How to Transform Linear Functions: Rules and Examples

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Related Topics

• How to Transform Quadratic Equations
• Transformation on the Coordinate Plane: Dilation
• Transformation on the Coordinate Plane: Rotation
• Transformation on the Coordinate Plane: Reflection
• Transformation: Rotations, Reflections, and Translations

Step-by-step to Transform Linear Functions

These rules all begin with \(f(x)=2x+1\). Keep the original function visible while comparing them.

Why the rotation center matters

A 90-degree counterclockwise rotation about the origin maps \((x,y)\) to \((-y,x)\). Rotating \(y=2x+1\) this way gives \(y=-\tfrac12x-\tfrac12\): the original point \((0,1)\) becomes \((-1,0)\). The equation \(y=-\tfrac12x+1\) instead describes a 90-degree rotation about the original y-intercept, \((0,1)\).

A negative reciprocal slope identifies a perpendicular direction when the original slope is nonzero; it does not specify a rotation center or intercept. A horizontal line rotated 90 degrees becomes vertical and fails the vertical-line test. Ordinary input/output shifts, nonzero scalings, and axis reflections preserve a nonvertical straight-line graph; arbitrary rotations require this extra check.

Reference: OpenStax: transformations of functions. The rotation equations above are verified by mapping points directly.

Transformations of the parent function y = x

In general, f(bx) compresses the graph horizontally when |b| > 1 and stretches it when 0 < |b| < 1. For y = x, several rules give the same line: 3f(x) and f(3x) both give y = 3x, and −f(x) and f(−x) both give y = −x.

Why horizontal and vertical shifts can look the same

A line has the same slope everywhere, so sliding it sideways can land it in the same place as sliding it up or down. For f(x) = mx + b, shifting right h gives f(x − h) = mx − mh + b. That is the original line moved down by mh.

Example: f(x) = 2x + 1. Shifting right 3 gives f(x − 3) = 2(x − 3) + 1 = 2x − 5. Shifting down 6 gives f(x) − 6 = 2x − 5. Same line.

Putting it together: describe g(x) = −2(x − 1) + 3 starting from f(x) = x. Shift right 1, stretch vertically by 2, reflect over the x-axis, then shift up 3. Simplified, g(x) = −2x + 5. Check with a point: (1, 1) on y = x moves right to (2, 1), stretches to (2, 2), reflects to (2, −2), and shifts up to (2, 1). The formula agrees: g(2) = −4 + 5 = 1.

Frequently Asked Questions

How do you transform linear functions?

Start with the original rule, write the change using f, then substitute and simplify. For f(x) = 2x + 1, f(x) + 3 shifts the line up 3 to give 2x + 4, while 2f(x) doubles every output to give 4x + 2. Keeping parentheses around the changed input or output, such as f(x − 3) = 2(x − 3) + 1, prevents sign slips.

How do you describe transformations of linear functions?

Name the type of change, its direction and its size, in the order they happen. For g(x) = −3f(x) + 2 with f(x) = x, say: a vertical stretch by a factor of 3, a reflection over the x-axis, then a shift up 2 units. The result is g(x) = −3x + 2, which is steeper than y = x, falls from left to right and crosses the y-axis at 2.

How do you move a linear function left and right?

Change the input: f(x − h) moves the line right h units and f(x + h) moves it left h units. The sign feels backwards because the new graph needs a larger x to reach the same output. With f(x) = 2x + 1, shifting right 3 gives f(x − 3) = 2x − 5. For a line, that same graph also equals a shift down 6.

What is a reflection of a linear function?

A reflection flips the line over an axis. Over the x-axis, use −f(x), which negates every output; over the y-axis, use f(−x), which negates every input. If f(x) = 3x − 2, then −f(x) = −3x + 2 and f(−x) = −3x − 2. Both reflected lines have slope −3, but they cross the y-axis at different points.

How do you graph transformations of linear functions?

Graph the parent line first, then move two easy points and redraw the line through them. For y = x, use (0, 0) and (1, 1). To graph g(x) = x − 4, drop both points 4 units to (0, −4) and (1, −3). To graph h(x) = ½x, halve each output to get (0, 0) and (1, ½). Checking two points shows the new slope and intercept at once.

Is there a transforming linear functions worksheet PDF?

Yes. A good transforming linear functions worksheet asks students to write the new rule, describe the shift, stretch or reflection in words, and graph it next to the original line. After finishing a page, compare answers by plugging in x = 0 and x = 1, since those two outputs quickly confirm both the slope and the y-intercept.