Pre-Calculus Chapter 2 Flashcards
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Questions and Answers

What is the standard form of a quadratic function?

f(x) = a(x-h)^2 + k, a does not = 0

What is the process of completing the square?

  • Use (b/2)^2 to find new c.
  • Add new c and shorten into squared expression.
  • Subtract new c.
  • All of the above. (correct)

    • If a > 0, where does the function have a minimum value?

    -b/2a

    If a < 0, where does the function have a maximum value?

    <p>-b/2a</p> Signup and view all the answers

    The graph of a polynomial function is discontinuous.

    <p>False</p> Signup and view all the answers

    What does the leading coefficient test determine for an odd-degree polynomial with a positive leading coefficient?

    <p>Left: Down, Right: Up</p> Signup and view all the answers

    What is a function's maximum number of real zeros?

    <p>The degree</p> Signup and view all the answers

    What is a function's maximum number of relative extrema?

    <p>The degree minus 1</p> Signup and view all the answers

    What do repeated zeros indicate in a polynomial?

    <p>(x - a)^k, if k is odd the graph crosses the x-axis at x = a, if k is even the graph touches but does not cross.</p> Signup and view all the answers

    What is the process of polynomial long division?

    <p>Put zeros in as placeholders, divide leading coefficients, multiply, subtract, bring down, repeat.</p> Signup and view all the answers

    What is synthetic division?

    <p>Can only be used with x-k, switch sign of k, multiply and carry coefficients.</p> Signup and view all the answers

    What is the remainder when a polynomial is divided by x-k?

    <p>r = f(k)</p> Signup and view all the answers

    What does the rational zero test involve?

    <p>p/q (factors of constant over factors of leading coefficient)</p> Signup and view all the answers

    What does PRZ stand for?

    <p>Possible rational zeros</p> Signup and view all the answers

    What are complex numbers?

    <p>a + bi, includes real and imaginary numbers.</p> Signup and view all the answers

    What is the addition formula for complex numbers?

    <p>(a + bi) + (c + di)</p> Signup and view all the answers

    If f(x) is a polynomial of degree n > 0, what does h have?

    <p>n linear factors</p> Signup and view all the answers

    If a polynomial has a factor a + bi, what else will it have?

    <p>A zero of a - bi</p> Signup and view all the answers

    What is the parent function of a rational function?

    <p>f(x) = 1/x</p> Signup and view all the answers

    How do you find vertical asymptotes?

    <p>Find zeros of the denominator</p> Signup and view all the answers

    How do you find horizontal asymptotes?

    <p>If degree of the numerator = n and degree of the denominator = m, if n &lt; m, there is no horizontal asymptote.</p> Signup and view all the answers

    How do you find the x-intercept of a function?

    <p>Find zeros of the numerator</p> Signup and view all the answers

    How do you find the y-intercept of a function?

    <p>Substitute zero for x</p> Signup and view all the answers

    What is a slant asymptote?

    <p>If the degree of the numerator is exactly 1 more than the degree of the denominator, divide numerator by denominator.</p> Signup and view all the answers

    Study Notes

    Standard Form of a Quadratic Function

    • Expressed as f(x) = a(x-h)² + k where a ≠ 0.

    Completing the Square

    • Method to transform a quadratic into vertex form, involving:
      • Calculating (b/2)² to find a new constant c.
      • Adding this new c, simplifying into a squared expression, then subtracting the new c.

    Minimum Value for Quadratics

    • Occurs at x = -b/(2a) when a > 0.

    Maximum Value for Quadratics

    • Occurs at x = -b/(2a) when a < 0.

    Continuity of Polynomial Functions

    • Graph of polynomial functions is continuous across its entire domain.

    Leading Coefficient Test (End Behavior)

    • Odd Degree, Positive Leading Coefficient: Left down, Right up.
    • Odd Degree, Negative Leading Coefficient: Left up, Right down.
    • Even Degree, Positive Leading Coefficient: Left up, Right up.
    • Even Degree, Negative Leading Coefficient: Left down, Right down.

    Maximum Number of Real Zeros in a Function

    • Determined by the function's degree.

    Maximum Number of Relative Extrema

    • Given by the degree minus one of the polynomial.

    Repeated Zeros

    • Expressed as (x-a)ᶦ, where k > 1.
      • If k is odd, the graph crosses the x-axis at x = a.
      • If k is even, it touches but does not cross the x-axis at x = a.

    Polynomial Long Division

    • Involves placeholder zeros, matching leading coefficients, multiplying, subtracting, and repeating with subsequent terms.

    Synthetic Division

    • Applicable only for divisors of the form x-k.
    • Requires switching the sign of k, using coefficients, and sequentially multiplying and adding.

    Remainder Theorem for Polynomial Division

    • The remainder when dividing by x-k is r = f(k).

    Rational Zero Test

    • Identifies potential rational zeros as p/q, where p is factors of the constant and q is factors of the leading coefficient.

    Possible Rational Zeros (PRZ)

    • All combinations of factors of the constant term over factors of the leading coefficient.

    Complex Numbers

    • Form a + bi, combining real numbers (both rational and irrational) and imaginary components.

    Addition and Subtraction of Complex Numbers

    • Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
    • Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i

    Linear Factors of a Polynomial

    • A polynomial of degree n>0 has exactly n linear factors.

    Complex Conjugates

    • If a polynomial has a factor of a + bi, it necessarily includes a zero of a - bi.

    Parent Function of Rational Functions

    • Represented by f(x) = 1/x.

    Finding Vertical Asymptotes

    • Determined by discovering the zeros of the denominator.

    Finding Horizontal Asymptotes

    • Compare degrees:
      • If degree of numerator (n) < degree of denominator (m), asymptote at y = 0.
      • If n = m, asymptote found at y = ratio of leading coefficients.
      • If n > m, no horizontal asymptote.

    Finding X-Intercepts

    • Derived by finding zeros of the numerator.

    Finding Y-Intercept

    • Found by substituting x = 0 into the function.

    Slant Asymptotes

    • Occur when the degree of the numerator is one more than that of the denominator; found via polynomial division.

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    Test your knowledge on quadratic functions with these flashcards from Pre-Calculus Chapter 2. Each card covers essential concepts such as the standard form of a quadratic and methods for finding minimum and maximum values. Perfect for quick revision and deepening your understanding.

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