PERT TEST

What are the distance and midpoint formulas? What is a good way to remember them?

Distance formula: d=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√d=(x2−x1)2+(y2−y1)2, which is just the Pythagorean formula c2=a2+b2c2=a2+b2 Midpoint formula: (x2+x1)2,(y2+y1)2(x2+x1)2,(y2+y1)2, which is just the average of the x-coordinates and the average of the y-coordinates Explanation: The distance between two points or coordinates use the Pythagorean formula for right triangles because the distance is actually the hypotenuse of a right triangle. Visualize this: the difference between the x values (run) is the horizontal side; the difference between the y values (rise) is the vertical side; the diagonal distance between the two points is the hypotenuse. To remember: Think of distance as the diagonal in a run-rise triangle. The mid-point is the middle of two points. This is clearly just the average of the x values and the average of the y values. To remember: Think of MID-point as MIDDLE = AVERAGE. Report a problem

What is a difference of squares and how can it be factored? What are the perfect square binomials and their factored forms?

The correct answer is: A binomial that is of the form x2−y2x2−y2 factors to (x−y)(x+y)(x−y)(x+y) (x+y)2=x2+2xy+y2(x+y)2=x2+2xy+y2 (x−y)2=x2−2xy+y2(x−y)2=x2−2xy+y2 Explanation: (a) The difference of the two squares x2x2 and y2y2: (x2−y2)(x2−y2) is factored out as (x−y)(x+y)(x−y)(x+y). (b) The perfect square binomial (x+y)2(x+y)2 is factored out into the trinomial x2+2xy+y2x2+2xy+y2. (c) The perfect square binomial (x−y)2(x−y)2 is factored out into the trinomial x2−2xy+y2x2−2xy+y2. Once you recognize these special binomials, they become very useful in simplifying equations. In (b) for instance, the first and last terms of the trinomial are perfect squares of xx and yy. The middle term is the product of xx and yy multiplied by 22. Let's try factoring the square binomial (3x+2y)2(3x+2y)2. To get the factored trinomial, square 3x3x and 2y2y: 9x29x2 + middletermmiddleterm + 4y24y2. To get the middle term, multiply: 3x⋅2y⋅2=12xy3x⋅2y⋅2=12xy. The factored out trinomial is: 9x2+12xy+4y29x2+12xy+4y2.

What are the four quadrants of the coordinate plane and the corresponding signs of the (x,y) coordinates?

The correct answer is: Counter clockwise from top right: Quadrant I, II, III, IV QI: (+x,+y) QII: (-x,+y) QIII: (-x,-y) QIV: (+x,-y) Explanation: Imagine the cartesian plane, with the x and y axes dividing the plane into four parts (each part is called a quadrant). They are named or numbered from I to IV, counting counterclockwise from top right. The coordinates in quadrant Q1 are (+x, +y); in QII are (-x, +y); in QIII are (-x, -y); and in QIV are (+y, -y).

How do you divide a polynomial by a monomial?

The correct answer is: Divide each term of the polynomial by the monomial, remember that xaxb=xa−bxaxb=xa−b. Explanation: For example: 18x4−9x3+6x23x2=6x2−3x+2

How do you divide a polynomial by a binomial?

The correct answer is: Do not separate the binomial. Completely factor the polynomial and binomial. Simplify where possible. Explanation: For example: x2−y24x+4y=(x+y)(x−y)4(x+y)=x−y4

How do you solve a system of linear equations by graphing? What do 0, 1 and infinite solutions represent?

The correct answer is: Graph the lines of both equations. The point of intersection is the solution. Lines that do not intersect have no solution. Lines that intersect once have 1 solution. Those that are the same line have infinite solutions. Explanation: A system of linear equations refers two or more lines working together. On the graph, you will see where they cross each other (point of intersection), or if they intersect at all. The coordinates at that point of intersection is the solution (or the values for x and y).

What are the different types of inequalities and how are they graphed?

The correct answer is: Greater than, or less than (> or <) are graphed with open circles (meaning the number is not part of the solution). Greater than or equal to, less than or equal to (≥or ≤) are graphed with closed or shaded circles (meaning the number is included in the solution). Explanation: When we graph, we make a visual representation of the inequality on the x and y axes. Points are labeled with the solved values for (x,y)(x,y). A point is represented by a tiny circle. An open circle (not shaded) means that the values of (x,y)(x,y) at that point are not part of the solution, but the values either less than or more than those values are included in the solution. A closed (shaded) tiny circle, on the other hand, means that the solution includes the values of x and y at that point.

What is the order of operations and how is it used?

The correct answer is: It is the order in which mathematical expressions are to be evaluated: Parentheses, exponents, and multiplication/division, addition/subtraction, performed left to right. Explanation: One easy way to remember it is to learn this phrase: Please Excuse My Dear Aunt Sally

How do you solve a system of linear equations using elimination?

The correct answer is: Multiply both equations by values such that when equation 1 is added to equation 2, one of the variables cancels. Explanation: 2x−3y=12x−3y=1 and 3x−4y=23x−4y=2 Multiply the 1st by −3−3 and the 2nd by 22 so that when added, −6x+9y=−3−6x+9y=−3 and 6x−8y=46x−8y=4 yield y=1y=1 Substitute this into either equation to solve for x.

What are the slope-intercept, point-slope, and standard form equations of a line?

The correct answer is: Slope-Int: y=mx+by=mx+b where mm is slope and bb is y-intercept. Point-Slope: (y−y1)=m(x−x1)(y−y1)=m(x−x1) where (x1,y1)(x1,y1) is a point on the line and mm is slope. Standard-Form: ax+by=cax+by=c. Explanation: The slope-intercept form of a linear equation shows how steep the line is (slope) and where it crosses the y-axis (intercept). The point-slope form shows that the slope (m) is the change in rise (up the y-axis) over the change in run (along x-axis). In the standard form (also called the general form), the constants aa,bb and cc must be integers, aa and bbmust be non-zero, and aa must always be positive.

How do you solve a system of linear equations using substitution? How do you verify your answer?

The correct answer is: Solve the first equation for variable x. Substitute the expression for variable x in the second equation to solve for variable y. Plug variable y back into the first equation. Verify answer by substituting both variables into both equations and confirming equivalence. Explanation: When you have 2 variables (such as xx and yy), there must be at least 2 equations for you to be able to solve the values of the variables.Start with the equation that looks the simplest of the two. Solve for one of the variables (this means: express xx in terms of yy). Plug in this expression to the variable xx in the second equation to get the value of yy. Plug in the value of yy to the first equation to get the value of x

What is the standard form of a parabola, its y-intercept, x-intercept(s), its axis of symmetry, and its vertex?

The correct answer is: f(x)=ax2+bx+cf(x)=ax2+bx+c Y-int: (0,c)(0,c) X-int: set f(x)=0f(x)=0 and solve for x by factoring (You can also use the quadratic formula.) Axis of symmetry: x=−b2ax=−b2a Vertex: (−b2a,f(−b2a))(−b2a,f(−b2a)) Explanation: In the standard equation of a parabola shown as f(x)=ax2+bx+cf(x)=ax2+bx+c, a positive aa indicates a parabola that opens upward with a vertical axis of symmetry given by the equation x=−b2ax=−b2a. The y-intercept is found by solving for yy when x=0x=0. We do this as follows: y=a⋅02+b⋅0+cy=a⋅02+b⋅0+c y=cy=c Therefore, the parabola intercepts the y-axis at (0,c)(0,c). The x-intercept is found by solving for xx when y=0y=0. ax2+bx+c=0ax2+bx+c=0. Solve for x by factoring or by using the quadratic formula.

What is the formula for the slope of a line using 2 of its points.

The correct answer is: m=y2−y1x2−x1m=y2−y1x2−x1, commonly referred to as "rise over run." Explanation: The slope or gradient of a line shows how steep the line is. This is computed by dividing the difference between two points along the y-axis (rise) by the difference between two points along the x-axis (run).

What is the quadratic formula and what is it used to find?

The correct answer is: x=−b±b2−4ac√2awhenax2+bx+c=0 x=−b±b2−4ac2awhenax2+bx+c=0 It is used to find the zeroes of a function, or where the function crosses the x-axis. Explanation: The quadratic formula is used to solve for the roots of a quadratic equation (aka "equation of degree 2") in the form ax2+bx+c=0ax2+bx+c=0 where xx is the variable and aa, bb and cc are constants. "Solving for the root" is another phrase for "solving for the zeroes of the function" or "solving for the value of the variable xx as the graph of the quadratic formula crosses the x-axis". A quadratic equation has two roots or two values for xx.

What are the rules for combining and simplifying expressions involving exponents with the same base?

The correct answer is: xa⋅xb=xa+bxa⋅xb=xa+b xa÷xb=xa−bxa÷xb=xa−b (xa)b=xa⋅b(xa)b=xa⋅b x−a=1xax−a=1xa x0=1,x≠0x0=1,x≠0 Explanation: To multiply expressions with exponents but the same base, simply copy the base and add the exponents. To divide, simply copy the base and subtract the exponents. To raise an expression to a power or exponent, copy the base and multiply the exponents. An expression with a negative exponent can be expressed another way as the reciprocal of the same base but with a positive exponent. A base raised to zero exponent is equal to 1, the base being a non-zero.