Algebra / English

Transforming Algebraic Functions: Shifting, Stretching, and Reflecting

Professor Dave Explains · YouTube · 7:51

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Transform function graphs using shifts, stretches and reflections, then combine the changes in a parabola.

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Chapters

0:00Inside versus outside shifts 3:00Stretch and reflect 4:30Combine transformations

Learning script

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Adding 2 outside x² moves the graph up two units. Replacing x by $x+2$ moves it left two units; input changes act differently from output changes.

Multiplying the function output by 2 doubles vertical distances from the horizontal axis. A negative outside multiplier also reflects the graph across that axis.

The graph of − $2(x-3)$ ²+4 comes from x² by shifting right three, stretching vertically by two and reflecting, then shifting up four. Its vertex is (3,4).

Knowledge cards

Inside versus outside shifts

Adding 2 outside x² moves the graph up two units. Replacing x by $x+2$ moves it left two units; input changes act differently from output changes.

Stretch and reflect

Multiplying the function output by 2 doubles vertical distances from the horizontal axis. A negative outside multiplier also reflects the graph across that axis.

Combine transformations

The graph of − $2(x-3)$ ²+4 comes from x² by shifting right three, stretching vertically by two and reflecting, then shifting up four. Its vertex is (3,4).

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Adding 2 outside x² moves the graph up two units. Replacing x by $x+2$ moves it left two units; input changes act differently from output changes. Multiplying the function output by 2 doubles vertical distances from the horizontal axis. A negative outside multiplier also reflects the graph across that axis. The graph of − $2(x-3)$ ²+4 comes from x² by shifting right three, stretching vertically by two and reflecting, then shifting up four. Its vertex is (3,4).