April 8, 2015

Today:

News & NotesImportant Points on a Parabola:

(What they are & how to find them)Graphing Parabolas From Quadratic Functions

Class Work:

8Ap

ril

V6Math has all the Khan Academy topics listed with the due dates for the entire 4th Quarter. You can download the schedule there. Due dates will now be Friday evenings.

It is strongly recommended that you take good notes throughout this unit. Also recommended: Bring a calculator everyday

But first, a sample of Additional Resources Available for download at the v6

math site:

Class Notes Section of Notebook, pls.

The first week's topics, due by April 17th are: 1. Understanding the process for solving Quadratic equations

2. Solving Quadratic Equations by factoring 23. Vertex of a Parabola

1st Period please leave your notebooks again today after class.

Textbook & Practice Problems/Quizzes with Answers

Most students say

this is the most

difficult unit of the year...

-use the resources

Class Zone

Let's get started..

Quadratic Equations vs. Functions

Remember, the standard form of a quadratic equation is:ax2 + bx + c = 0

Since the solutions/roots to a standard equation are where the line crosses the x-axis, the y value is always zero at this point. As such, we can substitute y for zero: y = ax2 + bx + c

Since the y variable is dependent on the x, or is a function of x, we can substitute the y for the function of x, or (f)x:

(f)x = ax2 + bx + c

Regardless of which form is presented, the problem is solved in the same way.

***Quadratic Equations are solved algebraically. Quadratic Functions are solved graphically.

1. To solve and graph a quadratic equation, we need to know where the graph either touches or crosses the x and y axis: These, of course, are the intercepts.

In order to graph a quadratic function, we must know to use the equation to plot the key parts of the parabola. Then, we basically connect the dots to complete the graph. Here are those key pieces and how to find them.

1. We will learn a number of ways to find various points, but for now we find them by creating a table of values and by factoringthe quadratic equation in standard form.

Graphing Parabolas & Parabola Terminology

The solutions are the x-intercept(s)

2.Axis of Symmetry:The axis of symmetry is the verticleor horizontal line which runs through the exact centerof the parabola.


Other Important points on a Parabola:

Another helpful point to remember about the

axis of symmetry is that is is always

halfway between two x-intercepts

3. Vertex: The vertex is the highest point (the maximum), or the lowest point (the minimum) on a parabola.

Notice that the axis of symmetry always runs through the

vertex.


If the value of a is negative, the parabola will open downward, and the vertex will be a vertex maximum

Vertex Minimums and Maximums

What do the vertex minimum or

maximum tell us in terms of the function's

domain and range.

The information regards the range of the function:No y value can be greater than the vertex

maximum, nor less than the vertex minimum.

Finding the Axis of Symmetry & Vertex

The center of the parabola crosses the x axis at -6. Since the axis of symmetry always runs through the vertex, the x coordinate for the vertex is -6 also.

The formula for finding the axis of symmetry x = - b/2a

Our quadratic function is: y = x2 + 12x + 32

−𝒃

𝟐𝒂=

−𝟏𝟐

𝟐𝒂= - 6

But, we still don't know where the vertex lies on the

vertical (y) axis.

To find the y-coordinate of the vertex, substitute the value of the x-coordinate back into the equation and find y.

Finding the Axis of Symmetry and Vertex

y = -62 + 12(-6) + 32. y = 36 - 72 + 32. y = -4

The bottom of the parabola (the vertex) is at -6 on the x axis, and -4 on the y axis.

Remember, the axis of symmetry always goes through the vertex; the AOS and the vertex are the same point.

Opens Up/Down?

Graphing Parabolas Using a Table of Values

0

-4-3

2

4

-2

Opens Up/Down?

Graphing Parabolas Using a Table of Values

0

-4-3

2

4

-2

Parabolas can be drawn using a table of values, but it is unlikely all of the important points on the parabola will be found with this method.

Finding the Axis of Symmetry(AOS) & the Vertex

Graphing a parabola accurately requires plotting the AOS & the vertex.

1. Find the axis of symmetry using the formula.....

a = 4 and b = -8

Example 3: Finding the Axis of Symmetry & Vertex when graphing.Consider the graph of y = 4x2 - 8x + 1

The axis of symmetry is ?2. Find the coordinates of the vertex.

The coordinates are: (1, ?)-3)

1

Since the graph curves up, the vertex is a maximum or

minimum?

Find the indicated information for each quadratic function.1. Opens:______Line of Symmetry:______Vertex:______Maximum/Minimum:_____Zeros:_______Domain:______Range:_______

Important Points on a Parabola









Class Work 4.1Due by end of class Friday for

full creditAs always, show work for any

credit

Finding the Axis of Symmetry and Vertex

Find the x-intercepts, axis of symmetry & vertex for the following:

x2 + 2x – 3 = 0 –2x2 + 6x + 56 = 0 2x2 + 2x = 3

Lastly,Solve a quadratic equation to

find the value of x

f(x) = -x2 -4x - 12

Today's Assignment: Graph equations, paying special attention to how the a, b, and c values change the shape of the parabola

2. The Microhard Corporation has found that the equationP = x2 - 7x - 94

describes the profit P, in thousands of dollars, for every x hundred computers sold. How many computers were sold if the profit was $50,000?

f(x) = x2 + 2x + 8

f(x) = 2x2 + 4x + 2

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April 8, 2015