A parabola is the graphic representation of a quadratic equation. The constant multipliers, or coefficients, in a quadratic equation determine the way a parabola looks when you graph it on the x-y plane. You can alter parabolic graphs by adjusting the constants in the equation. If you multiply the entire quadratic equation by a number, you can vertically shrink and stretch the parabola.
Vertical Shrinking and Stretching
In order to vertically shrink a function, you must multiply the entire function by a number between zero and one. This compresses the function; its y-values change slower than they did prior to the transformation. In the case of a quadratic equation, the parabola appears to widen across the coordinate plane. Conversely, if you wish to vertically stretch a function, you must multiply the entire function by a number greater than one. The function's y-values then change faster than they did prior to the transformation.
Shrinking a Parabola
When you vertically shrink a parabola, the x-intercepts of the parabola do not change; the parabola will still intersect the x-axis at the same values that it did prior to the shrinking. However, the value of the vertex does change. Consider the parabola represented by the equation y=x^2+2x-3. In this equation, the x-intercepts are 1 and -3. If you input 2 for x, the resulting y-value is 5. If you multiply the whole equation by 1/2, it becomes y=(1/2)x^2 + x - 3/2. The x-intercepts are still 1 and -3. However, if you input 2 for x, the resulting y-value is 5/2. The function's y-values change at a slower rate; the parabola has been vertically shrunk.
Graphing Your Parabola
In order to visualize the shrinking of your parabola, graph a number of different parabolas on the same coordinate plane. You can input a number of different functions into a graphing calculator or online graphing utility. If you are graphing by hand, use different colored writing utensils or differently-patterned lines in order to distinguish different graphs. Graph your initial equation along with other equations with diminishing multipliers. For example, if your function is y=x^2+2x-3, graph that along with y=(1/2)(x^2+2x-3) and y=(1/3)(x^2+2x-3) in order to visualize the shrinking effect. As the fractional value of the multiplying term become smaller, the parabola will shrink to a greater degree.
If you need to track the changes of your parabola beyond the visual shrinking or stretching, use a table. Write the x-values of your graph in one column of the table and write the resulting y-values in another column. This gives you a way to monitor the precise mathematical behavior of a parabola as you shrink it vertically.
- Comstock/Comstock/Getty Images