Quadratics are used everywhere!

Quadratics can be used in ways of finding the area of something or even the motion of an object, person, vehicle or something of the like.

The Graph

The shape of a quadratic function on a graph is a parabola that is either positive or negative.

Now For Real Life Examples!

Mikey decided to throw a ball through the air to his friend Mrs. Hilton across a field. Mikey throws the ball directly upward at 64ft/second while standing on a hill that is 20 feet high. Construct an equation and graph to find the maximum height the ball reaches in the air, the y-intercept on the graph, and when the ball reaches Mrs. Hilton.

Now the Graph!

**Quadratic Functions in the Real World.**

Quadratic Equation

ax^2 + bx + c

First-Make the Equation

f(x)= -16x^2+64x+20

A Quadratic is a polynomial function where quadratic stands for the fatc that 2 is the highest exponent of X.

An absolute maximum or minimum can also be found at the vertex of the parabola.

Each parabola can have the possibility of having one, two or no zeros.

Each parabola must have a y-intercept.

Key: Blue pins-Max's and Min's

Green pins- Y-intercept

Red pins- zeros

Factored Form: a(x+or-d)(x+or-e)

f(x) stands for the total height of the throw.

-16x^2 stands for the initial gravitational acceleration (in ft).

64x stands for the initial speed of the ball when it is thrown.

20 stands for the starting point where the ball is thrown.

Max:(2,84)

Y-int.:(0,20)

(-.291,0)

Zero:(4.291,0)

y-axis: Height (In Feet)

x-axis: Time (Seconds)

The max height is 84 feet at 2 seconds in the air.

The y-intercept is (20,0), which can be found on the graph and in the equation as +20.

The two zeros in this equation and graph are (-.291,0) and (4.291,0), but the only zero that matters in this word problem is the positive zero for a ball can be thrown back in time...

A good way to graph it is using a graphing calculator where one plugs in the said equation and hits graph to see the full image.

An appropriate window for the example problem would be :

Window:

xmin=-7 xmin=-100

xmax=7 xmax=150

xscl=1 yscl=1

xres=1

How to Graph on the Calculator!

First go to the "Y=" button and under "Y1=" plug in the equation.

Click the graph to check and see if you have an appropriate window to see the parabola correctly. if not adjust the window to the fit measures so you can see the parabola's zeros, vertex, and y-intercept.

Next, Time to Find the Required Elements of the Problem!

Finding the zeros:

First click the "2nd" button and then the trace button. Once the options appear before you, click the number 2 for zeros. You will be brought to the graph where it asks "Left Bound?" From the left side to the right of the part of the parabola intersecting with the x-axis. Click "Enter" on the left side where it is on one side of the x-axis. After that, move the point to the other side of the x-axis where it is right bound. Click enter twice and you have your zero! this goes for any real zero on the graph.

The Y-intercept...

Follow step one until reaching the options following the "Trace" button. Instead, click 1 or "value". It will bring you back to the graph where it asks what you want to plug in for x, write 0. Upon pressing enter, you will get your y-intercept!

Finding the Max/Min

Repeat the first instructions until the options at "Trace". depending on what you are looking for either click 3 or 4, "minimum" or "maximum". It will then bring you to to the graph where it again asks you "Left Bound?" Move the point to the left side of the vertex where it is close to it but not on it and click enter. Do the same for the right bound side and click enter twice to find your Max or Min!