Function or Relation?
Square Root Method
Factoring
Quadratic Formula
Completing the Square
100
Is it a function or not AND WHY??
{(-2, -2), (5, 5), (9,9), (24, 24)}
Yes! It is a function because the x values all go to unique y values. The x and y can be the same! This would represent the linear function y = x.
100
x2 = 81
x = 9 and x = -9
100
x2 + 7x - 10 = 0
(x + 5)(x - 2) = 0 so x = -5 and x = 2
100
What is the quadratic formula?
100
What should go in the blank to make a perfect square trinomial?
x2 - 18x _________
+ 81, since -18/2 = 9 and 9^2 = 81
200
What is the vertical line test? How do you use it?
The vertical line test is when you draw many vertical lines through a graph. If any line intersects the graph at more than one point/crosses the graph in multiple spots, then the graph is NOT a function. If they only intersects the graph at one point everywhere on the graph, it IS a function! This test checks to see if an x value repeats.
200
3x2 + 48 = 0
x = 4i and x = -4i
200
3x2 - 7x - 6 = 0
(3x + 2)(x - 3) = 0 so x = -2/3 and x = 3
200
Put this quadratic equation into standard form. Then, identify A, B, and C.
-5x + x2 - 4x + 3 = 13 + x
x2 - 10x - 10 so A = 1, B = -10 and C = -10
200
Complete the Square
x2 + 20x + 17 = 0
(x + 10)2 - 83 = 0
300
Draw a graph that IS NOT a function!
Answers may vary, but it must FAIL the vertical line test.
300
2(x - 3)2 - 128 = 0
x = 11 and x = -5
300
Factor completely and write in the most reduced form!
18x2 - 60x + 50 = 0
2(3x - 5)2 = 0 so x = 5/3
300
Use the Quadratic Formula to find the x intercepts:
x2 + 5x - 14 = 0
x = -7 and x = 2
300
Complete the square and find the vertex of the graph:
x2 - 14x - 10 = 0
(x - 7)2 - 59 = 0, so the vertex point is (7, -59).
400
Create a mapping diagram with the domain {-1, 0, 1, 2, 3} and range {3, 5, 11, 21} that IS a Function!
Other options will work, but so does this one:
400
When is the best time to use the square root method?
1. When the B coefficient is 0/the x term is missing.
2. If the equation is vertex form.
400
-1/2x2 - 3/2x - 1 = 0
-1/2(x2 + 3x + 2) = 0
-1/2(x + 2)(x + 1) = 0 so x = -2 and x = -1
400
Reduce completely!!
x2 + 2x - 2 = 0
x = -1 + sqrt(3) and x = -1 - sqrt(3)
400
Complete the square and then use the square root method to find the x intercepts.
x2 + 8x + 25 = 0
(x + 4)2 + 9 = 0
(x + 4)2 = -9
x + 4 = +/-3i
x = -4 +/- 3i
500
Explain why all quadratic equations are functions!
Students can say:
1. All x values have one unique y value because any number squared will only result in one specific y value. Ex: 3^2 = 9 and -3^2 = 9, so the y values repeat, but the x values do not.
2. All quadratic graphs make a U shape and those graphs will all pass the vertical lines test because the vertical lines will only touch the U at one point.
500
Reduce your answer completely!!
(5x + 10)2 - 75 = 0
x = -2 + sqrt(3) and x = -2 - sqrt(3)
500
The length of a rectangular garden is 4 feet less than 3 times the width. If the area is 20 square feet, what are the dimensions?
x(3x - 4) = 20
3x2 - 4x - 20 = 0
3x2 + 6x - 10x - 20 = 0
(3x - 10)(x + 2) = 0
so x = 10/3 and x = -2, but distance cannot be negative, so the width is 10/3 = 3 1/3 feet and length is 3*10/3 - 4 = 10 - 4 = 6 feet.
500
1/2x2 - 6x - 27 = 0
x = 6 +/- sqrt(36+54)/1 = 6 +/- sqrt(90)
x = 6 +/- 3sqrt(10)
500
A rocket is launched and its height is represented by the equation -x2 - 6x + 16 in thousands of feet. Graph this equation and determine the total distance that the rocket has moved from its launching spot to its landing point.
The graph should have a vertex at (-3, 25).
-(x2 + 6x - 4) = -(x + 3)2 + 25 = 0
-(x + 3)2 = -25 so (x + 3)2 = 25 so x + 3 = 5 and -5 so x = -3 + 5 and x = -3 - 5, and therefore, x = 2 and x = -8. The distance between these x intercepts is therefore 10 thousand feet between the launching and landing of the rocket.