Final answer:
The value of 'a' in the quadratic function f(x) = ax² affects the graph's width and the direction it opens. A positive 'a' results in an upwards opening parabola, with 'a' > 1 making it narrower and 0 < 'a' < 1 making it broader. A negative 'a' results in a downwards opening parabola, with the magnitude of 'a' affecting the width similarly.
Step-by-step explanation:
Understanding the Impact of 'a' in Quadratic Functions
The quadratic parent function is denoted by f(x) = x², which produces a symmetrical U-shaped curve called a parabola when graphed. This is the most basic form of quadratic functions. When introducing a coefficient a to the function, i.e., f(x) = ax², the graph of the function changes in two significant ways: the width of the parabola and its direction of opening.
When a is greater than 1, the parabola becomes narrower than the parent function. Conversely, if 0 < a < 1, the graph becomes broader. This is because as a increases in value, the y values change more rapidly for given x values.
If a is negative, the direction of the parabola flips, and the graph opens downward rather than upward. The magnitude of a still affects the width of the parabola, just as with positive values of a. If –a is less than 1 in magnitude, the parabola is wider; if it is greater than 1 in magnitude, the parabola is narrower.
Adjusting the value of a also stretches or compresses the graph of the parabola along the y-axis, demonstrating the changes in how quickly y values increase as x moves away from the vertex. In summary, manipulating the value of a allows you to change the quadratic function's growth rate, which is clearly visible in the graph's appearance.