1.5 Quadratic Functions

Introduction

Properties of Quadratic Functions

A water balloon is tossed vertically from a fifth story window. Its height, $h$, in meters, at time $t$, in seconds, is modeled by the function

.

Execute appropriate computations to complete both of the following tables: values of the function $h$ on the left, average rates of change for $h$ on the right.

$t$ | $h = q(t)$

$0$ | $q(0) = 25$

$1$ |

$2$ |

$3$ |

$4$ |

$5$ |

$[a,b]$ | $AV_{[a,b]}$

$[0,1]$ | $AV_{[0,1]} = 15$ m/s

$[1,2]$ |

$[2,3]$ |

$[3,4]$ |

$[4,5]$ |

Describe how changing the value of $a$ affects the graph of $q$.

Describe how changing the value of $b$ affects the graph of $q$.

Describe how changing the value of $c$ affects the graph of $q$.

Which parameter seems to have the simplest effect? Which parameter seems to have the most complicated effect? Why?

Is it possible to find a formula for a quadratic function that passes through the points $(0,8)$, $(1,12)$, $(2,12)$? If yes, do so; if not, explain why not.

Consider the quadratic function in standard form given by $y = q(x) = 0.25x^2 - x + 3.5$. Determine constants $a$, $h$, and $k$ so that $q(x) = a(x-h)^2 + k$, and hence determine the vertex of $q$. How does this alternate form of $q$ explain the symmetry in its graph?

How many quadratic functions have $x$-intercepts at $(-5,0)$ and $(10,0)$ and a $y$-intercept at $(0,-1)$? Can you determine an exact formula for such a function? If yes, do so. If not, explain why.

Suppose that a quadratic function $q$ has vertex $(-3,-4)$ and opens upward. How many $x$-intercepts can you guarantee the function has? Why?

In addition to the information in (b), suppose you know that $q(-1) = -3$. Can you determine an exact formula for $q$? If yes, do so. If not, explain why.

Does the quadratic function $p(x) = -3(x+1)^2 + 9$ have $0$, $1$, or $2$$x$-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist.

Does the quadratic function $w(x) = -2x^2 + 10x - 20$ have $0$, $1$, or $2$$x$-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist.