Graphing by Translation, Scaling and Reflection

For any function \( f(x) \), if we replace \( x \) with \( x + c \), the graph of \( f(x + c) \) will be the graph of \( f(x) \) shifted horizontally by \( c \) units:

• \( f(x + c) \) is the graph of \( f(x) \) shifted \( c \) units to the left for \( c \gt 0 \).
• \( f(x - c) \) is the graph of \( f(x) \) shifted \( c \) units to the right for \( c \gt 0 \).

Example A (Right Shift): Comparing \( f(x) = x^2 \) and \( g(x) = (x - 2)^2 \), the graph of \( (x - 2)^2 \) is that of \( x^2 \) shifted \( 2 \) units to the right.

\[ f(x) = x^2 \quad \text{vs} \quad g(x) = (x - 2)^2 \]

Example B (Left Shift): Comparing \( f(x) = \sqrt{x} \) and \( g(x) = \sqrt{x + 2} \), the graph of \( \sqrt{x + 2} \) is that of \( \sqrt{x} \) shifted \( 2 \) units to the left.

\[ f(x) = \sqrt{x} \quad \text{vs} \quad g(x) = \sqrt{x + 2} \]

If we add a constant \( c \) to \( f(x) \), the graph of \( f(x) + c \) will be the graph of \( f(x) \) translated vertically:

• If \( c \gt 0 \), the graph is translated up.
• If \( c \lt 0 \), the graph is translated down.

Example A (Shift Up): Comparing \( f(x) = x^2 \) and \( g(x) = x^2 + 2 \).

\[ f(x) = x^2 \quad \text{vs} \quad g(x) = x^2 + 2 \]

Example B (Shift Down): Comparing \( f(x) = x^2 \) and \( g(x) = x^2 - 1 \).

\[ f(x) = x^2 \quad \text{vs} \quad g(x) = x^2 - 1 \]

Taking any function \( f(x) \) and changing \( x \) to \( -x \), the graph of \( f(-x) \) will be the graph of \( f(x) \) reflected across the \( y \)-axis.

\[ f(x) = \sqrt{x} \quad \text{vs} \quad g(x) = \sqrt{-x} \]

Taking any function \( f(x) \) and negating it to \( -f(x) \), the graph of \( -f(x) \) will be the graph of \( f(x) \) reflected across the \( x \)-axis.

\[ f(x) = |x| \quad \text{vs} \quad g(x) = -|x| \]

For any function \( f(x) \), the graph of \( kf(x) \) (with \( k \gt 0 \)) will be:

• Expanded (stretched) vertically if \( k \gt 1 \).
• Compressed vertically if \( 0 \lt k \lt 1 \).

In the graph below, the blue curve represents \( f(x) = |x| \) and the red curve represents \( g(x) = 2|x| \) (vertically stretched by a factor of 2).