SECTION 4.1 AND 4.2 QUADRATIC FUNCTIONSDecember 1- December 5

Think back…Think back to the Absolute Value Function. On your own sheet of paper answer the following:

What is the parent function for the absolute value family?

What is the general form of the absolute value function? What is the vertex and axis of symmetry?

Make a prediction for the equation, vertex, and axis of symmetry of a quadratic function based on this observation.

What you predict will happen.

What actually happened

Parent Function and Vertex Form

The vertex form of a quadratic function is where

The axis of symmetry is a line that vertically divides the parabola into two mirror images, x=h.

The vertex of the parabola is the intersection of the parabola and its axis of symmetry, (h,k).

What is your prediction?• Take out a sheet of paper and fold it in half.

• On the left side, write “What I predict will happen” and on the right side, write “What actually happened”.

• On the left side, take a minute to write what you predict will occur with transformations for quadratic functions based on what you already know about transformations of linear and absolute value functions.

Exploration: Transformations of Quadratic Functions

• For each of the quadratic functions on the worksheet, graph the parent function, , and graph the transformation of the function given.

• Then, complete the table of values to check if your graph is correct.

How did we do?Check your graphs below. What questions do you have?

Notes

If a>0, the graph opens upward. The y-coordinate of the vertex is the minimum value of the function.

If a<0, the graph opens downward. The y-coordinate of the vertex is the maximum value of the function.

Elaboration: Quadratic Foldable You will receive 7 pieces of construction paper to make a Quadratic foldable.

You will create the book today and work on the first 4 pages. 1. The parent function: Graph it. What is the

vertex and axis of symmetry? Label it on the graph.

2. Transformations of the quadratic function (Translations)

3. Transformations of the quadratic function (Reflections, Vertical Stretch and Compression)

4. Minimum and maximum values of the quadratic function.

What you Predicted vs. What Happened

Take out your sheet of paper with your predictions. On the right side, take a minute to write what happened when your graphed transformations of quadratic functions. Were your predictions correct?

Engage: Let’s practice Expanding!

3. +4

Standard form

• The standard form of a quadratic function is where

• This form is useful when working with the graphing calculator because it can be easier to type.

Exploration: Vertex from Standard Form

Work on the activity provided to you going one step at a time. You will be working from Vertex form to standard form and discovering the values of the vertex through standard form.

When you complete the worksheet, speak with your partner on what the formula for the vertex (written as an ordered pair) could be from standard form.

How did we do?

Example

Given the following quadratic function in standard form, what is the vertex written as an ordered pair? Partner A can do part A and partner B can do part B, than you will teach your partner how to do your problem!

a) b)

Foldable: Vertex form vs. Standard Form

You will create a new foldable, which will be placed in their larger foldable which they started to create yesterday.

This will be a two-tab book. One tab will be the Vertex Form and one would be the Standard form.

You may write the equation, vertex, and axis of symmetry on the inside of the tabs.

Then, you may place this in your layered book and label the tab “Vertex and Standard Forms”.

Engage- Recall from yesterday!

What steps would we take to find the vertex from the standard form of a quadratic function? Take a few minutes to think and write down how you would do this. Be prepared to share your steps with the class.

So, what is the vertex from the standard form ?

Exploration: Standard form to Vertex formWork on the activity provided to you going one step at a time. You will be working from standard form to derive the equation in vertex form.

When you complete the worksheet, check with your partner to make sure you can agree on your answers. If not, collaborate to discover why your answers are different.

How did we do?

1.

2.

3.

Steps:

1. Find the a- and b-values of the standard form

2. Find the vertex of the equation using

3. Substitute the vertex and a-value into the vertex form of the quadratic equation.

4. Simplify

ROPES #11 Warm-upA ball is thrown into the air with an upward velocity of 44

ft/s. Its height h in feet after t seconds is given by the function , h = -16t 2 + 44t + 8.

How long does it take the ball to reach its maximum height? What is the ball’s maximum height? Round to the

nearest hundredth, if necessary.

ROPES #11A ball is thrown into the air with an upward velocity of 28

ft/s. Its height h in feet after t seconds is given by the function, h = -16t2 + 28t + 7. How long does it take the ball to reach its maximum height? What is the ball’s maximum

height? Round to the nearest hundredth, if necessary.

Let’s see the application!

The following worksheet has 3 application problems on it.

You will complete two of the problems on the worksheet.

You must show all work on your paper. This will be counted as a homework grade.

If you do not finish in class, you will need to finish the three problems for homework.

Exit Slip:

Find the vertex of the following quadratic functions.

1) 2) 3)

Convert the following from standard form to vertex form.

4)