Motivation Monday

Review Characteristics of a Parabola What happens when you don’t have a calculator?

Find the Vertex To find the vertex you will need to begin by using the formula X = Where does “a” and “b” come from? They come from the standard form of the quadratic function written as

Find the Vertex The “x” in the equation is the x-coordinate of the vertex . So if the formula gives us the x, then how do we find y? To find y, you take the x-coordinate and plug it into the original equation.

Example 1: Find the Vertex

Find the Vertex You Try! Find the Vertex You Try! Find the Vertex y =

Find the y - intercept This is super easy! To find the y - intercept without a calculator or a graph, you just need to look at the c - value in the equation. C will always be your y-interceept

Example 1: Find the Vertex Find the y - intercept Example 1: Find the Vertex Example 2: Find the Vertex

y = You Try! Find the Vertex You Try! Find the Vertex Find the y - intercept You Try! Find the Vertex You Try! Find the Vertex y =

Finding Domain and Range If you need to find the domain and range without a calculator or graph follow these simple rules: The domain will ALWAYS be all real numbers, no matter what! The range can be found by using the y-coordinate of the vertex. The range will always be ≤ or ≥ the y-coordinate.

Finding Domain and Range 3. Find the “a” value in the standard form of the equation. a > 0 (or positive) a > 0 (or positive) Parabola opens up Domain: y ≥ y - coordinate a < 0 (or negative) Parabola opens down Domain: y ≤ y - coordinate

Find the Domain and Range Example 1: Find the Vertex Example 2: Find the Vertex

Find the Domain and Range You Try! Find the Vertex You Try! Find the Vertex y =

Finding the Solutions When you find the solutions to a quadratic function, you are finding the values where the parabola crosses the x - axis. Synonyms for solutions: roots, x-intercepts, and zeros If you see any of those words, they are asking for where the parabola crosses the x-axis.

Finding the Solutions Here are three graphs to show what can happen:

Toothpicks and Marshmallows Objective: Your team is going to try to build the highest marshmallow and toothpick tower. You can only use toothpicks and marshmallows nothing else. You will be given 15 minutes to work with your group.

Tuesday Warm Up Get out your assignment sheet and warm up sheet. Find the characteristics of the quadratic: Vertex: Axis of Symmetry: Y-intercept: Domain: Range

Test Group Discussions In your groups Go through each problem as a group. Figure out what the right answer should be and how to get them. Everyone should completely rework each short answer problem on a seperate sheet of paper and turn it into the bin.

Wednesday Warm Up Get out your assignment sheet and warm up sheet. Turn your booklet to page 11 and work on #1-8

Homework Check!

A rocket carrying fireworks is launched from a hill 80 feet above a lake. The rocket will fall into the lake after exploding at its maximum height. The rocket’s height above the surface of the lake is given by the function h(t) = -162 + 64t + 80. What is the height of the rocket after 1.5 seconds? What is the maximum height reached by the rocket? After how many seconds after it is launched will the rocket hit the lake?

You Turn! A rock is thrown from the top of a tall building. The distance, in feet, between the rock and the ground t seconds after it is thrown is given by d(t) = -16t2 - 4t + 382. How long after the rock is thrown is it 370 feet from the ground?

You Turn! From 4 feet above a swimming pool, Susan throws a ball upward with a velocity of 32 feet per second. The height of the ball t seconds after Susan throws it is given by h(t) = -16t2 + 32t + 4. Find the maximum height reached by the ball and the time this height is reached. When was the ball at the same height as when it was thrown?

You Turn! Marta throws a baseball with an initial upward velocity of 70 feet per second. This equation h(t) = -16t2 + 70t models the situation. Ignoring Marta’s height, how long after she releases the ball, will it hit the ground? What is the maximum height of the baseball?

How long will it take the boulder to hit the ground? You Turn! A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. This is modeled by the equation h(t) = -16t2 + 240t How long will it take the boulder to hit the ground? How high was the boulder after 5 seconds?

Thursday Warm Up Marta throws a baseball with an initial upward velocity of 70 feet per second. This equation h(t) = -16t2 + 70t models the situation. Ignoring Marta’s height, how long after she releases the ball, will it hit the ground? What is the maximum height of the baseball? 2. A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. This is modeled by the equation h(t) = -16t2 + 240t. How long will it take the boulder to hit the ground? How high will the boulder be at 5 seconds??

Homework Check

Throwing and Dropping Stuff When an object is thrown upward in the air, its height over time forms the shape of a parabola. In fact a format can be used to create the equation.

Throwing and Dropping Stuff Let’s say you are throwing a ball into the air. You launch it at a velocity of 20 ft/s. Its initial height is 5 feet. Its equation would look like this: Graph it by creating a table of values or with your calculator.

Throwing and Dropping Stuff What is the initial velocity if you just drop an object? 0! Therefore the middle term in the equation would not be needed. If you drop a ball from a height of 5 feet, its equation looks like this: Graph it by creating a table of values or with your calculator.

Spread out the cards. Match each story with an equation and with a graph. Discuss the features of the graphs with your group.