unit 2

• What is the transformed graph for \( y = f(2(x - 3)) \) based on the original graph of \( y = f(x) \)?


  It is a horizontal dilation and translation 3 units to the right.
• What does the notation \( h(x) = g \circ g(x) \) represent?


  It represents the composition of the function \( g \) with itself.
• If \( g(-3) = 13 \), what is \( h(1) \) given \( h(x) = g(g(1)) \)?


  h(1) = 13
• What is the domain of the function \( g \)?


  [-7, 4]
• What is the range of the function \( g \)?


  [-5, 4]
• On which intervals does the function \( g \) increase?


  (-7, -4) \cup (2, 4)
• On which intervals is the function \( g \) concave down?


  (-7, -3)
• How do you calculate the average rate of change (AROC) of \( g \) on the interval (-7, -4)?


  AROC = \frac{g(-4) - g(-7)}{-4 - (-7)}
• Why is the rate of change of \( g \) at \( x = -4.2 \) less than at \( x = -6 \)?


  Because the graph is less steep at \( x = -4.2 \).
• What are the real zeros of the function \( h(x) = 3x^3 - 4x^2 + 1 \)?


  x = -0.434, x = 1, x = 0.768
• What does it mean for \( h(x) \) to have a point of inflection at \( x = -4 \)?


  It means the concavity of the graph changes at that point.
• What is the end behavior of \( h(x) \) as \( x \) decreases without bound?


  \(\lim_{x \to -\infty} h(x) = -\infty\)
• What must be true about the graph of \( h \) if it has a point of inflection at \( x = -4 \)?


  The graph of the rate of change of \( h \) crosses the x-axis at \( x = -4 \).
• What is the local maximum value of \( f \) at \( x = -5 \)?


  3
• What is the minimum degree of the polynomial function \( g \) with zeros at \( x = -1 - 6i, x = 3, x = 8 + \sqrt{11} \)?


  5
• What does it mean for a polynomial function to have rational coefficients?


  It means all coefficients of the polynomial are rational numbers.
• How can you determine if the degree of the function \( h(x) \) is even or odd?


  By observing the end behavior of the graph.
• What is the y-intercept of the function \( f(x) = 2(x - 3)(x - 1)(x + 4) \)?


  72
• What does it mean for \( f(x) \) to be concave up on an interval?


  It means the graph of \( f(x) \) is curving upwards on that interval.
• What is the average rate of change of \( f(x) \) on the interval [1, 3]?


  AROC = \frac{f(3) - f(1)}{3 - 1}
• What are the zeros of the function \( f(x) \) given its factors?


  x = -3, x = 2
• What does it mean for a rational function to have a vertical asymptote?


  It means the function approaches infinity or negative infinity as it approaches a certain x-value.
• What is the significance of the left and right limit statements as \( x \) approaches 1 for the function \( g(x) \)?


  They indicate the behavior of the function as it approaches that point from either side.
• What is the equation of the slant asymptote of \( f(x) = 2x^3 + 8x^2 - 5x \)?


  y = x + 6
• What is the domain of the function \( r(x) = x^3 + 8x^2 + 12x \)?


  All real numbers except \( x = -6, -2, 0 \)
• What does it mean for a function to be undefined at a certain point?


  It means the function does not have a value at that point.
• What can be inferred if the graph of \( f \) is negative for \( x < 1 \) and positive for \( x > 6 \)?


  There must be at least one zero between \( x = 1 \) and \( x = 6 \).
• What must be true for the rational function \( g(x) \) if it has a limit approaching infinity?


  It indicates the function has a vertical asymptote or a hole at that point.
• What are the criteria for sketching a polynomial function \( f \)?


  • Leading coefficient is negative
  • \( (x + 3) \) is a factor of \( f(x) \)
  • \( f(x) \) has an odd degree
  • \( f(x) \) has an x-intercept at \( x = 5 \) with multiplicity 3
• What are the characteristics of the polynomial function \( g \) with rational coefficients and zeros at \( x = -1 - 6i, x = 3, x = 8 + \sqrt{11} \)?


  • Minimum degree is 5
  • Must include complex conjugate \( -1 + 6i \)
  • Must include rational zeros
• What is the significance of the average rate of change (AROC) in determining concavity?


  • AROC indicates the slope of the secant line between two points.
  • If AROC is increasing, the function is concave up.
  • If AROC is decreasing, the function is concave down.
• What are the implications of the end behavior of a rational function?


  • Describes the behavior of the function as \( x \) approaches infinity or negative infinity.
  • Helps identify horizontal and vertical asymptotes.
  • Indicates the overall trend of the function.
• What is the relationship between the degree of a polynomial and its end behavior?


  • Even degree: both ends go in the same direction.
  • Odd degree: ends go in opposite directions.
  • Leading coefficient determines the direction of the ends.
• What is the value of \( f(2) \) in the expression \( 23 + 822 - 5 \)?


  It is not explicitly calculated in the material.
• What is the slant asymptote of the function given in the study material?


  The slant asymptote is \( y = x + 6 \).
• What is the answer choice for the existence of a slant asymptote for the function \( f \)?


  1. \( y = x + 6 \)
• What is the domain of the function \( r(x) = x^3 + 8x^2 + 12x \)?


  The domain is all real numbers \( x \) where \( x \neq -6, x \neq -2, x \neq 0 \).
• What is the correct answer choice for the domain of \( r(x) \)?


  1. all real numbers \( x \) where \( x \neq -6, x \neq -2, x \neq 0 \)
• What can be inferred about the rational function \( f \) based on the given table?


  • \( f \) has exactly one real zero.
  • The graph of \( f \) has a maximum at \( x = 1 \).
  • The graph of \( f \) has either a vertical asymptote or a hole at \( x = 6 \).
• Which answer choices are true for the rational function \( f \)?


  B, C, E