Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Vertical Shifts

1. Use the Desmos graphing tool or technology outside the course. Graph the following linear functions.

$f (x) = x + 4$
$f (x) = x + 2$
$f (x) = x$
$f (x) = x - 2$
$f (x) = x - 4$

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Compare your work:

A graph showing five diagonal lines, each representing a linear function: f of x equals x plus 4 is purple, fof x equals x plus 2 is orange, f of x equals x is blue, f of x equals x minus 2 is red, and f of x equals x minus 4 is green.

2. What are the slopes of each line?

The slope of all the lines equals 1.

The parent function of a linear function is $f (x) = x$ .

The $y$ -intercept is (0,0).
The slope is 1.

3. How can the parent function, $f (x) = x$ be shifted to overlap $f (x) = x + 2$ ?

Compare your answer:

Shift up 2.

4. How can the parent function, $f (x) = x$ , be shifted to overlap $f (x) = x - 4$ ?

Compare your answer:

Shift down 4.

In $f (x) = m x + b$ , the $b$ acts as the vertical shift. This is a type of transformation to the graph of $f (x) = x$ .

Vertical Shift of a Function

A vertical shift “transforms” the parent function into another function by moving the graph up or down $d$ units.

Vertical Shift → $f (x) + d$

If the $d$ value is positive, the graph of the function shifts up. If the $d$ value is negative, the graph of the function shifts down.

Self Check

What transformation takes place from the graph of $y=4x-3$ to $y=4x +5$ ?

Left 8
Up 8
Right 8
Down 8

Additional Resources

Vertical Shifts and $y$ -Intercepts

In the equation $f (x) = m x + b$ :

$b$ is the $y$ -intercept of the graph and indicates the point $(0, b)$ at which the graph crosses the $y$ -axis.

$m$ is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points.

This graph shows how to calculate the rise over run for the slope on an x, y coordinate plane. The x-axis runs from negative 2 to 7. The y-axis runs from negative 2 to 5. The line extends right and upward from point (0,1), which is the y-intercepts. A dotted line extends two units to the right from point (0, 1) and is labeled Run equals 2. The same dotted line extends upwards one unit and is labeled Rise equals 1.

The $y$ -intercept tells where the parent function, $f (x) = x$ , has shifted up or down.

Example 1

Tell how $f (x) = x + 7$ has shifted from the parent function $f (x) = x$ .

Solution Since $b = 7$ , the $y$ -intercept is +7, so the graph shifts up 7 from the parent function.

Example 2

Tell how $f (x) = x - 9$ has shifted from the parent function $f (x) = x$ .

Solution Since $b = - 9$ , the $y$ -intercept is –9, so the graph shifts down 9 from the parent function.

Example 3

Tell how $f (x) = 2 x + 1$ has shifted from $f (x) = 2 x + 5$ .

Solution Since $b = 1$ , the $y$ -intercept is 1. This point is 4 vertical units below the $y$ -intercepts of the original line that had a $y$ -intercept of 5. So, the graph shifts down 4 units from $f (x) = 2 x + 5$ to arrive at $f (x) = 2 x + 1$ .