The graph of a function f is shown in the accompanying...

The graph of a function $f$ is shown in the accompanying figure. Sketch the graphs of the following equations.
(a) $y=f(x)-1$
(b) $y=f(x-1)$
(c) $y=\frac{1}{2} f(x)$
(d) $y=f\left(-\frac{1}{2} x\right)$

Key Concepts

Reflection

A reflection transformation flips the graph of a function across a specified axis. In the transformation y = f(-1/2 x), the negative sign within the function argument results in a reflection across the y-axis. Reflection changes the orientation of the graph while preserving distances along the reflected axis.

Horizontal Scaling

Horizontal scaling occurs when the function’s input is multiplied by a constant factor, leading to a stretch or compression along the x-axis. In the equation y = f(-1/2 x), the factor -1/2 affects the horizontal components, resulting in a horizontal stretch by the reciprocal of the absolute value and a reflection due to the negative sign.

Combined Transformations

Some function transformations involve a combination of the above operations, such as both scaling and reflecting. In the case of y = f(-1/2 x), the function experiences a horizontal stretch (by a factor of 2, since the scale factor is the reciprocal of 1/2) and a reflection across the y-axis. Understanding the order and effect of each transformation is crucial in accurately sketching the resultant graph.

Horizontal Translation

A horizontal translation moves the graph left or right. This happens when a constant is added or subtracted from the function’s input, as seen in y = f(x - 1), where the function is shifted one unit to the right. The value inside the function argument determines the horizontal displacement.

Vertical Translation

A vertical translation shifts the entire graph of a function up or down without altering its shape. When a constant is added or subtracted to the function’s output, such as in y = f(x) - 1, it moves the graph vertically. In this example, subtracting 1 shifts the graph one unit downward.

Vertical Scaling (Stretching/Compressing)

Vertical scaling involves multiplying the function's output by a constant factor, which stretches or compresses the graph vertically. For example, y = (1/2)f(x) compresses the graph toward the x-axis because each y-value is halved. This scaling changes the amplitude of the graph without altering the x-coordinates.

Please give Ace some feedback