Compressed or stretched
Identify vertical and horizontal compressions and stretches of functions in Grade 10 algebra, understanding that y=af(x) stretches vertically by factor a and y=f(bx) compresses horizontally.
Key Concepts
When $0 < |a| < 1$, the parabola is vertically compressed , so it is wider. When $|a| 1$, the parabola is vertically stretched , so it is narrower.
The graph of $y = 4x^2$ (where $|a|=4 1$) is narrower, or vertically stretched, compared to $y = x^2$. The graph of $y = \frac{1}{4}x^2$ (where $|a|=\frac{1}{4} < 1$) is wider, or vertically compressed, compared to $y = x^2$. Comparing $y = 5x^2$ and $y = 0.2x^2$, the graph of $y = 5x^2$ is narrower because $| 5| | 0.2|$.
The absolute value of 'a' is the parabola's personal trainer! If $|a|$ is greater than 1, the parabola is tall and skinny (stretched). If $|a|$ is a fraction between 0 and 1, the parabola is short and wide (compressed). It's all about how intense the curve gets, and 'a' is the intensity dial.
Common Questions
How does y=a·f(x) stretch or compress the graph of y=f(x)?
If |a|>1, the graph stretches vertically (taller). If 0<|a|<1, it compresses vertically (shorter). If a<0, it also reflects across the x-axis.
How does y=f(bx) stretch or compress the graph of y=f(x)?
If |b|>1, the graph compresses horizontally (narrower). If 0<|b|<1, it stretches horizontally (wider). The period of periodic functions changes as 1/b times the original.
What is the effect of y=sin(2x) compared to y=sin(x)?
The period becomes π instead of 2π (compressed by factor 2 horizontally). The graph completes two full cycles in the same horizontal space as one cycle of y=sin(x).