1.13: Graphs of Functions based on Rules
What if you were given a function rule like
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CK12 Foundation: 0113S Graph a Function from a Rule
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Once you’ve had some practice graphing functions by hand, you may want to use a graphing calculator to make graphing easier. If you don’t have one, you can also use the applet at http://rechneronline.de/functiongraphs/. Just type a function in the blank and press Enter. You can use the options under Display Properties to zoom in or pan around to different parts of the graph.
Guidance
Of course, we can always make a graph from a function rule, by substituting values in for the variable and getting the corresponding output value.
Example A
Graph the following function:
Solution
Make a table of values. Pick a variety of negative and positive values for



4 

3 

2 

1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

It is wise to work with a lot of values when you begin graphing. As you learn about different types of functions, you will start to only need a few points in the table of values to create an accurate graph.
Example B
Graph the following function:
Solution
Make a table of values. We know



0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Note that the range is all positive real numbers.
Example C
The post office charges 41 cents to send a letter that is one ounce or less and an extra 17 cents for each additional ounce or fraction of an ounce. This rate applies to letters up to 3.5 ounces.
Solution
Make a table of values. We can’t use negative numbers for
Watch this video for help with the Examples above.
CK12 Foundation: Graph a Function from a Rule
Vocabulary
We represent functions graphically by plotting points on a coordinate plane (also sometimes called the Cartesian plane). The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at a point called the origin. The horizontal number line is called the
The
Guided Practice
Graph the following function:
Solution
Make a table of values. Even though
\begin{align*}x\end{align*} 
\begin{align*}y = f(x) = \sqrt{x^2}\end{align*} 

2  \begin{align*}\sqrt{(2)^2} = 2\end{align*} 
1  \begin{align*}\sqrt{(1)^2} = 1\end{align*} 
0  \begin{align*}\sqrt{0^2} = 0\end{align*} 
1  \begin{align*}\sqrt{1^2} = 1\end{align*} 
2  \begin{align*}\sqrt{2^2}=2\end{align*} 
Note that the range is all positive real numbers, and that this looks like an absolute value function.
Practice
Graph the following functions.
 Vanson spends $20 a month on his cat.
 Brandon is a member of a movie club. He pays a $50 annual membership and $8 per movie.
 \begin{align*}f(x) = (x  2)^2\end{align*}
 \begin{align*}f(x) = 3.2^x\end{align*}
 \begin{align*}f(t) = 27tt^2\end{align*}
 \begin{align*}f(w) = \frac{w}{4}+5\end{align*}
 \begin{align*}f(x) = t+2t^2+3t^3\end{align*}
 \begin{align*}f(x) = (x1)(x+3)\end{align*}
 \begin{align*}f(x) = \frac{x}{3}+\frac{x^2}{5}\end{align*}
 \begin{align*}f(x) = \sqrt{2x}\end{align*}
Coordinate Plane
The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.coordinate point
A coordinate point is the description of a location on the coordinate plane. Coordinate points are written in the form (x, y) where x is the horizontal distance from the origin, and y is the vertical distance from the origin.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .quadrant
A quadrant is onefourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the topright, and increasing counterclockwise.Image Attributions
Description
Learning Objectives
Here you'll learn how to graph a function from a given rule.