FUNCTIONS · GRAPH INTUITION

Function transformations

Transformations let you predict how a graph changes without plotting every point again. A single parent function can generate an entire family of related graphs.

f(x)=x²g(x)=(x−1)²+1

1. The master transformation form

Starting from a parent function \(y=f(x)\), a broad family of transformations can be written as

\[g(x)=a\,f\bigl(b(x-h)\bigr)+k\]

The parameters act in different directions: \(h\) and \(b\) affect the input and therefore the graph horizontally; \(a\) and \(k\) act on the output and therefore vertically.

Inside is horizontal; outside is vertical. The horizontal effects often feel reversed: replacing \(x\) with \(x-h\) shifts the graph to the right by \(h\).

2. Horizontal and vertical shifts

Vertical shift

\(g(x)=f(x)+k\)

Every output increases by \(k\). The graph moves up by \(k\) when \(k>0\), and down when \(k<0\).

Horizontal shift

\(g(x)=f(x-h)\)

The same output now occurs at an input \(h\) units larger. The graph moves right by \(h\).

3. Stretches and compressions

\(a f(x)\)Vertical scaling by factor \(|a|\). If \(|a|>1\), the graph stretches away from the x-axis; if \(0<|a|<1\), it compresses.
\(f(bx)\)Horizontal scaling by factor \(1/|b|\). Thus \(|b|>1\) compresses the graph horizontally.

The reciprocal in the horizontal rule comes from solving \(bx=u\), so \(x=u/b\): features that originally occurred at \(u\) now occur at \(u/b\).

4. Reflections

Across the x-axis

\(g(x)=-f(x)\)

Every y-value changes sign.

Across the y-axis

\(g(x)=f(-x)\)

Every input changes sign.

5. Transformation order matters

When several transformations are combined, read the formula structurally rather than applying a memorized visual order. For

\[g(x)=-2f\bigl(3(x-4)\bigr)+5\]
\(x-4\)shift right 4
factor 3 insidehorizontal compression by \(1/3\)
factor −2 outsidereflect across x-axis and vertically stretch by 2
+5shift up 5

6. Worked example

Transform the parent parabola \(f(x)=x^2\) into

\[g(x)=2(x-3)^2-4\]

The graph moves right 3, stretches vertically by factor 2, and moves down 4. The parent vertex \((0,0)\) becomes \((3,-4)\), while the axis of symmetry becomes \(x=3\).

This is faster and more informative than calculating a large table of values from scratch.

7. Common mistakes

Horizontal sign reversal: \(f(x-3)\) moves right, not left. Horizontal scale: \(f(2x)\) compresses by \(1/2\), not stretches by 2. Missing parentheses: \(f(2(x-3))\) and \(f(2x-3)\) describe different transformations.

8. Quick practice

Describe the transformation from \(f(x)\) to \(g(x)= -\tfrac12 f(x+2)+3\).

Solution

Shift left 2, vertically compress by factor \(1/2\), reflect across the x-axis, then shift up 3.

Related topics

Function fundamentals →Inverse & composition →Quadratic equations →