March 18

Today:

Warm-UpTest Review

Khan Academy Results/ScheduleBegin Unit on Quadratic Equations

March 18th

Khan Academy:

Saturday/Sunday -- 1409 minutes = 23.48 Hours

Topics for March 24th:Graphing Parabolas in Standard

FormSolving Quadratics by Factoring

1

Number Sense: Space & Volume

3D Sphere


Test Review:

Top 4 missed questions from Friday's test: v.14th; (44% correct) #8.

32x2 = 50 3rd; (42% correct) #10. x3 - 121x = 0 2nd; (40% correct) #4. -3x3 - 12x2 = 0

1st; (37%) #3. The product of (9 - 4t)(9 + 4t) results in:

Today's Objectives:

1. Understand the characteristics of Quadratic Equations, (What they are, and what they aren't).

2. Recognize the Graph of a Quadratic Equation

3. Describe the Differences between Quadratic &

Linear Equations

4. Solve Quadratic Equations by factoring5. Listen Carefully, take notes, ask questions when needed.

Quadratic Equations:

1. What is a Quadratic Equation? From the Latin 'quad', as in quaduplets, quadrilaterals, and quarters...

Quad means 4. A square has four sides. A variable in a quadratic equation can have an exponent of 2, but no higher.An exponent of 2 is a number 'squared'....

The following are all examples of quadratic equations:

x2 = 25, 4y2 + 2y - 1 = 0, y2 + 6y = 0, x2 + 2x - 4 = 0

The standard form of a quadratic is written as: ax2 + bx + c = 0, where only a cannot = 0


We have been solving quadratic equations recently without actually calling them Quadratics. Let's review. Solve: x2 - 13x = 0

x( x - 13) = 0x = 0, or x = 13

One more example. Solve: y = x2 - 4x - 5. To find the x-intercepts, we set the equation to x2 - 4x - 5 = 0

( x - 5)( x + 1) = 0x = 5 or x = -1Which brings us to: What do Quadratic Equations look like and how are they different from linear equations?


Y = 2x + 0 is a linear equation.Linear Equations are straight lines and cross the x and y axis only one time. For each 'y', there is only one 'x'. The greatest degree of any exponent in a linear equation is 1. The relationship between x and y is constant; the slope stays the same.

Linear Equations:

Linear vs. Quadratic Equations

A. The graphs of quadratics are not straight lines, they are always in the shape of a Parabola.B. Parabolas can cross an axis more than once.

C. Unlike linear equations, each value of Y in a quadratic equation has more than one value of x. Because Y is 0 at the X-intercept, when we set the equation = to 0, we get the values of the x-intercepts.D. The slope of a quadratic is not constant. The

slope-intercept formula will not work with parabolas.

Parabolas:...In Sports

Parabolas:...In Archeticture

Parabolas:...In Nature

Parabolas:...Everywhere

Finally, the most importantParabola of all

Objective 4: Solving Quadratic Equations by Factoring

There are 2 ways to factor Quadratic Equations and we have done both already. Let's review:

Method 1: Set the equation = to 0 and solve:Example A. x2 + 6x + 9

x2 + 6x + 9 = 0; (x + 3) (x + 3) = 0, x = -3.This is a perfect square trinomial, and the parabola only crosses the x axis at -3 and would be in this shape:

Objective 4: Solving Quadratic Equations by Factoring

Example B. x2 + 16x + 48 = 0

(x + 12) (x + 4) = 0; x = -12, x = -4. This parabola is to the left of the Y axis

Method 2: Solve x2 = 64. Remember the standard form? ax2 + bx + c = 0, where only a cannot = 0 In this case, b is 0, and c is 64.

We can solve by taking the square root of both sides. The result is x = + 8; x = 8, and x = -8

Factoring Quadratic Equations

From the warm-up exercises, we have seen the variousways to factor quadratic equations. The solutions, or roots, tell us where the graph crosses the x axis.

Given this information, we can begin to plot the graph. However, there is still more information we need to complete the graph.

Remember, all Quadratic Equations are in the form of a Parabola. Parabolas are in one of these forms:

To solve and graph a quadratic equation, we need to know where the graph crosses the x and y axis:

Graphing Parabolas & Parabola Terminology

Important points on a Parabola:

1.Axis of Symmetry:The axis of symmetry is the verticle or horizontal line which runs through the exact centerof the parabola.


Important points on a Parabola:2. 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.


Finding the Axis of Symmetry and Vertex

1. Finding the Axis of Symmetry: The formula is: x = - b/2a Plug in and solve for y = x2 + 12x + 32

We get - 12/2; = -6. 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.

There is one more point left to find and that is they-coordinate of the vertex. To find this, plug in the value of the x-coordinate back into the equation and find y. y = -12 + 12(4) + 32. Y = 1 + 48 + 32. Y = 81.

The bottom of the parabola is at -1 on the x axis, and way up at 81 on the y axis.

Finding the Axis of Symmetry and Vertex

Warm- Up Exercises

The slope is 2,

which is positive

and the Y-intercept

is -2Therefore, the correct graph is

A

Warm- Up Exercises

The Y-intercept is:0

Write the equation for the line above

The slope is:2

The equation of the line is: Y = 2x + 0

Warm- Up Exercises

3. Write the inequality for the graph below

The Y-intercept is:2

The slope is: -3The line is solid, not dotted. The equation is:

Y < -3x + 2

Class Work:

90% of 90 girls and 80% of 110 boys have shown up in the concert hall on time.

How many children are late?

Parabolas

A parabola with -x2

March 18

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Transcript of March 18