Midterm Notes

• Order of Operation (if parentheses, square brackets, or fraction bars are present :) 1. Work separately above and below any division bar. 2. Use the rules that follow within each set of parentheses or square brackets. Start with the innermost set and work outward. Order of Operation (if no parentheses or brackets are present :) 1. Evaluate all power and roots. 2. Do any multiplications or divisions in the order in which they occur, working from left to right. 3. Do any additions or subtractions in the order in which they occur, working from left to right. EXAMPLE: simplify 4 ⋅ 32 + 7 − (2 + 8) SOLUTION: Work inside the parentheses first. 4 ⋅ 32 + 7 − (2 + 8) = 4 ⋅ 32 + 7 − 10 Simplify powers and roots. Since 32 = 3 ⋅ 3 = 9 4 ⋅ 32 + 7 − 10 = 4 ⋅ 9 + 7 − 10 Do all multiplications or divisions, working from left to right. 4 ⋅ 9 + 7 − 10 = 36 + 7 − 10 36 + 7 −10 = 43 − 10 = 33

• Application 1. Translate from word expressions to mathematical expression. 2. Write equations from given information EXAMPLE Verbal Sentence

Equation

Twice a number, decreased by 3, is 42

2x-3=42

If the product of number and 12 is decreased by 7, the result is 105.

12x-7=105

A number divided by the sum of 4 and the number is 28

x = 28 4+ x x + x = 10 4

The quotient of a number and 4, plus the number is 10.

When you are given a word problem: Step1: Determine what you are asked to find. Step2: Write down any other pertinent information Step3: Write an equation. Step4: Solve the equation Step5: Answer the question(s) of the problem. Step6: Check

• Distance and Midpoint d 2 = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 m =(

x1 + x2 y1 + y2 , ) 2 2

Note: It is customary to leave the distance in radical form. Do not use a calculator to get an approximation unless you are specifically directed to do so. EXAMPLE: Find the distance between (-3,5) and (6,4) SOLUTION: When using the distance formula to find the distance between two point, designating the points as ( x1 , y1 ) and ( x2 , y2 ) , such as: ( x1 , y1 ) = (-3,5) and ( x2 , y2 ) = (6,4).

d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 = (6 − ( −3) 2 + (4 − 5) 2 = 9 2 + ( −1) 2 = 82 In geometry, points that lie on the same straight line are said to be collinear. The distance formula can be used to determine if three points are collinear. For instance, three points A, B, and C can be obtained by calculating distance between A and B and distance between B and C and adding the two distances together.

• Circle A Circle is the set of all points in a plane that lie a fixed distance from a fixed point. The fixed point is called the center and the fixed distance is called the radius. The distance formula can be used to find an equation of a circle.

circle: r 2 = ( x − h) 2 + ( y − k ) 2

with center (h ,k ) and radius r

• Linear Equation Type Standard Form Point-Slope Form Slope-Intercept Form

Equation Ax + By = C y − y1 = m( x − x1); given point ( x1 , y1 ) y = mx + b

Intercepts: Let y=0 to find the x-intercept; let x=0 to find the y-intercept. Slopes of Parallel Lines: Two nonvertical lines with the same slope are parallel; two nonvertical parallel lines have the same slope. Slopes of Perpendicular Lines: If neither is vertical, perpendicular lines have slopes that are negative reciprocals; that is, their product is -1. Also, lines with slope that are negative reciprocals are perpendicular. a EXAMPLE: Line A has slope of . Line B is perpendicular to line A; hence the slope b b of line B is − . a

• Graph a line given its slope and a point on the line EXAMPLE: graph the line that has slope 2/3 and goes through the point (-1,4). SOLUTION: First locate the point (-1,4) on a graph. Then, from the definition of slope Change in y 2 m= = Change in x 3 Move up 2 units in the y-direction and then 3 units to the right in xdirection to locate another point on the graph. Connect the first and second point. Now, there is a line.

• Linear Inequalities in Two Variables Step1: Draw the boundary. Draw the graph of the straight line that is the boundary. Make the line solid if the inequality involves ≤ or ≥ ; make the line dashed if the inequality involves < or >. Step2: Choose a test point. Choose any point not on the line as a test point. Step3: Shade the appropriate region. Shade the region that includes the test point if it satisfies the original inequality; otherwise, shade region on the other side of the boundary line. If y > mx + b , shade above the boundary line; If y < mx + b , shade below the boundary line. When graphing the intersection of two linear inequalities: A pair of inequalities joined with the word “and” is interpreted as the intersection of the solutions of the inequalities. The graph of the intersection of two or more inequalities is the region of the plane where all points satisfy all of the inequalities at the same time.