4.11.3 Horizontal Shifts

Horizontal Shifts

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

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

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In linear functions, when a value is added to or subtracted from the input value, or the $x$-values, the graph acts as if a horizontal shift is applied. This is another type of transformation to the graph.

If the $c$ value is positive, the graph of the function shifts right. If the $c$ value is negative, the graph of the function shifts left.

But, be careful! The sign of the $c$ value is preceded by subtraction in the equation. When the equation is simplified, the sign of the $c$ value is changed.

Horizontal Shifts and $x$-intercepts

In addition to being shifted vertically, graphs can also be shifted to the right or left. For linear functions, this is sometimes hard to identify because a line that is shifted horizontally may appear to have a vertical shift.

Let’s shift the parent function, $f(x)=x$, horizontally to the left 7 units.

If the line shifts horizontally 7 units to the left, the $x$-intercept is shifted 7 units to the left, as well.

The graph of the parent function, $f(x)=x$, is represented by the dashed orange line. The graph of the solid blue line represents the line that we horizontally shift 7 units to the left.

To determine the equation of the shifted line, we use the slope of 1 and $(-7,0)$ as a point on the line.

Substituting these values into point-slope form, we get:

$$\begin{gathered} (y-y_1)=m(x-x_1) \\ (y-0)=1(x-(-7)) \\ y=1(x+7) \end{gathered}$$

So, the equation of a line that has been shifted 7 units to the left from the parent function, $f(x)=x$, is $y=x+7$

But, doesn’t it look like the function $f(x)=x+7$ was shifted up 7 units, too? This is because $f(x)=x+7$ is equivalent to $f(x)=1(x+7)$. And this is what makes horizontal shifts difficult to see.

Example 1

How has the function $f(x)=3(x-4)$ been transformed from $f(x)=3x$?

Solution

To answer this question, first consider how the x-values, including the $x$-intercept, are being changed. The function says to subtract 4 from each x-value. This means the function is being shifted 4 units.

Next, we need to determine which direction the x-values are being shifted. Let’s look at the graphs.

The solid blue line represents the graph of $f(x)=3(x-4)$. The dashed orange line represents the graph of $f(x)=x$.

On the graph, we can look at specific points on the parent function like the $x$-intercept (0, 0) and (1, 3). The transformed line is to the right of our parent function. To determine how far to the right the line has been shifted, we count the number of horizontal units between the lines.

• We see that (0, 0) corresponds to the point (4, 0) on the transformed line.
• The point (1, 3) corresponds to the point (5, 3) on the transformed line.
• Each point on the parent function has been shifted 4 units to the right.

Therefore, we conclude the line representing $f(x)=3(x-4)$ appears to have been shifted 4 units to the right from the function $f(x)=3x$.

Example 2

Use the $x$-intercept to describe how the function $f(x)=x+9$ has been transformed from the parent function $f(x)=x$.

Solution

We know the shift is 9 units, but we need to consider which direction the $x$-intercept has been moved.

The orange dashed line represents the parent function, $f(x)=x$ and the solid blue line represents the shifted function.

Carefully looking at the $x$-intercepts, we see the line representing $f(x)=x+9$ has been shifted 9 units to the left from the parent function, $f(x)=x$.

Often the horizontal shift is denoted using parentheses to explicitly identify how the values of the independent variable, or domain, are being changed.

Example 3

Describe how the function $f(x)=½(x+4)$ is horizontally transformed to $g(x)=½(x-2)$.

Solution

Let’s rewrite the equations in point-slope form, $(y-y_1)=m(x-x_1)$:

$$\begin{array}{rl}f(x)& =\frac{1}{2}(x+4)\\ (y-0)& =\frac{1}{2}(x+4)\\ (y-0)& =\frac{1}{2}(x-(-4))\end{array}$$

So, the $x$-intercept is $(-4,0)$.

$$\begin{array}{rl}g(x)& =\frac{1}{2}(x-2)\\ (y-0)& =\frac{1}{2}(x-2)\end{array}$$

So, the $x$-intercept is $(2,0)$.

The $x$-intercept of $f(x)$ starts at $(-4,0)$ and moves to $(2,0)$, so this represents a shift of 6 units to the right.

We can also check this using graphs.

Examining the $x$-intercept for $f(x)=½(x+4)$, on the orange dashed line, we see that $(-4,0)$ gets shifted 6 units to the right and lands at $(2,0)$ on $g(x)=½(x-2)$, represented by the blue solid line.

So, we can identify horizontal shifts using either graphs or algebraic equations.

If the $c$ value is positive, the graph of the function shifts right. If the $c$ value is negative, the graph of the function shifts left. But, be careful! The sign of the $c$ value is preceded by subtraction in the equation. When the equation is simplified, the sign of the $c$ value is changed.