Quadratic Functions

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


  • Compass
  • 50 pennies


  • 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.

Penny circle table

Graph -Quad1

  • 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


  • 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.


f(x)= x ^{2}

f(x)= 5x ^{2}

f(x)= x ^{2}+x+1

       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 x ^{2}).

A quadratic function is a function in the form:

f(x)=a x ^{2}+bx+c

Where ab, 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 ab, and c do? Lets have a look…

The Simplest Quadratic:  f(x)=x ^{2}  and its graph:


When we introduce the “a” value: f(x)=ax ^{2}


  • 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 Form: (Yes, you have seen this before). This is the most common way of writing a quadratic function.



Where ab, 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:

hidden quads

We can also write them in Vertex Form.

f(x)=a(x-h) ^{2}+k

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?

  • h=\frac{-b}{2a}   and   k=f(h)
  • Find h, then solve f(h) to find k



Day 2 Assignment:

Converting and Graphing Vertex Form

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)

Fireworks example

Linear versus quadratic

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

    1. Two dollars for each bag of leaves filled,
    2. 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.
  1. 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?
  2. How many bags of leaves would Celia have to fill before method 2 pays more than method 1?



Linear versus quadratic task

Linear versus Quadratic task 1-2