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# Quadratic Functions: Breanna Rice and Essence Ware

The theory of relativity explains that the time and position of an event is relative to the frame of reference of an observer. This prezi simplifies the theory of relativity using beautiful visuals.

by

Tweet## Breanna Rice

on 2 October 2012#### Transcript of Quadratic Functions: Breanna Rice and Essence Ware

But not to plethora

mathematicians Q E i t a r d a u c There once was a time when

finding the solutions of any quadratic equation was unfathomable to us scholars, because the zeros/roots are found on the x-axis when the function crosses the x-axis. X-intercepts are similar to that of zeros/roots. Except, the y-intercepts are the y-values that cross the y-axis. X = -b + Such as the Chinese, the Egyptians, the famous Pythagorus, Muhammad bin Al-Khwarismi(spreaded the knowledge throughout Europe), and Abraham bar Hiyya(spreaded the knowledge throughout Barcelonia.) They all contributed in the discovery in their own way. They found the 'solutions' by using the q u a t i o n The standard form of a quadratic equation is ax2+bx+c=0, where 'a' determines the shrink or stretch of the graph, 'b' stands for the linear term, and 'c' stands for the constant term and where the function crosses the y-axis. When finding roots it will be used to find exactly where the function crosses the x-axis without physically drawing the graph on a coordinate plane.

In order to find the roots of an equation, we have plethora of ways, but specifically using the quadratic formula and taking the square roots of an equation. b -4ac 2a An example using the quadratic formula is to find the solutions is: 2x-4x+6 2=a

-4=b

6=c x=-(-4) + (-4) -4(2)(6) 2(2) x=4 + 16-48 4 x= 4+ 4 2 2 2 2 1 Answer: 4 2 & 2 This is the Quadratic Formula: a formula used to find the solutions from the quadratic equation ax^2+bx+c=0 where a≠0. The formula is x=(-b±√(b^2-4ac))/2a. The quadratic formula is Another method to finding roots of a quadratic function is by taking the square roots of the equation. (x-3) = 16 A example of taking the square roots of a quadratic equation is: The goal in taking the square root if to isolate the x onto one side of the equation to find the root(s), being that 'x' represents what the root is. Step 1: Square both sides (x-3)= 16 Step 2: Simplify x-3= + 4 Step 3: move everything to one side by putting x on one side of the equation x=-3+4 x=-3-4 x=1 x=-7 To graph a quadratic function there are characterisctics that you should considered Such as the axis of symmetry, vertex, y- intercept, and maximimin/ mininium value. Discriminant: b²-4ac which is found in the quadratic formula and determines whether or not the solutions will be imaginary.

b²-4ac=0=2 real solutions

b²-4ac=0=1 real solution

b²-4ac<0=2 imaginary solutions(no real solutions) An example of b²-4ac>0, has 2 real solutions An exmple of b²-4ac=0, has 1 real solution Writinng a Quadratic Function 5 After solving for the solutions/roots, you are able to begin the process of graphing the function. Create a table for the function to see where the function lies.

x y

-3 15

-2 8

-1 5 vertex(critical value)

0 6

1 11

2 20

3 30 2x2+3x+6

When using just the table, it’s necessary to identify the axis of symmetry, which is a line that divides the graph in half by passing through the vertex (the min/max. point of a quadratic function, also known as the critical value). Once, the table is drawn, graph the equation. Axis of Symmetry is a line that divides the graph in half by passing through the vertex. Vertex: the minimum or maximum point on a quadratic function, also known as the critical value. Find the vertex of the function (-1, 5) and draw a line that divides the graph in half by passing through the vertex.

The vertex could either be the min. /max. point (highest/lowest point (x, y) on a quadratic function) Also, the vertex could be the min. /max. value which is the y-value that is the lowest/highest on a quadratic function that could also be the y-intercept, where the y-values cross the y-axis. Maximum/Minimum Point: the highest or lowest point (x,y) on a quadratic function. Maximum/Minimum Value: The y-value that is the lowest or highest on a quadratic or absolute function. b - 4ac>0=2 real solutions

To be a real solution, you lie on the x-axis on a coordinate plane (x,y) An example of where you might find 2 real solutions ona coordinate plane would be: 49 49 is a perfect square and is greater than 0, thus there will be 2 real solutions Answer: +7 x=-2+ 144

2 x=-2+12

2 x=-2+ 12

2 Answer: 5 and -7, 144 id greater than thus there is 2 real solutions. b -4ac=0 1 real solution The greaph will only show one real solutions if the discrimant is equal to 0 because it will lie on the x-axis. An example where there wil only be one real solution would be: 9x2 + 12x + 4 = 0. a = 9, b = 12, and c = 4, the Quadratic Formula gives:

x=-12+ 0

18

x=-12+0

18

Then the answer is x = –2/3 Standard form to Vertex Form Vertex Form to Standard Form The standard form of a quadratic function is ax^2+bx+c=0. The vertex form is a(x-h)^2+k

h=-b k= f(-b)

2a 2a An example of changing a quadratic function to vertex form is: 2x +9x+7=0 a=2

b=9

c=7 The key to finding the components of vertex form. h=-(9)/2(2)=-2.25

k=f(-2.25)=37.375 Plug into vertex form: 2(x+2.25) +37.375 THIS IS THE VERTEX FORM OF THE QUADRATIC FUNCTION As told before, the vertex form is a(x-h) +k and standard form of a quadratic function is ax^2+bx+c=0 An example changing from vertex form to standrad form is:

(x+4) -5 a=1

h= -4

k=-5 First solve in parenthaesis.

Full transcriptmathematicians Q E i t a r d a u c There once was a time when

finding the solutions of any quadratic equation was unfathomable to us scholars, because the zeros/roots are found on the x-axis when the function crosses the x-axis. X-intercepts are similar to that of zeros/roots. Except, the y-intercepts are the y-values that cross the y-axis. X = -b + Such as the Chinese, the Egyptians, the famous Pythagorus, Muhammad bin Al-Khwarismi(spreaded the knowledge throughout Europe), and Abraham bar Hiyya(spreaded the knowledge throughout Barcelonia.) They all contributed in the discovery in their own way. They found the 'solutions' by using the q u a t i o n The standard form of a quadratic equation is ax2+bx+c=0, where 'a' determines the shrink or stretch of the graph, 'b' stands for the linear term, and 'c' stands for the constant term and where the function crosses the y-axis. When finding roots it will be used to find exactly where the function crosses the x-axis without physically drawing the graph on a coordinate plane.

In order to find the roots of an equation, we have plethora of ways, but specifically using the quadratic formula and taking the square roots of an equation. b -4ac 2a An example using the quadratic formula is to find the solutions is: 2x-4x+6 2=a

-4=b

6=c x=-(-4) + (-4) -4(2)(6) 2(2) x=4 + 16-48 4 x= 4+ 4 2 2 2 2 1 Answer: 4 2 & 2 This is the Quadratic Formula: a formula used to find the solutions from the quadratic equation ax^2+bx+c=0 where a≠0. The formula is x=(-b±√(b^2-4ac))/2a. The quadratic formula is Another method to finding roots of a quadratic function is by taking the square roots of the equation. (x-3) = 16 A example of taking the square roots of a quadratic equation is: The goal in taking the square root if to isolate the x onto one side of the equation to find the root(s), being that 'x' represents what the root is. Step 1: Square both sides (x-3)= 16 Step 2: Simplify x-3= + 4 Step 3: move everything to one side by putting x on one side of the equation x=-3+4 x=-3-4 x=1 x=-7 To graph a quadratic function there are characterisctics that you should considered Such as the axis of symmetry, vertex, y- intercept, and maximimin/ mininium value. Discriminant: b²-4ac which is found in the quadratic formula and determines whether or not the solutions will be imaginary.

b²-4ac=0=2 real solutions

b²-4ac=0=1 real solution

b²-4ac<0=2 imaginary solutions(no real solutions) An example of b²-4ac>0, has 2 real solutions An exmple of b²-4ac=0, has 1 real solution Writinng a Quadratic Function 5 After solving for the solutions/roots, you are able to begin the process of graphing the function. Create a table for the function to see where the function lies.

x y

-3 15

-2 8

-1 5 vertex(critical value)

0 6

1 11

2 20

3 30 2x2+3x+6

When using just the table, it’s necessary to identify the axis of symmetry, which is a line that divides the graph in half by passing through the vertex (the min/max. point of a quadratic function, also known as the critical value). Once, the table is drawn, graph the equation. Axis of Symmetry is a line that divides the graph in half by passing through the vertex. Vertex: the minimum or maximum point on a quadratic function, also known as the critical value. Find the vertex of the function (-1, 5) and draw a line that divides the graph in half by passing through the vertex.

The vertex could either be the min. /max. point (highest/lowest point (x, y) on a quadratic function) Also, the vertex could be the min. /max. value which is the y-value that is the lowest/highest on a quadratic function that could also be the y-intercept, where the y-values cross the y-axis. Maximum/Minimum Point: the highest or lowest point (x,y) on a quadratic function. Maximum/Minimum Value: The y-value that is the lowest or highest on a quadratic or absolute function. b - 4ac>0=2 real solutions

To be a real solution, you lie on the x-axis on a coordinate plane (x,y) An example of where you might find 2 real solutions ona coordinate plane would be: 49 49 is a perfect square and is greater than 0, thus there will be 2 real solutions Answer: +7 x=-2+ 144

2 x=-2+12

2 x=-2+ 12

2 Answer: 5 and -7, 144 id greater than thus there is 2 real solutions. b -4ac=0 1 real solution The greaph will only show one real solutions if the discrimant is equal to 0 because it will lie on the x-axis. An example where there wil only be one real solution would be: 9x2 + 12x + 4 = 0. a = 9, b = 12, and c = 4, the Quadratic Formula gives:

x=-12+ 0

18

x=-12+0

18

Then the answer is x = –2/3 Standard form to Vertex Form Vertex Form to Standard Form The standard form of a quadratic function is ax^2+bx+c=0. The vertex form is a(x-h)^2+k

h=-b k= f(-b)

2a 2a An example of changing a quadratic function to vertex form is: 2x +9x+7=0 a=2

b=9

c=7 The key to finding the components of vertex form. h=-(9)/2(2)=-2.25

k=f(-2.25)=37.375 Plug into vertex form: 2(x+2.25) +37.375 THIS IS THE VERTEX FORM OF THE QUADRATIC FUNCTION As told before, the vertex form is a(x-h) +k and standard form of a quadratic function is ax^2+bx+c=0 An example changing from vertex form to standrad form is:

(x+4) -5 a=1

h= -4

k=-5 First solve in parenthaesis.