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Quadratic Functions: Breanna Rice and Essence Ware
Transcript of Quadratic Functions: Breanna Rice and Essence Ware
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
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.
-1 5 vertex(critical value)
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 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:
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
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
k=-5 First solve in parenthaesis.