College Algebra - Final
Milestone with Answers!!
You passed this Milestone
23 questions were answered correctly.
2 questions were answered
... [Show More] incorrectly.
1
Which of the following equations is correctly calculated?
•
-2 × -7 = -14
•
81 ÷ -9 = -9
•
-36 ÷ 4 = 9
•
4 × 12 = -48
RATIONALE
This is correct. The quotient of a positive and negative
number is always negative.
This is incorrect. The product of two negative numbers is
always positive. The correct product is 14.
CONCEPT
Multiplying and Dividing Positive and Negative Numbers
2
Select the solution to the following system of equations:
•
•
correct
•
•
RATIONALE
This is incorrect. The quotient of a positive and a
negative number is always negative. The correct quotient
is -9.
This is incorrect. The product of two positive numbers is
always positive. The correct product is 48.
When we add the two equations, the term will
cancel, leaving . Now we can solve for by
dividing both sides by .
Dividing by gives a solution of .
Once we divide both sides by , we get a
solution of . We can solve for by plugging
in for in either equation.
Let's use the second equation . Once
is plugged in for , evaluate and solve for .
Here, by multiplying by , the result
contains a term, which will cancel when
combined with the second equation. Now, we can
add the two equations together.
Before you can add equations, sometimes you
must multiply entire equations by a scalar value for
the method to work as intended. For this system,
we can multiply the second equation by .
Subtracting a negative is the same as adding a
postive . Next, subtract from both sides.
On the right side, minus equals . Now we
can solve for by dividing both sides by .
CONCEPT
Solving a System of Linear Equations using the Addition Method
3
Perform the following operations and write the result as a single
number.
[4 + 8 × (5 – 3)] ÷ 5 + 6
•
10 correct
•
10.8
•
2
•
1.8
RATIONALE
This is the solution to the system of equations.
Following the Order of Operations, we must first
evaluate everything in parentheses and grouping
symbols. When there are brackets or braces, evaluate
the innermost operations first. Here, we must evaluate
5 minus 3 first.
5 minus 3 is 2. There are still operations inside
grouping symbols to evaluate. Multiplication comes
before addition, so we must evaluate 8 times 2 next.
CONCEPT
Introduction to Order of Operations
4
Consider the quadratic function .
What do we know about the graph of this quadratic equation,
based on its formula?
•
The vertex is and it opens upward.
•
The vertex is and it opens downward.
•
The vertex is and it opens upward.
8 times 2 is 16. Next, we add 4 and 16 to complete the
operations inside parentheses.
4 plus 16 is 20. Now there is just division and
subtraction. Division comes before subtraction in the
Order of Operations, so we divide 20 by 5 next.
20 divided by 5 is 4. Lastly, add 4 and 6.
4 plus 6 is 10.
•
The vertex is and it opens downward. correct
RATIONALE
Take note of the sign in the numerator. Evaluate the
division to get the x-coordinate of the vertex.
The sign of tells us if the parabola opens upward
or downward. If is positive, the parabola opens
upward. If is negative, the parabola opens
downward.
Compare the given equation to the general
. The values of and in particular
give us useful information about the graph.
In this case, since is negative, we know the
parabola opens downward. Next, we can use the
values of and to find the x-coordinate of the
vertex.
The values and can be plugged into this formula
to give us the x-coordinate of the vertex.
From the given equation, plug in for and for
. Simplify the denominator.
CONCEPT
Introduction to Parabolas
5
Suppose and .
Find the value of .
•
This is the y-coordinate to the parabola's vertex.
Return to the original equation, but write in the
calculated x-coordinate, , for every instance of .
Then, evaluate the equation.
squared is . Next multiply this by the
coefficient, .
times is . Finally, evaluate the addition and
subtraction.
From the equation, we know that the parabola's
vertex is at and opens downward.
The x-coordinate of the vertex is . To determine
the y-coordinate of the vertex, plug this x-value into
the original equation and solve for .
times is . Next, evaluate times .
•
•
•
correct
RATIONALE
CONCEPT
Function of a Function
6
To evaluate this composite function, focus on the
innermost function first. Evaluate first by plugging
in for the variable in the function .
Once has been replaced with , evaluate the
expression.
The function evaluates to . To evaluate ,
use the value of , which is , as the input for the
function .
Once has been replaced with , evaluate the
expression.
This tells us that is equal to [Show Less]