gallons.

500

_{The table below shows the number of grams of carbohydrates, x, and the number of Calories, y, of six different foods.  To the nearest tenth, state (1) Linear equation (2) correlation coefficient & explain.}

1) y=14.1x + 5.8

2) r=0.9977... rounded to nearest tenth is r=1.0. This suggests a very strong positive correlation between the number of grams of carbohydrates and the number of calories in food because r is very close to 1.

Rational vs. Irrational

100

How do you define a rational number? Give examples. 

A rational number is any number that can be written as a fraction of two integers. A rational Number can be simplified to an integer, a terminating decimal, or a repeating decimal. 

(For example 4/1=4, -3/5= -0.6 or  -2/11=-0.1818181818...)

200

How do you define an irrational numer?

An irrational number is a non-repeating and non-terminating decimal. (It cannot be expressed as a fraction of two integers).

EX: 

sqrt(5)=2.23606798...

300

State if statement below is rational or irrational. Explain.

7+sqrt(4) 

Option 1:  

7+sqrt(4)=9

Therefore the sum is rational because 9 can be written as 9/1

Option 2:

7=R and sqrt(4)=2=R

Therefore the sum is rational because R+R=always

400

Explain if the product of

3/7

and

-3/5

is rational or irrational.

Because 3/7=R and -3/5=R,  the product of those numbers is also Rational because (R)(R)=R.

**If you use the definition you will get this number when you MULTIPLY the fractions: -0.25714285714285714285714... which doesn't look rational at first but there is a repeating pattern of "285714"**

500

Determine if the number  sqrt(7)/10  is rational or irrational. Explain. 

sqrt(7)/10=0.264575131106...

is irrational because the number is a non-repeating AND non-terminating decimal.

Option 2:  sqrt(7)  is irrational and 10 is rational. Therefore the statement is also irrational because I/R=I. 

Exponentiaal Problems

100

In an exponential problem:  y=a(b)^x 

What do the "a" the "b" and the "x" represent?

a=the initial value (like initial $ or population)

b=the growth factor OR the decay factor

x= represents the time (sometimes we use "t")

200

Give at least 3 words that mean we are dealing with a growth exponential function such that b=1+rate

grows, increases, inflation, saves (bank), twice, three times, etc.

300

Give at least 3 words that mean we are dealing with a decay exponential function such that b=1-rate

decreases, decays, depreciates, loses, half, third, etc.

400

A high school sponsored a basketball tournament. After each round, one-half of the players were eliminated. If there were 64 players at the start of the tournament, which equation models the number of players left after t rounds?

Step 1: rate=

1/2=0.5

Step 2: initial value=64 and 

b=1-0.5=0.5

Step 3: Create the equation 

y=a(b)^t

y=64(0.5)^t

500

Rhonda deposited $3000 in an account in the Merrick National Bank, earning 4.2% interest, compounded annually. She made no deposits or withdrawals. Write an equation that can be used to find B, her account balance after t years.

Step 1:  convert rate=4.2% to a decimal

(4.2)/100=0.042

Step 2:  initial value=3000  and 

b=1+0.042 = 1.042

Step 3:  Create the equations 

B=a(b)^t

B=3000(1.042)^t