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# 1.13: Graphs of Functions based on Rules

Difficulty Level: At Grade Created by: CK-12
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Practice Graphs of Functions based on Rules

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What if you were given a function rule like \begin{align*}f(x) = \sqrt{2x^2 + 1}\end{align*}. How could you graph that function? After completing this Concept, you'll be able to create a table of values to graph functions like this one in the coordinate plane.

### Try This

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/function-graphs/. 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: \begin{align*}f(x) = |x - 2|\end{align*}

Solution

Make a table of values. Pick a variety of negative and positive values for \begin{align*}x\end{align*}. Use the function rule to find the value of \begin{align*}y\end{align*} for each value of \begin{align*}x\end{align*}. Then, graph each of the coordinate points.

\begin{align*}x\end{align*} \begin{align*}y = f(x) = \mid x - 2 \mid\end{align*}
-4 \begin{align*}\mid -4 - 2 \mid = \mid -6 \mid = 6\end{align*}
-3 \begin{align*}\mid -3 - 2 \mid = \mid -5 \mid = 5\end{align*}
-2 \begin{align*}\mid -2 - 2 \mid = \mid -4 \mid = 4\end{align*}
-1 \begin{align*}\mid -1 - 2 \mid = \mid -3 \mid = 3\end{align*}
0 \begin{align*}\mid 0 - 2 \mid = \mid -2 \mid = 2 \end{align*}
1 \begin{align*}\mid 1 - 2 \mid = \mid -1 \mid = 1\end{align*}
2 \begin{align*}\mid 2 - 2 \mid = \mid 0 \mid = 0\end{align*}
3 \begin{align*}\mid 3 - 2 \mid = \mid 1 \mid = 1\end{align*}
4 \begin{align*}\mid 4 - 2 \mid = \mid 2 \mid = 2\end{align*}
5 \begin{align*}\mid 5 - 2 \mid = \mid 3 \mid = 3\end{align*}
6 \begin{align*}\mid 6 - 2 \mid = \mid 4 \mid = 4\end{align*}
7 \begin{align*}\mid 7 - 2 \mid = \mid 5 \mid = 5\end{align*}
8 \begin{align*}\mid 8 - 2 \mid = \mid 6 \mid = 6\end{align*}

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: \begin{align*}f(x)=\sqrt{x}\end{align*}

Solution

Make a table of values. We know \begin{align*}x\end{align*} can’t be negative because we can't take the square root of a negative number. The domain is all positive real numbers, so we pick a variety of positive integer values for \begin{align*}x\end{align*}. Use the function rule to find the value of \begin{align*}y\end{align*} for each value of \begin{align*}x\end{align*}.

\begin{align*}x\end{align*} \begin{align*}y = f(x) = \sqrt{x}\end{align*}
0 \begin{align*}\sqrt{0} = 0\end{align*}
1 \begin{align*}\sqrt{1} = 1\end{align*}
2 \begin{align*}\sqrt{2} \approx 1.41\end{align*}
3 \begin{align*}\sqrt{3} \approx 1.73\end{align*}
4 \begin{align*}\sqrt{4} = 2\end{align*}
5 \begin{align*}\sqrt{5} \approx 2.24\end{align*}
6 \begin{align*}\sqrt{6} \approx 2.45\end{align*}
7 \begin{align*}\sqrt{7} \approx 2.65\end{align*}
8 \begin{align*}\sqrt{8} \approx 2.83\end{align*}
9 \begin{align*}\sqrt{9} = 3\end{align*}

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 \begin{align*}x\end{align*} because it doesn’t make sense to have negative weight. We pick a variety of positive values for \begin{align*}x\end{align*}, making sure to include some decimal values because prices can be decimals too. Then we use the function rule to find the value of \begin{align*}y\end{align*} for each value of \begin{align*}x\end{align*}.

\begin{align*}& x \quad 0 \quad 0.2 \quad 0.5 \quad 0.8 \quad 1 \quad 1.2 \quad 1.5 \quad 1.8 \quad 2 \quad 2.2 \quad 2.5 \quad 2.8 \quad 3 \quad 3.2 \quad 3.5\\ & y \quad 0 \quad 41 \quad \ 41 \quad \ 41 \quad 41 \quad 58 \quad 58 \quad \ 58 \quad 58 \quad 75 \quad 75 \quad \ 75 \quad 75 \quad 92 \quad 92\end{align*}

Watch this video for help with the Examples above.

### 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 \begin{align*}x-\end{align*}axis and the vertical line is called the \begin{align*}y-\end{align*}axis.

The \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}axes divide the coordinate plane into four quadrants. The quadrants are numbered counter-clockwise starting from the upper right, so the plotted point for (a) is in the first quadrant, (b) is in the second quadrant, (c) is in the fourth quadrant, and (d) is in the third quadrant.

### Guided Practice

Graph the following function: \begin{align*}f(x)=\sqrt{x^2}\end{align*}

Solution

Make a table of values. Even though \begin{align*}x\end{align*} can’t be negative inside the square root, because we are squaring \begin{align*}x\end{align*} first, the domain is all real numbers. So we integer values for \begin{align*}x\end{align*} which are on either side of zero. Use the function rule to find the value of \begin{align*}y\end{align*} for each value of \begin{align*}x\end{align*}.

\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.

1. Vanson spends $20 a month on his cat. 2. Brandon is a member of a movie club. He pays a$50 annual membership and \$8 per movie.
3. \begin{align*}f(x) = (x - 2)^2\end{align*}
4. \begin{align*}f(x) = 3.2^x\end{align*}
5. \begin{align*}f(t) = 27t-t^2\end{align*}
6. \begin{align*}f(w) = \frac{w}{4}+5\end{align*}
7. \begin{align*}f(x) = t+2t^2+3t^3\end{align*}
8. \begin{align*}f(x) = (x-1)(x+3)\end{align*}
9. \begin{align*}f(x) = \frac{x}{3}+\frac{x^2}{5}\end{align*}
10. \begin{align*}f(x) = \sqrt{2x}\end{align*}

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

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 $x$, there is only one value for $y$.

A quadrant is one-fourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the top-right, and increasing counter-clockwise.

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