SOPHIA| College Algebra - Final Milestone with Answers!!| SPRING 2021

SOPHIA| College Algebra - Final Milestone with Answers!!| SPRING 2021 You passed this Milestone 23 questions were answered correctly. 2 questions were answered 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. 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. CONCEPT Multiplying and Dividing Positive and Negative Numbers 2 Select the solution to the following system of equations: • • correct • • RATIONALE 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 . 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. When we add the two equations, the term will cancel, leaving . Now we can solve for by dividing both sides by . 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 . 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 . Dividing by gives a solution of . This is the solution to the system of equations. 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 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. 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. 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? • • • • RATIONALE Compare the given equation to the general . The values of and in particular give us useful information about the graph. 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. 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. Take note of the sign in the numerator. Evaluate the division to get the x-coordinate of the vertex. 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 . 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 . Next, evaluate times . times is . Finally, evaluate the addition and subtraction. This is the y-coordinate to the parabola's vertex. From the equation, we know that the parabola's vertex is at and opens downward. CONCEPT Introduction to Parabolas 5 Suppose and . Find the value of . • • • • correct RATIONALE 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 . CONCEPT Function of a Function 6 Consider the following system of two linear equations: Select the graph that correctly displays this system of equations and point of intersection. • •