Function Transformations

Function transformations allow us to manipulate the graph of a function to produce a new graph. The basic functions can be transformed using shifts, stretches, and reflections. Understanding these transformations is crucial for solving SAT math problems.

Types of Transformations

1. Vertical Shifts: Adding or subtracting a constant shifts the graph up or down.

   If $f(x)$ is the original function, then:

   • $f(x) + k$ shifts the graph up by $k$.
   • $f(x) - k$ shifts the graph down by $k$.

2. Horizontal Shifts: Adding or subtracting inside the function shifts the graph left or right.

   • $f(x + k)$ shifts the graph left by $k$.
   • $f(x - k)$ shifts the graph right by $k$.

3. Vertical Stretch/Compression: Multiplying the function by a constant stretches or compresses the graph.

   • $a imes f(x)$ where $|a| > 1$ stretches the graph.
   • $0 < |a| < 1$ compresses the graph.

4. Reflection: Reflecting the graph over the x-axis or y-axis.

   • $-f(x)$ reflects over the x-axis.
   • $f(-x)$ reflects over the y-axis.

Example

Let’s apply these transformations to the function $f(x) = x^{2}$:

1. Start with the function: $f(x) = x^{2}$

2. Shift up by 3: $g(x) = f(x) + 3 = x^{2} + 3$

3. Reflect over the x-axis: $h(x) = -g(x) = -(x^{2} + 3) = -x^{2} - 3$

4. Stretch vertically by a factor of 2: $j(x) = 2h(x) = 2(-x^{2} - 3) = -2x^{2} - 6$

Thus, the final transformed function is: $j(x) = -2x^{2} - 6$

Key Questions

• What is the effect of adding a constant to a function?
• How does multiplying a function by a factor greater than 1 affect its graph?
• What happens to the graph when we reflect it over the y-axis?
• How can we determine the new vertex of a transformed quadratic function?
• What transformation occurs if we replace $x$ with $x - 4$ in the function?

Keywords

• Transformations
• Shifts
• Stretches
• Reflections
• Quadratic Functions