Multiplying a quadratic function by a number greater than 1 results in a transformation known as a vertical stretch.
Here's a detailed, step-by-step explanation:
1. Understanding the Quadratic Function:
A quadratic function typically takes the form [tex]\( f(x) = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants and [tex]\( a \neq 0 \)[/tex].
2. Effect of Multiplying by a Constant Greater Than 1:
When you multiply this function by a constant [tex]\( k \)[/tex] where [tex]\( k > 1 \)[/tex], the new function can be represented as [tex]\( g(x) = k \cdot f(x) = k \cdot (ax^2 + bx + c) \)[/tex].
3. Transformation Interpretation:
- For each point on the graph of [tex]\( f(x) \)[/tex], the corresponding point on the graph of [tex]\( g(x) \)[/tex] will have its [tex]\( y \)[/tex]-coordinate multiplied by [tex]\( k \)[/tex].
- Since [tex]\( k > 1 \)[/tex], every [tex]\( y \)[/tex]-coordinate is increased proportionally, causing the graph to stretch away from the x-axis.
4. Graphical Representation:
- Originally, if [tex]\( f(x) \)[/tex] passes through a point [tex]\( (x_1, y_1) \)[/tex], then [tex]\( g(x) \)[/tex] will pass through [tex]\( (x_1, k \cdot y_1) \)[/tex].
5. Conclusion:
Therefore, the function's graph will undergo a vertical stretch. This means it will become narrower since the y-values (heights) are increased while x-values (widths) remain unchanged.
Thus, multiplying a quadratic function by a number greater than 1 results in a vertical stretch.
