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College Algebra: Functions and Graphs Review

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Termini in questo insieme (20)

  • What is the definition of a function?
    A function assigns exactly one output value for each input value.
  • How do you determine if a graph represents a function?
    Use the vertical line test: if any vertical line intersects the graph more than once, it is not a function.
  • What is the domain of a function?
    The domain is the set of all possible input values (x-values) for which the function is defined.
  • What is the range of a function?
    The range is the set of all possible output values (y-values) of the function.
  • How do you find the slope of a line given two points?
    Slope \(m\) is change in y over change in x: \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
  • What is the slope-intercept form of a linear equation?
    The form is \(y=mx+b\), where m is slope and b is the y-intercept.
  • How do you find the x-intercept of a function?
    Set \(y=0\) and solve for \(x\).
  • How do you find the y-intercept of a function?
    Set \(x=0\) and solve for \(y\).
  • What is the general form of a quadratic function?
    A quadratic function is \(f(x)=ax^2+bx+c\) where a, b, and c are constants and a ≠ 0.
  • How do you find the vertex of a parabola given by \(f(x)=ax^2+bx+c\)?
    The vertex is at \(\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\).
  • What does the sign of a in a quadratic function indicate?
    If a > 0, the parabola opens upward; if a < 0, it opens downward.
  • How do you determine if a function is increasing or decreasing from its graph?
    A function is increasing where the graph rises as x increases, and decreasing where it falls as x increases.
  • What is the formula for the midpoint between two points \((x_1,y_1) and (x_2,y_2)\)?
    Midpoint is \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\).
  • How do you write the equation of a line given a point and slope?
    Use point-slope form: \(y - y_1 = m(x - x_1)\).
  • What is the difference between a relation and a function?
    A relation is any set of ordered pairs; a function has exactly one output for each input.
  • How do you find the inverse of a function?
    Swap \(x\) and \(y\) in the equation and solve for \(y\).
  • What is the horizontal line test used for?
    To determine if a function has an inverse that is also a function.
  • How do you graph a function with a vertical asymptote?
    The graph approaches but never touches the vertical line where the function is undefined.
  • What is the effect of adding a constant to a function, \(f(x)+k\)?
    The graph shifts vertically by k units.
  • What is the effect of replacing \(x\) by \(x-h\) in a function?
    The graph shifts h units to the right.