march 18
TRANSCRIPT
![Page 1: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/1.jpg)
Today:
Warm-UpTest Review
Khan Academy Results/ScheduleBegin Unit on Quadratic Equations
March 18th
![Page 2: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/2.jpg)
Khan Academy:
Saturday/Sunday -- 1409 minutes = 23.48 Hours
Topics for March 24th:Graphing Parabolas in Standard
FormSolving Quadratics by Factoring
1
![Page 3: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/3.jpg)
Number Sense: Space & Volume
![Page 4: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/4.jpg)
Number Sense: Space & Volume
![Page 5: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/5.jpg)
Number Sense: Space & Volume
![Page 6: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/6.jpg)
Number Sense: Space & Volume
![Page 7: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/7.jpg)
3D Sphere
Number Sense: Space & Volume
![Page 8: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/8.jpg)
Number Sense: Space & Volume
![Page 9: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/9.jpg)
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:
![Page 10: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/10.jpg)
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:
![Page 11: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/11.jpg)
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
Quadratic Equations:
![Page 12: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/12.jpg)
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?
Quadratic Equations:
![Page 13: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/13.jpg)
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:
![Page 14: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/14.jpg)
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.
![Page 15: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/15.jpg)
Parabolas:...In Sports
![Page 16: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/16.jpg)
Parabolas:...In Archeticture
![Page 17: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/17.jpg)
Parabolas:...In Nature
![Page 18: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/18.jpg)
Parabolas:...Everywhere
Finally, the most importantParabola of all
![Page 19: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/19.jpg)
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:
![Page 20: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/20.jpg)
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
![Page 21: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/21.jpg)
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.
![Page 22: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/22.jpg)
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
![Page 23: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/23.jpg)
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.
Graphing Parabolas & Parabola Terminology
![Page 24: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/24.jpg)
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.
Graphing Parabolas & Parabola Terminology
![Page 25: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/25.jpg)
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.
![Page 26: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/26.jpg)
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
![Page 27: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/27.jpg)
![Page 28: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/28.jpg)
![Page 29: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/29.jpg)
Warm- Up Exercises
The slope is 2,
which is positive
and the Y-intercept
is -2Therefore, the correct graph is
A
![Page 30: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/30.jpg)
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
![Page 31: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/31.jpg)
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
![Page 32: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/32.jpg)
Class Work:
![Page 33: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/33.jpg)
90% of 90 girls and 80% of 110 boys have shown up in the concert hall on time.
How many children are late?
![Page 34: March 18](https://reader036.vdocument.in/reader036/viewer/2022062418/55495a47b4c905f74e8b52ed/html5/thumbnails/34.jpg)
Parabolas
A parabola with -x2