Lesson 3.5: Transformation of Functions

New Concept

Function transformations allow us to take a basic 'toolkit' function and modify its graph. By systematically shifting, reflecting, stretching, or compressing the graph, we can create new functions to model a wide variety of real-world scenarios.

What’s next

Next, you'll master each transformation—shifts, reflections, and stretches—through interactive examples and practice cards. Soon, you'll be combining them like a pro.

Property

Given a function $f(x)$, a new function $g(x) = f(x) + k$, where $k$ is a constant, is a vertical shift of the function $f(x)$. All the output values change by $k$ units. If $k$ is positive, the graph will shift up. If $k$ is negative, the graph will shift down. To apply a vertical shift from a table, add the constant $k$ to each output value.

Examples

• The function $f(x) = x^2$ shifted up 4 units becomes $g(x) = x^2 + 4$. The vertex moves from $(0,0)$ to $(0,4)$.

• The function $f(x) = \sqrt{x}$ shifted down 1 unit becomes $h(x) = \sqrt{x} - 1$. The graph's starting point moves from $(0,0)$ to $(0,-1)$.

Property

Given a function $f$, a new function $g(x) = f(x - h)$, where $h$ is a constant, is a horizontal shift of the function $f$. If $h$ is positive, the graph will shift right. If $h$ is negative, the graph will shift left. To apply a horizontal shift from a table, add the value of the shift $h$ to each input value.

Examples

• The function $f(x) = x^3$ shifted right by 2 units becomes $g(x) = (x-2)^3$. The point of inflection at $(0,0)$ moves to $(2,0)$.

• The function $f(x) = |x|$ shifted left by 3 units becomes $h(x) = |x+3|$. The vertex moves from $(0,0)$ to $(-3,0)$.

Property

Given a function $f(x)$, a new function $g(x) = -f(x)$ is a vertical reflection of the function $f(x)$, sometimes called a reflection about the $x$-axis. To apply this, multiply all output values by $-1$.

Given a function $f(x)$, a new function $g(x) = f(-x)$ is a horizontal reflection of the function $f(x)$, sometimes called a reflection about the $y$-axis. To apply this, multiply all input values by $-1$.

Examples

• The graph of $f(x) = x^3$ reflected across the x-axis is $g(x) = -x^3$. The point $(2,8)$ on the original graph becomes $(2,-8)$ on the new graph.

Property

A function is called an even function if for every input $x$, $f(x) = f(-x)$. The graph of an even function is symmetric about the $y$-axis.

A function is called an odd function if for every input $x$, $f(x) = -f(-x)$. The graph of an odd function is symmetric about the origin.

Examples

• The function $f(x) = 3x^4 + 5$ is even because $f(-x) = 3(-x)^4 + 5 = 3x^4 + 5 = f(x)$.

Property

Given a function $f(x)$, a new function $g(x) = a f(x)$, where $a$ is a constant, is a vertical stretch or vertical compression of the function $f(x)$.

• If $a > 1$, then the graph will be stretched.
• If $0 < a < 1$, then the graph will be compressed.
• If $a < 0$, then there will be a combination of a vertical stretch or compression with a vertical reflection.

Examples

• To stretch the graph of $f(x) = x^2$ vertically by a factor of 4, you use $g(x) = 4x^2$. The point $(1,1)$ on the original graph moves to $(1,4)$.

• To compress the graph of $f(x) = |x|$ vertically by a factor of $1/3$, you use $h(x) = \frac{1}{3}|x|$. The point $(3,3)$ on the original graph moves to $(3,1)$.

Property

Given a function $f(x)$, a new function $g(x) = f(bx)$, where $b$ is a constant, is a horizontal stretch or horizontal compression of the function $f(x)$.

• If $b > 1$, then the graph will be compressed by $\frac{1}{b}$.
• If $0 < b < 1$, then the graph will be stretched by $\frac{1}{b}$.
• If $b < 0$, then there will be a combination of a horizontal stretch or compression with a horizontal reflection.

Examples

• To compress the graph of $f(x) = x^2$ horizontally by a factor of $1/2$, use $g(x) = (2x)^2$. The point $(1,1)$ on the original graph moves to $(1/2, 1)$.

• To stretch the graph of $f(x) = \sqrt{x}$ horizontally by a factor of 4, use $h(x) = \sqrt{\frac{1}{4}x}$. The point $(16,4)$ on the original graph moves to $(64,4)$.

Property

When combining vertical transformations written in the form $af(x) + k$, first vertically stretch by $a$ and then vertically shift by $k$.

When combining horizontal transformations written in the form $f(b(x - h))$, first horizontally stretch by $\frac{1}{b}$ and then horizontally shift by $h$.

To handle forms like $f(bx + p)$, factor inside the function to get $f(b(x + \frac{p}{b}))$. Horizontal and vertical transformations are independent and can be performed in either order.