Algebra 2 – linear functions

Q:  f(x) is a linear function with a y-intercept of 6. If f(-2) = 9, what is f(2)?

Answer:   3

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Explanation:

f(x) = mx + b              (because it is linear)

m is the slope of the line passing through (0,6) and (-2,9).

m = (9-6) / (-2-0)

m = – 3/2

b is the y-intercept.  b = 6.

So, f(x) = – 3/2 x + 6

Now, find f(2):

f(2) = (- 3/2)(2) + 6 = 3

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Algebra 1, 2 – solving quadratic equations

Q: The square of a positive number is equal to 5 more than the product of 4 and the same number. What is the number?

Answer: 5

Explanation:

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Algebra 1 – Find the equation of a line

Q:  Find the equation of the line that has the following intercepts:

x-intercept: 5,   y-intercept: -3

Answer:  y = 3/5 x – 3

Explanation:  The x-intercept is the point (5, 0).  The y-intercept is the point (0, -3).  Calculate the slope of the line passing through these points:

m= (0- -3)/(5 – 0) = 3/5

Using slope intercept form (y = mx + b), the equation is y = 3/5 x + b.  We just need to find b.  Or do we?  We could plug in an ordered pair and solve for b, but it’s not necessary.  We already have b, because in slope-intercept form, b always stands for the y-intercept.  So, b = -3.

The equation is: y = 3/5 x – 3

My TI-84 Plus program ‘LINES’ will find this equation for you, given the two intercepts.

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SAT – Percent change

Q: If the length of a rectangle is increased by 30%, while the width is decreased by 10%, by how much will the area increase? Give the answer as a percent.
 
Answer: 17%
Explanation:  Choose the length and width of the original rectangle to be 10.  Then, the length and width of the new rectangle are 13 and 9 (by increasing 30% and decreasing 10%).  The original area is 10 x 10 = 100.  The new area is 13 x 9 = 117.  The percent increase from 100 to 117 is (117-100)/100 = 17/100 = 17%.
Note that if this were an SAT problem with answer choices, you should immediately eliminate any ‘too easy’ answers, such as those related to the numbers 30 and 10 too directly.  Eliminate 20% and 40%.
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SAT / Algebra 1: Ratios

Q:  In my box of cereal, the ratio of flakes to raisins is 9:2. If there are a total of 220 flakes and raisins in the box, how many of these are flakes?
 
Answer:  180
Explanation:  Solve the equation: 9x + 2x = 220 to get x = 20.  ‘Flakes’ are the 9x part of the equation.  9(20) = 180.  Or, use my TI-84 Plus program ‘RATIO’.
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Algebra 1, 2 – functions

Q: Which of the following represents a function?  (Choose all correct answers.)
 
a. {(4,2), (6,2), (-1,5), (3,-2), (-4,-7)}
b. {(4,2), (5,1), (-1,5), (5,-2), (4,-2)}
c. {(-1,-1), (1,1), (4,4), (3,3), (6,6)}
d. {(1,2), (3,2), (5,2), (7,2), (9,2)}
 
Answer: a, c, and d
Explanation:  A function must have only one y-value for each x-value.  Choice b has two points with the same x:  (4,2) and (4, -2) (and also two others).  All of the other choices have no repeated values for x, so they are functions.
 
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SAT – systems of equations

Q: A phone call costs c cents for the first 10 minutes and n cents per minute for additional minutes. A 20-minute phone call costs 50 cents, and a 40-minute phone call costs 90 cents. What is the value of c?
 
Answer: 30
Explanation:  A 20-minute call costs c cents for the first 10 minutes and n cents per minute for each of the next 10 minutes.  So, c+10n=50.  A 40-minute call costs c cents for the first 10 minutes and n cents per minute for each of the next 30 minutes: c+30n=90.  Solve this system of equations.  Use my TI-84 Plus program CRAMER or an algebraic method such as substitution, elimination, Cramer’s Rule, etc.  c=30.
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Algebra 1, 2 – systems of equations

Q: 100 people are to be seated at 22 tables. Some tables seat 4 people and some seat 6 people. If no seats are empty, how many tables seating 4 people are there?
 
Answer: 16
Explanation:  Solve the system: x + y = 22; 4x + 6y = 100.  My ‘CRAMER’ program will solve this for you! The number of 4-person tables is the x-value of the solution.
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Algebra 1 – distance

Q: To the nearest tenth, what is the distance between the points (-6,2) and (6,5)?
 
Answer:  12.4 
(Use the distance formula, and round to the nearest tenth.)
My TI-84 Plus program DISTANCE will find this value.
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Algebra 1, 2 – systems of equations

Two cubes and one pyramid weigh 9 lb. Seven cubes and two pyramids weigh 24 lb. All cubes weigh the same, and all pyramids weigh the same. What is the weight of one pyramid?

 
Answer: 5 lb

Explanation:  Let x be the weight of a cube.  Let y be the weight of a pyramid. 

The first sentence becomes: 2x + y = 9. 

The second sentence is: 7x + 2y = 24. 

Solve this as a system of equations by any method you choose… substitution, elimination, Cramer’s Rule, etc.  Or use my TI-84 Plus program ‘CRAMER’.

x = 2, y = 5

‘y’ is the weight of one pyramid, so the answer is 5.

 

 

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