Skip to main content
Back

College Algebra: Functions, Graphs, and Models

Control buttons has been changed to "navigation" mode.
1/20

Terms in this set (20)

  • What is a function?

    A function is a relation where each input has exactly one output.

  • How do you determine if a graph represents a function?

    Use the vertical line test: if any vertical line crosses the graph more than once, it is not a function.

  • What is a piecewise defined function?

    A piecewise defined function is defined by different expressions over different intervals of the domain.

  • Name common types of functions in the library of functions.

    Common types include linear, quadratic, absolute value, square root, cubic, cube root, and rational functions.

  • What are the basic transformations of functions?

    Transformations include translations (shifts), reflections, stretches, and compressions of the graph.

  • How does adding a constant outside the function affect its graph?

    Adding a constant k outside the function, \(f(x)+k\), shifts the graph vertically by k units.

  • How does adding a constant inside the function argument affect its graph?

    Adding a constant h inside the argument, \(f(x+h)\), shifts the graph horizontally by -h units.

  • What is the general form of a linear function?

    The general form is \(f(x)=mx+b\), where m is the slope and b is the y-intercept.

  • How do you interpret the slope of a linear function?

    The slope represents the rate of change or how much y changes for a unit change in x.

  • What are key properties of linear functions?

    Linear functions have a constant slope, their graphs are straight lines, and they have no curvature.

  • What is the standard form of a quadratic function?

    The standard form is \(f(x)=ax^2+bx+c\), where a, b, and c are constants and a \(\neq\) 0.

  • How does the coefficient a affect the graph of a quadratic function?

    If a > 0, the parabola opens upward; if a < 0, it opens downward. The value of a also affects the width.

  • What is the vertex of a quadratic function?

    The vertex is the highest or lowest point on the parabola, representing the maximum or minimum value.

  • How can you find the vertex of a quadratic function in standard form?

    The vertex's x-coordinate is \(-\frac{b}{2a}\). Substitute this into the function to find the y-coordinate.

  • What is the axis of symmetry in a quadratic function?

    The axis of symmetry is the vertical line through the vertex, given by \(x=-\frac{b}{2a}\).

  • What is the effect of the constant term c in a quadratic function?

    The constant c represents the y-intercept, where the graph crosses the y-axis.

  • What is a quadratic model used for?

    Quadratic models describe situations with a maximum or minimum, such as projectile motion or area optimization.

  • How do you graph a piecewise function?

    Graph each piece on its specified domain interval, using open or closed dots to indicate inclusion or exclusion of endpoints.

  • 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.