Transforming Graphs
Aligned to F-BF.B.3 — Common Core State Standards for Mathematics.
What this lesson teaches
Changing a function shifts or stretches its graph: f(x) + k moves it up k units, f(x + k) moves it left k units, k·f(x) stretches it vertically, and a negative sign reflects it.
Worked examples
Example 1
How does y = x^2 + 3 compare to y = x^2? Adding 3 moves every point up 3 units, so the vertex shifts from (0, 0) to (0, 3).
Example 2
y = (x + 3)² moves LEFT 3, not right. Inside the function the shift is opposite to the sign.
Example 3
y = −x² reflects over the x-axis and y = 2x² stretches it. Vertical changes behave the way you expect; horizontal ones do not.
Practice questions
- How does y = x^2 − 5 compare to y = x^2?
- Describe the shift from y = x^2 to y = (x + 2)^2.
- What does the negative sign do to the graph of y = −x^2?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Transformar Gráficas
Cambiar una función desplaza o estira su gráfica: f(x) + k la sube k unidades, f(x + k) la mueve k unidades a la izquierda, k·f(x) la estira verticalmente, y un signo negativo la refleja.
Ejemplo 1: ¿Cómo se compara y = x^2 + 3 con y = x^2? Sumar 3 mueve cada punto 3 unidades hacia arriba, así que el vértice pasa de (0, 0) a (0, 3).
Ejemplo 2: y = (x + 3)² se mueve 3 a la IZQUIERDA, no a la derecha. Dentro de la función el desplazamiento es contrario al signo.
Ejemplo 3: y = −x² refleja sobre el eje x y y = 2x² la estira. Los cambios verticales se comportan como esperas; los horizontales no.
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