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# Graphs of Linear Functions

## Graph f(x) = ax +b

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Practice Graphs of Linear Functions

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Use Function Notation to Graph Functions

Therese has just decided to plant a cherry tree in the backyard of her summer cottage. A worker at Farmer Clem’s sold her one that was 30 inches tall. He also sold her the name of a fertilizer that would make the tree grow at a rate of 3 inches a day. How tall will the cherry tree be at the end of three weeks?

Therese does not want to wait for the tree to grow to measure its height. How can she use a graph to predict the height of the tree?

In this concept, you will learn to use function notation to graph functions.

### Graphing Functions

The equation of a straight line can be written in slope-intercept form such that (x,y)\begin{align*}(x, y)\end{align*} are the coordinates of a point on the line, ‘m\begin{align*}m\end{align*}’ is the slope of the line and ‘b\begin{align*}b\end{align*}’ is the y\begin{align*}y\end{align*}-intercept of the line. The equation is y=mx+b\begin{align*}y=mx +b\end{align*}. For this function, the y\begin{align*}y\end{align*}-value can be represented using function notation f(x)\begin{align*}f(x)\end{align*}. When function notation is used to name a function, then the function will be expressed as:

f(x)=mx+b\begin{align*}f(x) = mx + b\end{align*}

When the function expressed in function notation is used for graphing, the y\begin{align*}y\end{align*}-axis should be labeled as the f(x)\begin{align*}f(x)\end{align*} axis. The dependent variable is ‘y\begin{align*}y\end{align*}’ f(x)\begin{align*}f(x)\end{align*} and the independent variable is ‘x\begin{align*}x\end{align*}’. This means that the value for the independent variable is the ‘input’ number and the value for the dependent variable is the ‘output’ number. The value of ‘y\begin{align*}y\end{align*}’ depends upon the value of ‘x\begin{align*}x\end{align*}’.

Let’s look at an example.

Rewrite the following graph using function notation and graph the new function on a Cartesian grid.

y=23x+6\begin{align*}y = -\frac{2}{3} x +6\end{align*}

First, replace the ‘y\begin{align*}y\end{align*}’ with the function notation f(x)\begin{align*}f(x)\end{align*}.

yf(x)==23x+623x+6\begin{align*}\begin{array}{rcl} y &=& -\frac{2}{3} x +6 \\ f(x) &=& - \frac{2}{3} x +6 \end{array}\end{align*}

Next, plot the function on the Cartesian grid. Plot the point (0,6)\begin{align*}(0, 6)\end{align*} on the f(x)\begin{align*}f(x)\end{align*} axis. From this point move horizontally 3 units right, downward 2 units and plot a point.

f(x)=23x+6\begin{align*}f(x) = - \frac{2}{3} x +6\end{align*}

Then, join the two plotted points with a straight line.

A linear function can also be used to represent the ordered pairs given in a t-table. Joey works for a call center where he is paid 30.00 a day plus a fixed bonus for each new internet contract he sells. The following t-table shows the number of internet contracts he sold for the first five days he worked and the amount of money he earned.  x\begin{align*}x\end{align*} (contracts sold) y\begin{align*}y\end{align*} (money earned) 10 77.50 15 101.25 20 125.00 25 148.75 30 172.50

First, choose and name two ordered pairs from the table and use them to calculate the equation of the line.

(x1,20,y1125) and (x2,25,y2148.75)\begin{align*}\begin{pmatrix} x_1, & y_1 \\ 20, & 125 \end{pmatrix} \ \text{and} \ \begin{pmatrix} x_2, & y_2 \\ 25, & 148.75 \end{pmatrix}\end{align*}

Next, using the two named points substitute the values into the formula to calculate the slope of the line.

mm==y2y1x2x1148.75125.002520\begin{align*}\begin{array}{rcl} m &=& \frac{y_2 - y_1}{x_2 - x_1} \\ m &=& \frac{148.75 - 125.00}{25-20} \end{array}\end{align*}

Next, evaluate the formula to determine the value of the slope ‘m\begin{align*}m\end{align*}’.

\begin{align*}\begin{array}{rcl} m &=& \frac{23.75}{5} \\ m &=& 4.75 \end{array}\end{align*}

Next, substitute into the equation of a line written in slope-intercept form, the value of the slope.

\begin{align*}\begin{array}{rcl} y &=& mx +b \\ y &=& 4.75 x +b \end{array}\end{align*}

Next, substitute the value of the \begin{align*}y\end{align*}-intercept into the equation. The value of ‘\begin{align*}b\end{align*}’ is 30.00 per day.

\begin{align*}\begin{array}{rcl} y &=& mx +b \\ y &=& 4.75 x +b \\ y &=& 4.75 x +30 \end{array}\end{align*}

Then, express the equation of the line using function notation and graph it.

\begin{align*}\begin{array}{rcl} y &=& 4.75 x +30 \\ f(x) &=& 4.75 x +30 \end{array}\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Therese and the cherry tree. She wants to show the height of the tree using a graph. How can Therese do this?

Therese can use a function expressed in function notation and plot the function on a Cartesian grid.

First, write down the given information.

\begin{align*}y\end{align*} = height of the tree at the end of three weeks

\begin{align*}m\end{align*} = rate the tree grows each day \begin{align*}\rightarrow \frac{3 \ \text{inches}}{1 \ \text{day}} = 3\end{align*}

\begin{align*}x\end{align*} = number of days

\begin{align*}b\end{align*} = height of tree when purchased \begin{align*}\rightarrow 30 \ \text{inches}\end{align*}

Next, write an equation, in slope-intercept form, to represent the given information.

\begin{align*}\begin{array}{rcl} y &=& mx +b \\ y &=& 3x+30 \end{array}\end{align*}

Next, express the equation using function notation. Replace ‘\begin{align*}y\end{align*}’ using \begin{align*}f(x)\end{align*} notation.

\begin{align*}\begin{array}{rcl} y &=& 3x+30 \\ f(x) &=& 3x+30 \end{array}\end{align*}

Then, graph the function

\begin{align*}f(x) = 3x+30\end{align*}

From the graph it can be seen that the height of the tree at the end of three weeks will be slightly greater than 90 inches. Remember three weeks is equivalent to 21 days.

#### Example 2

Rewrite the linear function using function notation and graph the line.

\begin{align*}y = 3x+2\end{align*}

First, replace the ‘\begin{align*}y\end{align*}’ using function notation \begin{align*}f(x)\end{align*}.

\begin{align*}\begin{array}{rcl} y &=& 3x+2 \\\ f(x) &=& 3x+2 \end{array}\end{align*}

Then, graph the function.

\begin{align*}f(x) = 3x+2\end{align*}

#### Example 3

Identify the slope and the \begin{align*}y\end{align*}-intercept for the following linear function.

\begin{align*}y = \frac{8}{9} x - \frac{1}{3}\end{align*}

First, name the form in which the function is written.

Slope-intercept form

Next, write the general form for a function written in slope-intercept form.

\begin{align*}y= mx+b\end{align*}

Then, state the slope and \begin{align*}y\end{align*}-intercept of the line.

\begin{align*}\begin{array}{rcl} y &=& {\color{blue}m}x +{\color{red}b} \\ y &=& \frac{{\color{blue}8}}{{\color{blue}9}} x {\color{red}-}{\color{red}\frac{1}{3}} \end{array}\end{align*}

The answer is \begin{align*}\frac{8}{9}\end{align*} and \begin{align*}-\frac{1}{3}\end{align*}.

The slope of the line is \begin{align*}\frac{8}{9}\end{align*} and the \begin{align*}y\end{align*}-intercept is \begin{align*}-\frac{1}{3}\end{align*}.

#### Example 4

Express the following equation using function notation.

\begin{align*}y = -\frac{3}{4} x +9\end{align*}

First, replace the ‘\begin{align*}y\end{align*}’ with function notation.

\begin{align*}\begin{array}{rcl} y &=& - \frac{3}{4} x +9 \\ f(x) &=& - \frac{3}{4} x +9 \end{array}\end{align*}

The answer is \begin{align*}f(x) = -\frac{3}{4} x +9\end{align*}.

#### Example 5

Identify the slope and the \begin{align*}y\end{align*}-intercept for the following linear function.

\begin{align*}f(x) = \frac{1}{2} x -8\end{align*}

First, name the form in which the function is written.

Slope-intercept form using function notation.

Next, write the general form for a function written in slope-intercept form.

\begin{align*}f(x) = mx+b\end{align*}

Then, state the slope and \begin{align*}y\end{align*}-intercept of the line.

\begin{align*}\begin{array}{rcl} y &=& mx+b \\ f(x) &=& {\color{blue}m} x+{\color{red}b} \\ f(x) &=& \frac{{\color{blue}1}}{{\color{blue}2}} x {\color{red}-8} \end{array} \end{align*}

The answer is \begin{align*}\frac{1}{2}\end{align*} and -8.

The slope of the line is \begin{align*}\frac{1}{2}\end{align*} and the \begin{align*}y\end{align*}-intercept is -8.

### Review

Graph each function.

1. \begin{align*}f(x) = 3x+1\end{align*}

2. \begin{align*}f(x) = 2x+2\end{align*}

3. \begin{align*}f(x) = 5x-1\end{align*}

4. \begin{align*}f(x) = x-3\end{align*}

5. \begin{align*}f(x) = -2x+1\end{align*}

6. \begin{align*}f(x) = -2x-5\end{align*}

7. \begin{align*}f(x) = -4x+9\end{align*}

8. \begin{align*}f(x) = 4x+8\end{align*}

9. \begin{align*}f(x) = x-10\end{align*}

10. \begin{align*}f(x) =2x+6\end{align*}

Use what you have learned to solve each problem.

11. A migrating monarch butterfly travels 1100 miles. If it flies 30 miles per day, the distance \begin{align*}d\end{align*} it still has to travel is a function of days \begin{align*}t\end{align*} it has traveled. Write a function rule for the situation.

12. What is the slope of the situation?

13. A writer gets paid a writer’s fee of $3000 plus$1.50 for each copy of the book that is sold. Create a function rule for this situation.

14. What is the slope of this situation?

15. How many books does the writer need to sell to earn \$10,000 total?

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### Vocabulary Language: English

dependent variable

The dependent variable is the output variable in an equation or function, commonly represented by $y$ or $f(x)$.

independent variable

The independent variable is the input variable in an equation or function, commonly represented by $x$.