In the applet, move the m slider and the b slider.

- What happens to the graph when the value of m changes?
- What happens to the graph when the value of b changes?

The slope of a linear function, m, refers to the steepness of the line, and recall we can find the slope using the formula:

\displaystyle m= \dfrac{\text{change in }y}{\text{change in }x}=\dfrac{y_2-y_1}{x_2-x_1}

\bm{m}

slope

\bm{\left(x_1,y_1\right)}

a point on the line

\bm{\left(x_2,y_2\right)}

a second point on the line

The y-intercept, b, represents the point where the function crosses the y-axis.

There are several **function families**that share the same general shape but may have been shifted or reflected in some way. As we examine linear functions, we notice that we can transform the **parent function** f\left(x\right)= x by translating, dilating, or reflecting it vertically to create new functions.

Transformations on the parent function f\left(x\right) = x can be used to graph and write equations.

Determine what transformation has occurred from the parent function f(x) = x.

Worked Solution

Consider the graph of the parent function f\left(x\right) = x. Graph the function after a vertical stretch of a factor of 3, and a vertical translation of -2 units. Write the equation of the transformed line.

Worked Solution

By first identifying the slope and y-intercept, describe the transformations of the following lines from the parent line f\left(x\right) = x.

a

g\left(x\right) = 5x

Worked Solution

b

g\left(x\right) = x - 2

Worked Solution

c

g\left(x\right) = -\dfrac{2}{3}x + 4

Worked Solution

The drama club is raising money for a field trip to see a Broadway musical. To raise the money, they plan to set up a face-painting stand during the high-school football game, and charge \$4 per person. The function {R\left(x\right)=4x} represents their revenue in dollars where x represents the number of faces painted.

a

The club members spent \$45 on face-painting supplies. Write the function P\left(x\right) that represents their profit.

Worked Solution

b

Describe the transformation applied to R\left(x\right) to get P\left(x\right).

Worked Solution

c

The drama team realizes that they will need to paint 11 faces to break even at the current rate, so they decide to increase the cost per person to \$8. The function N\left(x\right)=8x represents their new revenue function.

Describe the transformation from the original revenue function, R\left(x\right), to the new revenue function, N\left(x\right).

Worked Solution

Use transformations from the parent function f(x) = x to graph the function f(x) = 3x-4.

Worked Solution

Idea summary

The slope of a line is represented by

\displaystyle m= \dfrac{\text{change in }y}{\text{change in }x}=\dfrac{y_2-y_1}{x_2-x_1}

\bm{m}

slope

\bm{\left(x_1,\,y_1\right)}

a point on the line

\bm{\left(x_2,\,y_2\right)}

a second point on the line

The parent function f\left(x\right) = x can be transformed to write new functions with changes to the slope and y-intercept.

A vertical shift represented by f\left(x\right) + k will translate the graph of a total of k units up if k \gt 0 or k units down if k \lt 0

A horizontal shift represented by f\left(x +k\right) will translate the graph of a total of k units right if k \gt 0 or k units left if k \lt 0

A transformation of a vertical dilation kf\left(x\right) will stretch a graph's slope if k \gt 1 or compress the graph if 0 \lt k \lt 1

A vertical reflection of k will change the sign of the slope.