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Review for College Algebra Midterm 1 1.2 Functions and graphs • Vocabulary: function, graph, domain, range, Vertical-Line Test. • Determine whether a given correspondence/mapping is a function. • Evaluate a function at a given input. • Use the vertical line test to determine whether a graph in the plane is the graph of a function. • Find the domain and range of a function, either from the algebraic definition of a function or from its graph. 1.6 The algebra of functions • Vocabulary: composition of functions • Compute the sum, difference, product, or quotient of a pair of functions, and find the domain of the resulting function. (Don’t simplify before finding the domain.) For x in the domain of the two functions f and g, (f + g)(x) = f (x) + g(x) (f − g)(x) = f (x) − g(x) f g(x) = f (x)g(x) (f /g)(x) = f (x)/g(x), provided g(x) 6= 0. • Compute the composition of two functions f and g, and find the domain of the resulting function. (Don’t simplify before finding the domain.) For x in the domain of g and g(x) in the domain of f , (f ◦ g)(x) = f (g(x)). • Given a composite function h(x), find two functions f and g such that h(x) = (f ◦ g)(x). 1.7 Symmetry and transformations • Vocabulary: symmetric about the x-axis/y-axis/origin, even/odd function, transformation, reflection, translation, scaling/stretching/shrinking • Determine whether the graph of an equation is symmetric with respect to the x-axis, the y-axis, and/or the origin. • Determine whether a function f is even: f (−x) = f (x), graph is symmetric about y-axis, or odd: f (−x) = −f (x), graph is symmetric about the origin. • Know the basic graphs on pp. 151, 152 and determine how other graphs can be obtained from them by a series of transformations (reflections, translations/shifts, and scaling). Know how to obtain these transformations by altering the expression defining a function. 2.1 Linear equations, functions, and models • Vocabulary: linear equation, solve, solution set • A linear equation can be written in the form mx + b = 0, where m and b are real numbers and m 6= 0. • Solve linear equations, and use linear equations to model and solve word problems. 2.2 The complex numbers • Vocabulary: complex number, real part, imaginary part, complex conjugate • Simplify expressions involving complex numbers (particularly products or quotients of complex numbers). To simplify a quotient, multiply the numerator and denominator by the conjugate of the denominator. 2.3 Quadratic equations, functions, and models • Vocabulary quadratic equation, quadratic function, zero/root, solution, completing the square, quadratic formula, discriminant • A quadratic equation in standard form is as follows, for some numbers a, b, and c: ax2 + bx + c = 0. • Solve a quadratic equation (i.e. find the zeros of a quadratic function) by factoring, by completing the square, and using the quadratic formula. • Determine whether a quadratic equation has 0, 1, or 2 real-number solutions by examining the sign of its discriminant, b2 − 4ac. • Solve an equation which can be reduced to a quadratic equation by a simple substitution. 2.4 Analyzing graphs of quadratic functions • Vocabulary: parabola, vertex, axis of symmetry, maximum value, minimum value • The graph of a quadratic function of the form f (x) = a(x − h)2 + k, a 6= 0, is a parabola with vertex (h, k) and axis of symmetry x = h. If a is positive, the parabola opens up, while if a is negative, the parabola opens down. • The graph of a quadratic function f (x) = ax2 + bx + c in standard form has vertex −b −b , f( ) . 2a 2a • The maximum or minimum value of a quadratic function is the y-coordinate of the vertex of the corresponding parabola. • Determine how a given quadratic function is obtained from the basic one, f (x) = x2 , by a series of transformations. • Find the range of a quadratic function and the intervals on which it is increasing and decreasing. • Solve application problems involving maximizing or minimizing quadratic functions. Write down what is known, what is asked, obtain a quadratic function in one variable, find the vertex, and use it to answer whichever questions are asked. 2.5 More equation solving • Vocabulary: rational equation, radical equation, absolute value equation • To solve a rational equation (an equation involving fractions), multiply both sides of the equation by the least common divisor (LCD) to clear the denominators. This may introduce extra solutions. You must check your solutions at the end to make sure that they satisfy the original equation. • To solve a radical equation (an equation involving nth roots), isolate one of the nth roots and raise both sides to the nth power. If any nth roots remain, repeat the process. If you raise both sides to an even power, then you may introduce additional solutions. You must check your solutions to be sure that they satisfy the original equation. • Let X be an algebraic expression and a a positive integer. Then |X| = a is equivalent to X = −a or X = a. If a is a negative integer, then |X| = a has no solutions. 2.6 Solving linear inequalities • Vocabulary: compound inequality, linear inequality, union, absolute value inequality • To solve a linear inequality, proceed as if solving a linear equation, but change the direction of the inequality sign when multiplying or dividing by a negative number. • Use linear inequalities to model and solve word problems. • A compound inequality such as −2 < 3x + 4 ≤ 16 is equivalent to the two inequalities −2 < 3x + 4 and 3x + 4 ≤ 16. Note the use of the word “and” rather than “or”. The solution set to this compound inequality can be written in set notation as {x| − 2 < x ≤ 4}, in interval notation as (−2, 4], or on a number line. • Inequalities involving absolute value come in four cases. (We consider only < and >. A similar result holds for ≤ and ≥.) Let X be an algebraic expression. The number 3 can be replaced below by any positive number. 1. |X| < 3 is equivalent to −3 < X < 3, which is equivalent to (−3 < X and X < 3). 2. |X| > 3 is equivalent to (X < −3 or 3 < X). 3. |X| < −3 has no solutions. 4. |X| > −3 is true for any value of X. 3.1 Polynomial functions and models • Vocabulary: polynomial function, leading term, leading coefficient, degree, end behavior, Leading-Term Test, continuous function, multiplicity of a zero • A general polynomial function is of the form f (x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 , with a 6= 0. The degree of such a polynomial is n, the leading term is an xn , and the leading coefficient is an . • Polynomials of degree 0, 1, 2, 3, and 4 are called constant, linear, quadratic, cubic, and quartic, respectively. • Determine the end behavior of a polynomial function using the Leading-Term Test. • Find the zeros of a polynomial function and their multiplicities. • If a zero has even multiplicity, the graph is tangent to the x-axis at the corresponding xintercept. If a zero has odd multiplicity, the graph crosses the x-axis at the corresponding x-intercept. 3.2 Graphing polynomial functions • Vocabulary: The Intermediate Value Theorem (IVT) • Graph a polynomial function f using the following steps: 1. Find the zeros of f (and thus the x-intercepts of its graph). 2. Determine the sign of f on the intervals between the zeros (it suffices to test the sign of f at one point in each interval, because f is continuous). 3. Find f (0) (and thus the y-intercept of the graph of f ). 4. Sketch the graph of f . You can check the end behavior using the Leading-Term Test, and you can check the behavior at x-intercepts using the multiplicity of the corresponding zeros. • Determine whether or not the IVT guarantees the existence of a zero of a function f between two x-values a and b. 3.3 Polynomial division • Vocabulary: (linear) factor, polynomial long division, synthetic division, The Remainder Theorem, The Factor Theorem • Use polynomial long division or synthetic division to divide one polynomial by another. • If dividing a polynomial f (x) by a linear factor (x − c) yields a quotient Q(x) and a remainder R, then f (x) = (x − c)Q(x) + R. • The Remainder Theorem: The value f (c) is equal to the remainder R above. • The Factor Theorem: (x − c) is a factor of the polynomial f (x) if and only if f (c) = 0. 3.4 Zeros of polynomial functions • Vocabulary: The Fundamental Theorem of Algebra, (complex/real/nonreal/rational/irrational/integer) zeros, The Rational Zeros Theorem • The Fundamental Theorem of Algebra: Every polynomial function with complex coefficients of degree n, with n ≥ 1, has exactly n complex zeros if they are counted with multiplicity. • For a polynomial function f (x) with real coefficients, if a + bi or a − bi is a zero of f (a and b real, b 6= 0), then so is the other one. √ √ • For a polynomial function f (x) with rational coefficients, if a + c b or a − c b is a zero of f (a,b, and c rational, b not a perfect square), then so is the other one. • The Rational Zeros Theorem: Any rational zero of the polynomial function f (x) = an xn + · · · + a1 x + a0 with integer coefficients is of the form p/q, where p divides the constant term a0 and q divides the leading coefficient an . • Use the Rational Zeros Theorem to find the rational zeros of a polynomial function f (x). Then divide by the corresponding linear factors and find the zeros of the quotient. Use the resulting factorization of the original polynomial f (x) to sketch its graph. • Determine a polynomial function of minimal degree with certain zeros and a specified type of coefficients.