Math Mid Exam Review

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    Math Mid-Exam Review

    Relations are a set of ordered pairs. Eg, {(2,3), (3,5), (3,7), (4,9)}

    Functions is a type of relation where for every x value, there can only be one yvalue. In the Eg, it is not a function because when x = 2, y = 5 or y = 7.

    Vertical Line Test is a test to determine if a relation is a function. If the line passes

    more than one points, the relation is not a function.

    Domain is the first set of elements (x) in the relation. Eg, D = {2, 3, 4}

    Range is the second set of elements (y) in the relation. Eg, R = {3, 5, 7, 9}

    A function can be written in function notation. The y in the equation is replaced by

    f(x).

    F(x) means f at x, where x can be replaced by a number. For example, f(2) means

    f at 2.

    Mapping Diagram:

    If one arrow leads from each value in domain to range, then this relation is a

    function. In this case, this relation is a function.

    There are THREE ways to find the vertex and the max/min of a quadratic equation.

    1. Completing the Square

    a. Factor the coefficient of x2, then divide the x coefficient by two, and

    square it. Then simplify to y = a(x-h)2 + k

    b. The vertex is (h,k)

    c. The function has a max value of k if a < 0

    d. The function has a min value of k if a > 0

    e. The max/min occurs as x = h

    2. Calculate the x-intercepts by factoring. *Only works if the equation is

    factorable*

    a. You have to let f(x) = 0 and then solve for x.

    b. Average the two answers to get the middle of the parabola, or h.

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    c. Substitute that value back in the equation to find the y-value of the

    vertex.

    d. If there is one answer, that is the x-coordinate vertex.

    e. To find a, just look at the coefficient of x2

    3. Partial Factoring

    a. When given f(x), say let g(x) equal to the equation without the vertical

    translation. If y = ax2 + bx + c, change g(x) by taking out c.

    b. Find the x-intercept, let g(x) = 0

    c. Average the answer if two answers are found.

    d. Substitute the value back into the ORIGINAL formula, with the vertical

    translation.

    e. Another way to find x is to use the formula: x = -b2a

    X-intercepts are also called the zeros or roots.

    Quadratic functions may have 2 real roots, 1 real root, or no real roots.

    To solve for x, there is an option of factoring, or the Quadratic Formula.

    The Quadratic Formula is used if factoring is not an option.

    The Quadratic Formula is: x = -bb2-4ac2afrom the original standard equation.

    To determine the number of roots, use b2-4ac =?

    a. If ? > 0, the answer is 2 real roots.

    b. If ? = 0, the answer is 1 real root.c. If ? ?< 0, the answer is no real roots.

    To simplify radicals, find the biggest perfect square number than can reduce the

    answer. Eg, 45= 95=35

    To Multiply/Divide radicals, smash together like bases and front numbers. Then

    reduce.

    To Add/Subtract radicals, add/minus LIKE radicals. Keep the number inside the

    square root the same. If no like radicals are found, the answer is simplified.

    To divide all terms out, make sure the common number can be divided to ALLnumbers, not including the radical. The number in front of the radical, however,

    cannot be 1.

    A secant is a line that intersects a curve at two points.

    A tangent is a line that intersects a curve at one point; it is not through the curve,

    but rather still outside the curve, barely touching it.

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    There is no name for a line that cannot intersect a curve.

    A rational expression is a quotient where the numerator and denominators are

    polynomials.

    f(x) =p (x)q (x)where q(x) 0

    For rational expressions, determine all values that make the denominator equal to

    zero. Those values are called restrictions. Even if you flip a fraction for a division,

    state ALL of the restrictions whenever the number is on the bottom. So in that case,

    you will state both the bottom and top of that fraction.

    Invariant points are points that are unaltered, stayed the same, in a transformation.

    In general: y = f(x-h) + k

    -h is the units to the move the graph horizontally

    k is the units to the move the graph vertically

    In general : y =-f(-x)

    -f is a reflection on the x-axis

    -x is a reflection on the y-axis

    In general: y = af(ax)

    af is a vertical stretch. If a > 1, the graph will stretch. If a 1, the graph will compress by a factor of 1a.

    If a < 1, the graph will stretch by a factor of 1a.

    It is important to factor anything before describing any transformation. Translations

    may be different compared to not simplifying it.

    The inverse of the function is found when you switch x and y around in the

    equation. That means the domain and range will be mirrored for x and y. The f(x)

    will be changed to f-1(x). To get there, solve for y in the inverse equation.

    Note that you can only interchange x and y when there is no powers connected to

    it. If there is, change the equation to vertex form.

    Transformations must be described in the following order:

    Stretches, Reflections, and Translations

    When multiplying like bases add exponents and keep the base the same.

    When dividing like bases subtract exponents and keep the base the same.

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    When a power is raised to an exponent, multiply the exponents and keep the base

    the same.

    If a term with more than 1 factor is raised to an exponent, the exponent applies to

    each factor.

    If a fraction is raised to an exponent, the exponent applies to the numerator and

    denominator.

    Anything to the power of zero is always 1.

    Negative powers need to be flipped to its reciprocal.

    The square root of a number is equal to that number to the power of one half.

    a1n=na where , a (N for natural, R for real numbers)

    Flower power is above, and the roots are below.

    amn=namwhere m (integers)

    y = 12xis the same as y = 2-x

    Remember when graphing, dot line the asymptote if possible.

    If the negative goes on the power, it is a reflection on the yaxis.

    If the negative goes on the base, it is a reflection on the x-axis.

    Change bases if necessary to figure out the reflections.

    Intervals of Inc/Dec just means if the values inc/dec from left to right for x.

    For the function of y = cax

    y represents the total amount or number

    c represents the initial amount or number

    a represents decay or growth factor

    x represents the number of growth or decay factors

    Remember for word problems to make a let statement and a concluding statement.

    For reviewing in general, do the handouts and questions on the textbook.

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