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Algebra 1

4.11.2 Vertical Shifts

Algebra 14.11.2 Vertical Shifts

4.11.2 Vertical Shifts

4.11.2 • Vertical Shifts

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 + 4
  • f ( x ) = x + 2 f ( x ) = x + 2
  • f ( x ) = x f ( x ) = x
  • f ( x ) = x − 2 f ( x ) = x − 2
  • f ( x ) = x − 4 f ( x ) = x − 4
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2. What are the slopes of each line?

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

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

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

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

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

Vertical Shift → f ( x ) + d f ( x ) + d

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

Self Check

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

  1. Left 8
  2. Up 8
  3. Right 8
  4. Down 8

Additional Resources

Vertical Shifts and y y -Intercepts

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

  • b b is the y y -intercept of the graph and indicates the point ( 0 , b ) ( 0 , b ) at which the graph crosses the y y -axis.
  • m m is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points.

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

Example 1

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

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

Example 2

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

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

Example 3

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

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

Vertical Shift of a Function

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

Vertical Shift → f ( x ) + d f ( x ) + d

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

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