*Making sense of quadratic functions by examining their graphs*

**Graphing Quadratic Functions**

*Introduction: Will it hit the hoop?*

*Collecting data: Penny Circle *

Introduction: Dan Meyer – Penny Circle

Materials:

- Compass
- 50 pennies

Instructions:

**Step 1**: Draw 5 circles using a compass. Use diameters 1, 2, 3, 4, and 5 inches. Place as many pennies as you can in each circle, making sure that each penny is completely within the circle (Penny Circles)**Step 2**: Record your findings in a table like the one shown. Also include a prediction for the number of pennies that would fit in a circle with a diameter of zero inches.

**Step 3**: Make a scatter plot of your predictions. Fit a line to your scatter plot. Describe what your line looks like.**Step 4**: Use your graph to predict how many pennies will fit in a 6 inch circle. Then, Draw a 6 inch circle and test your prediction. How reasonable was your prediction?**Beyond**: Predict how many pennies will be in the huge circle in the video.

*Examining the structure of a quadratic function and relating it to its graph.*

**Getting to know Quadratics**

Desmos:

- Polygraph
- Marbleslides

*What is a quadratic function?*

First, what is a function? (Hint: it is in your notes). A function defines a relation for which each input has one (and only one) output.

So a quadratic function is a still a relation for which each input has one output, except the rule which defines the relation contains a variable that is squared. It is said to have a degree of 2.

Ex:

* Write your own quadratic function…*

*What to know about quadratic functions*

- The name quadratic comes from “quad” meaning square, because the variable gets squared (such as ).

A quadratic function is a function in the form:

Where **a**, **b**, and **c** are real numbers and **a** is not equal to zero.

- What are some examples of real numbers?
- Why cannot
**a**be equal to zero?

*So, what do a, b, and c do? *

*Lets have a look…*

The Simplest Quadratic: ** **and its graph:

*When we introduce the “a” value: *

- Larger values of
**a**squeeze the curve - Smaller values (closer to zero) of
**a**expand the graph - And negative values of
**a**flip it upside down

*Using the different forms of quadratic functions*

**The Different Forms of Quadratics**

A quadratic function is a function in **Standard ****F****orm**: (Yes, you have seen this before). This is the most common way of writing a quadratic function.

Where **a**, **b**, and **c** are real numbers and **a** is not equal to zero.

Sometimes a quadratic is not in standard form, but it is still a quadratic. For example:

We can also write them in **Vertex Form**.

But first, what is a vertex?

- The vertex is the point representing the maximum or minimum of a parabola, depending of it is opening up or down.

- We can convert from standard for to vertex form

*Converting between standard form and vertex form*

- First, why would we use vertex form? Well,
**h**and**k**represent the maximum or minimum, so if we know that point, we can graph the function easier. - Also, the curve is symmetrical (mirror image) about the axis that passes through

**x = h**

*How do we find h and k?*

- and
- Find
**h**, then solve f(h) to find**k**

*Day 2 Assignment:*

**Practice with Quadratics**

- PH 9-2
- Axis of symmetry
- Graphing quadratics from standard form
- Graphing quadratic inequalities
- Matching graphs with equations
- Assignment: P. 436-437 (1-31, 38-39)

**Linear versus quadratic**

Mr. Wiggins gives his daughter Celia two choices of payment for raking leaves:

- Two dollars for
*each*bag of leaves filled, - She will be paid for the number of bags of leaves she rakes as follows: two cents for filling one bag, four cents for filling two bags, eight cents for filling three bags, and so on, with the amount doubling for each additional bag filled.

- If Celia rakes enough to five bags of leaves, should she opt for payment method 1 or 2? What if she fills ten bags of leaves?
- How many bags of leaves would Celia have to fill before method 2 pays more than method 1?

**Linear versus quadratic task**