Chapter 1.2, Problem 54PS

Solution

To find: The constraints on a quadratic function that define the height of the inverted parabola above the x-axis.

The y -coordinate of the vertex of the parabolic path affects the maximum height of the jump.

Given data: The paths of jumps, on a pogo stick on conventional spring and a bow spring are inverted parabolas are $y=-0.5{\left(x-6\right)}^2+18$ and $y=-1.17{\left(x-6\right)}^2+42$

shown here:

  

Method/Formula used:

A quadratic function $y\left(x\right)=a{\left(x-h\right)}^2+k$ represents a parabola with $x=h$ as the axis of symmetry, $\left(h,k\right)$ the vertex and ‘ a’ normal to the focus to that parabola. If $a<0$ , the parabola is inverted. For the vertex k is the height of parabola above x -axis.

Calculation:

The given functions (of parabola) are $y=-0.5{\left(x-6\right)}^2+18$ , and $y=-1.17{\left(x-6\right)}^2+42$ .

If $a<0$ , the parabola is negative. For both the parabola $x=6$ is the axis of symmetry. The vertex of parabola $y=-0.5{\left(x-6\right)}^2+18$ from conventional spring is $\left(h,k\right)\equiv \left(6,18\right)$ and that of the parabola $y=-1.17{\left(x-6\right)}^2+42$ from bow spring is $\left(h,k\right)\equiv \left(6,42\right)$

Therefore, the factor k , the y -coordinate of the vertex of the parabolic path affects the maximum height of the jump.