1.5 Quadratic Functions (continued)

Modeling falling objects

How quadratic functions change

Summary

Determine a formula, $s(t)$, for the function that models the height of the water balloon at time $t$.

Plot the function in Desmos in an appropriate window. Sketch a copy of the graph here.

Use the graph to estimate the time the water balloon lands.

Use algebra to find the exact time the water balloon lands.

Determine the exact time the water balloon reaches its highest point and its height at that time.

Compute the average rate of change of $s$ on the intervals $[1.5, 2]$, $[2, 2.5]$, $[2.5,3]$. Include units on your answers and write one sentence to explain the meaning of the values you found. Sketch appropriate lines on the graph of $s$ whose respective slopes are the values of these average rates of change.

Two quadratic functions, $f$ and $g$, are determined by their respective graphs in Figure.

• How does the information provided enable you to find a formula for $f$? Explain, and determine the formula. - How does the information provided enable you to find a formula for $g$? Explain, and determine the formula. - Consider an additional quadratic function $h$ given by $h(x) = 2x^2 - 8x + 6$. Does the graph of $h$ intersect the graph of $f$? If yes, determine the exact points of intersection, with justification. If not, explain why. - Does the graph of $h$ intersect the graph of $g$? If yes, determine the exact points of intersection, with justification. If not, explain why.

Consider the quadratic function $f$ given by $f(x) = \frac{1}{2}(x-2)^2 + 1$.

• Determine the exact location of the vertex of $f$. - Does $f$ have $0$, $1$, or $2$$x$-intercepts? Explain, and determine the location(s) of any $x$-intercept(s) that exist. - Complete the following tables of function values and average rates of change of $f$ at the stated inputs and intervals. - What pattern(s) do you observe in Table and ?

A water balloon is tossed vertically from a window on the fourth floor of a dormitory from an initial height of $56.3$ feet. A person two floors above observes the balloon reach its highest point $1.2$ seconds after being launched.

• What is the balloon's exact height at $t = 2.4$? Why? - What is the exact maximum height the balloon reaches at $t = 1.2$? - What exact time did the balloon land? - At what initial velocity was the balloon launched?