PC - Function Graphs Lesson

Function Graphs

Standard Parent Graphs

A parent function is the simplest form of a function. It is the template from which variations or transformations are generated.

Absolute Value, Polynomial, and Greatest Integer

Parent functions for polynomial, absolute value, and greatest integer functions are:

Linear: y = x
Absolute value: $LaTeX: y=\left|x\right|$ $y=\left|x\right|$
Quadratic: $y=x^2$
Cubic: $y=x^3$
Greatest Integer: $LaTeX: y=int\left(x\right)\:or\:y=\left[\left[x\right]\right]$ $y=int\left(x\right)\:or\:y=\left[\left[x\right]\right]$

Click HERE to view the video below to review parent graphs and characteristics of elementary functions. Links to an external site.

Square Root, Rational, Exponential, and Logarithmic

Parent functions for square root, rational, exponential, and logarithmic are:

Square Root: $LaTeX: y=\sqrt[]{x}$ $y=\sqrt[]{x}$
Rational: $LaTeX: y=\frac{1}{x}$ $y=\frac{1}{x}$
Exponential Growth: $LaTeX: y=b^x,\:b>1$ $y=b^x,\:b>1$
Exponential Decay: $LaTeX: y=b^x,\:0<b<1$ $y=b^x,\:0<b<1$
Logarithmic: $LaTeX: y=\log x$ $y=\log x$
Natural Logarithmic: $LaTeX: y=\ln x$ $y=\ln x$

Standard logarithmic functions use either base 10 for y = log x or base e for y = ln x.

Click HERE to view the video below to review parent graphs and characteristics of square root, rational, exponential, and logarithmic functions. Links to an external site.

Recall that a rational function f is a ratio of two polynomials, equation image indicator $LaTeX: f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ $f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ where P and Q are polynomials. The domain of f(x) consists of all values of x for which $LaTeX: Q\left(x\right)\ne0$ $Q\left(x\right)\ne0$ . As noted above, the simplest rational function is the reciprocal function f(x) = 1/x.

Transformations of Standard Parent Graphs

View the videos below to investigate transformations involving vertical and horizontal shifts as well as reflections across an axis.

Piecewise

A piecewise-defined function is a function whose definition changes depending on the value of the independent variable. Its graph appears as distinct pieces. View the video below to learn how piecewise functions are defined and graphed.

Symmetry with Respect to an Axis and the Origin

The graph of an equation is symmetric with respect to the y-axis if replacing x by -x produces an equivalent equation.

image of graph with parabola opening up, equation of y=0; 5x^2-3

The graph of an equation is symmetric with respect to the x-axis if replacing y by -y produces an equivalent equation.

image of graph with parabola opening right

The graph of an equation in x and y is symmetric with respect to origin if replacing x by -x and y by -y produces an equivalent equation.

graph with curved line going up then down then up again

Function Graphs Practice

Determine whether the following functions are symmetric with respect to the x-axis, y-axis, origin, or none of these.

Drag the labels from the bottom to the correct slots.

Function Graphs: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of function graphs.

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