Intermediate Algebra: Functions and Graphs

Subsection Glossary

• function

• input variable

• output variable

• cube root

• absolute value

• proportional

• direct variation

• inverse variation

• constant of variation

• concavity

• scaling

• horizontal asymptote

• vertical asymptote

Subsection Key Concepts

1. A function can be described in words, by a table, by a graph, or by an equation.

2. Function Notation.

   
3. Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.
4. The point \((a, b)\) lies on the graph of the function \(f\) if and only if \(f(a)=b\text{.}\)
5. Each point on the graph of the function \(f\) has coordinates \((x, f(x))\) for some value of \(x\text{.}\)

6. The Vertical Line Test.

   A graph represents a function if and only if every vertical line intersects the graph in at most one point.

7. We can use a graphical technique to solve equations and inequalities.
8. \(b\) is the cube root of \(a\) if \(b\) cubed equals \(a\text{.}\) In symbols, we write
   \begin{equation*} \blert{b=\sqrt[3]{a}~~~~\text{if}~~~~b^3=a} \end{equation*}

9. Absolute Value.

   The absolute value of \(x\) is defined by

   \begin{equation*} \abs{x} = \begin{cases} x \amp \text{if } x\ge 0\\ -x \amp \text{if } x\lt 0 \end{cases} \end{equation*}
10. Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.
11. The maximum or minimum of a quadratic function occurs at the vertex.

12. Eight basic functions and their graphs are important in applications:

    \begin{gather*} f(x)=x~~~~~~f(x)=\abs{x}~~~~~~f(x)=x^2~~~~~~f(x)=x^3\\ f(x)=\sqrt{x}~~~~~f(x)=\sqrt[3]{x}~~~~~~f(x)=\dfrac{1}{x}~~~~~~f(x)=\dfrac{1}{x^2} \end{gather*}
13. Two variables are directly proportional if the ratios of their corresponding values are always equal.

14. Direct Variation.

    \(y\) varies directly with \(x\) if

    \begin{equation*} y = kx \end{equation*}

    where \(k\) is a positive constant called the constant of variation.

15. Direct variation defines a linear function of the form
    \begin{equation*} y = f (x) = kx \end{equation*}
    The positive constant \(k\) in the equation \(y = kx\) is just the slope of the graph.
16. Direct variation has the following scaling property: increasing \(x\) by any factor causes \(y\) to increase by the same factor.

17. Direct Variation with a Power.

    \(y\) varies directly with a power of \(x\) if

    \begin{equation*} y = kx^n \end{equation*}

    where \(k\) and \(n\) are positive constants.

18. If the ratio \(\dfrac{y}{x^n}\) is constant, then \(y\) varies directly with \(x^n\text{.}\)

19. Inverse Variation.

    \(y\) varies inversely with \(x\) if

    \begin{equation*} y = \dfrac{k}{x}\text{, }x \ne 0 \end{equation*}

    where \(k\) is a positive constant.

20. Inverse Variation with a Power.

    \(y\) varies inversely with \(x^n\) if

    \begin{equation*} y = \frac{k}{x^n}\text{, }x \ne 0 \end{equation*}

    where \(k\) and \(n\) are positive constants.

21. If the product \(~yx^n~\) is constant and \(n\) is positive, then \(y\) varies inversely with \(x^n\text{.}\)
22. A graph that bends upward is called concave up, and one that bends downward is concave down.