In Game 3 of the 1970 NBA championship series, the L.A. Lakers were down by two points with three seconds left in the game. The ball was inbounded to Jerry West, whose image is silhouetted in today's NBA logo. He launched and made a miraculous shot from beyond midcourt, a distance of 60 feet, to send the game over time (there was no 3-point line at that time).

Through careful analysis of the game tape, one could determine the height at which Jerry West released the ball, as well as the amount of time that elapsed between the time the ball left his hands and the time the ball reached the basket. This information could then be used to write a rule for the ball's height $h$ in the fort as a function of time in flight $t$ in seconds.

Write a rule giving $h$ as a function of $t$.

Write each of the following quadratic expressions in equivalent factored or expanded form.

$$9 x^2+72 x$$

Quadratic equations, like linear equations, can often be solved more easily by algebraic reasoning than by estimation using a graph or table of values for the related quadratic function.

Use algebraic methods to find coordinates of the maximum and minimum points on graphs of these quadratic functions.

i. $y=-5 x^2-2$

ii. $y=5 x^2+3$

iii. $y=x^2-8 x$

iv. $y=-x^2+2 x$

The physical forces that determine the shape of a suspension bridge and business factors that determine the graph of profit prospects for a concert apply to other situations as well. For example, the parabolic reflectors that are used to send and receive microwaves and sounds have shapes determined by quadratic functions. Suppose that the profile of one such parabolic dish is given by the graph of $y=0.05 x^2-1.2 x$, where dish width $x$ and depth $y$ are in feet.

What is the maximum depth of the dish and at what distance from the edge will that occur? Label the point (with coordinates) on your graph of $y=0.05 x^2-1.2 x$.

Use what you have learned about the quadratic formula to complete the slowing tasks.

Suppose that a group of students made these statements in a summary of what they had learned about quadratic functions and equations. With which would you agree? For those that you don't believe to be true, what example or argument would you offer to correct the proposer's thinking?

i. "Every quadratic equation has two solutions."

ii. "The quadratic formula cannot be applied to an equation like $-7 x+8+x^2=0$.

iii. "To use the quadratic formula to solve $-7 x+8+x^2=0$, you let $a=-7; b=8$, and $c=1^{\text {." }}$

iv. "To use the quadratic formula to solve $x^2-6 x+8=0$, you let $a=0$, $b=6$, and $c=8$."

v. "To use the quadratic formula to solve $3 x^2+5 x-7=8$, you let $a=3$, $b=5$, and $c=-7^{. "}$

Use what you have learned about the quadratic formula to complete the slowing tasks.

Solve each of these equations. Show or explain the steps in each solution process.

i. $x^2-6 x+8=0$

ii. $3 x^2-8=-2$

iii. $x^2-2 x+8=2$

iv, $-7 x+8+x^2=0$

v. $15=7+4 x^2$

vi. $x^2=36$

Use your understanding of equivalent expressions to help complete the following tasks.

Which of the following pairs of algebraic expressions are equivalent, and how do you know?

i. $x^2+5 x$ and $x(x+5)$

ii. $m(100-m)+25$ and $-m^2+100 m+25$

iii. $43-5(x-10)$ and $33-5 x$

Quadratic equations, like linear equations, can often be solved more easily by algebraic reasoning than by estimation using a graph or table of values for the related quadratic function.

Solve each of these quadratic equations algebraically.

i. $3 x^2=36$

ii. $-7 x^2=28$

iii. $5 x^2+23=83$

iv. $-2 x^2+4=4$

v. $7 x^2-12 x=0$

vi. $x^2+2 x=0$

The physical forces that determine the shape of a suspension bridge and business factors that determine the graph of profit prospects for a concert apply to other situations as well. For example, the parabolic reflectors that are used to send and receive microwaves and sounds have shapes determined by quadratic functions. Suppose that the profile of one such parabolic dish is given by the graph of $y=0.05 x^2-1.2 x$, where dish width $x$ and depth $y$ are in feet.

Sketch a graph of the function $y=0.05 x^2-1.2 x$ for $0 \leq x \leq 25$. Then write calculations, equations, and inequalities that would provide answers for parts i-iv. Use algebraic, numeric, or graphic reasoning strategies to find the answers, and label (with coordinates) the points on the graph corresponding to your answers.

i. If the edge of the dish is represented by the points where $y=0$, how wide is the dish?

ii. What is the depth of the dish at points 6 feet from the edge?

iii. How far in from the edge will the depth of the dish be 2 feet?

iv. How far in from the edge will the depth of the dish be at least 3 feet?

In Game 3 of the 1970 NBA championship series, the L.A. Lakers were down by two points with three seconds left in the game. The ball was inbounded to Jerry West, whose image is silhouetted in today's NBA logo. He launched and made a miraculous shot from beyond midcourt, a distance of 60 feet, to send the game over time (there was no 3-point line at that time).

Through careful analysis of the game tape, one could determine the height at which Jerry West released the ball, as well as the amount of time that elapsed between the time the ball left his hands and the time the ball reached the basket. This information could then be used to write a rule for the ball's height $h$ in the fort as a function of time in flight $t$ in seconds.

Suppose the basketball left West's hands at a point 8 feet above the ground. What does that information tell about the rule giving $h$ as a function of $t$?

Write each of the following quadratic expressions in equivalent factored or expanded form.

$$9 x(4 x-5)$$

In this investigation, you discovered some facts about the ways that patterns in tables and graphs of quadratic functions $y=a x^2+b x+c(a \neq 0)$ are determined by the values of $a, b$, and $c$.

What does the sign of a tell about the patterns of change and graphs of quadratic functions given by rules in the form $y=a x^2$ ? What does the absolute value of a tell you?

The Pythagorean Theorem says that in any right triangle with legs of length $a$ and $b$ and hypotenuse of length $c, c^2=a^2+b^2$. Write and solve quadratic equations that provide answers to these questions.

a. What is the length of the hypotenuse of a right triangle with legs of length 7?

b. If a right triangle has one leg of length 9 and the hypotenuse of length 15, what is the length of the other leg?

c. If a right triangle has a hypotenuse of length 20 and one leg of length 10, what is the length of the other leg?

d. What are the lengths of the sides of a square that has a diagonal length of 15?

Explore the following examples and look for explanations of the patterns observed.

a. The diagram at the right gives graphs for three of the four quadratic functions below.

$$\begin{aligned} & y=x^2-4 x \\ & y=x^2-4 x+6 \\ & y=-x^2-4 x \\ & y=x^2-4 x-5 \end{aligned}$$

Without using graphing technology:

i. determine the function with the graph that is missing on the diagram.

ii. match the remaining functions to their graphs, and be prepared to explain your reasoning.

b. Without using graphing technology, sketch the pattern of graphs you who expect for the next set of quadratic functions, Explain your reasoning in making the sketch. Then check your ideas with the help of technology.

$$\begin{aligned} & y=x^2+4 x \\ & y=x^2+4 x-6 \\ & y=x^2+4 x+5 \end{aligned}$$

c. How would a sketch showing graphs of the following functions be similar to and different from those in Parts a and b? Explain your reasoning. Check your ideas with the help of technology.

$$\begin{aligned} & y=-x^2+4 x \\ & y=-x^2+4 x-6 \\ & y=-x^2+4 x+5 \end{aligned}$$

d. How can properties of the special quadratic functions $y=a x^2, y=a x^2+c$ and $y=a x^2+b x$ help in reasoning about the shape and location of graphs for functions in the form $y=a x^2+b x+c$?