Quadratic Equations Test Part 1, ALGEBRA 1 SECTION 6: QUADRATIC EQUATIONS AND FUNCTIONS PART2, Math, Quadratic Equations
How do you find the vertex in general form?
(-b/2a, f(-b/2a))
How do you find the y-intercept in general form?
(0, c)
How do you find the vertex in standard form?
(h,k)
Factor x^2 + 20x + 100
(x + 10)(x + 10)
Factor x^2 + 10x + 25
(x + 5)(x + 5)
Trinomial
A polynomial with three terms
Dependent variable
A variable whose variation depends on that of another (output, range)
Quadratic Equation
Polynomial with degree 2
Compressing a function
When a function is multiplied by a coefficient greater than zero but less than 1
Formula for the discriminant
b^2-4ac
The discriminant tells you...
how many solutions a quadratic equation has.
ZPP
if you multiply two numbers and get 0 then at least one of the numbers must equal 0.
If a = a positive number then the vertex is a _________.
minimum
Describe the solutions when the discriminant equals 0
one real rational solution
What are other ways of saying solutions?
roots, x-intercepts, zeroes
complex number
system of numbers adding i to a number Ex. a+bi
i=
√-1
System of equations
A collection of two or more equations that are related and have the same set of variables or unknowns
Parabola
A curved symmetrical U-shaped graph that is the graphic representation of a quadratic function; it may open up or down
Quadratic function
A non-linear function in which the independent variable is raised to the second power; when graphed, forms a curved line called a parabola
Ordered pairs
A pair of numbers used to locate a point or a location on a coordinate plane; the first number (usually x) tells how far to move horizontally and the second number (usually y) tells how far to move vertically
One solution
A parabola is said to have one solution when its vertex is on the x-axis
Two solutions
A parabola that intercepts the x-axis twice is said to have solutions
Binomial
A polynomial with two terms
Successive approximations
A problem-solving method used to find/approximate a solution of a system of equations, where the output values are examined to find the smallest absolute difference measured to the given unit
Coordinate plane
A two-dimensional surface on which points, lines and curves can be plotted; it has two scales, called the x-axis (a horizontal number line) and y-axis (a vertical number line) and four quadrants with positive and negative numbers in both axes
Independent variable
A variable whose variation does not depend on that of another (input, domain)
Axis of Symmetry
A vertical line that goes through the vertex of a quadratic function and divides the parabola into two congruent halves, that are mirror images of each other (in f(x)= 〖ax〗^2+bxb+c the axis of symmetry is x=-b/2a)
Area formula
A=length times width
Shifting a function
Adding or subtracting a constant to the independent variable of a function will shift the function left or right; Adding or subtracting a constant to/from the dependent variable of a function will shift the function up or down;
Solution of a system
Any ordered pair that makes all equations in the system true
Roots/Zeroes/Solutions
Can be found where the graph crosses the x-axis
Zero(s) of a function
Equivalent to the x-intercept, the value(s) of x that make(s) the value of the function equal to zero; the solutions to a function is set to zero.
The X intercepts of the graph, or the zeros of the quadratic function correspond to the solutions, or roots, of the quadratic equation
For example, you can solve x²-5x+6=0 by graphing the corresponding function f(x)= x²-5x+6 and determining the x intercepts. The X intercepts of the graph and the zeros of the function are two and three. So, the roots of the equation are two and three
Vertex form
Given a(x-h)^(2 )+k,(h,k) is the vertex of a parabola
Tables of values
Input and output values that are true for a given function and organized in a table
Stretching a function
Multiplying a function by a coefficient greater than 1
Complex solutions
Non- real solutions; a parabola that does not intercept the x-axis is said to have complex solutions
x-intercept
On a graph, the value of x at the point where the line crosses the x-axis; this happens when the value of y is zero
Y-intercept
On a graph, the value of y at the point where the line crosses the y-axis; this happens when the value of x is zero
Discriminant
The expression located under the radical in the quadratic formula. The discriminant helps us to determine the nature of the roots of the polynomial
A quadratic equation will have two solutions when...
The graph crosses the x axis in two places
Maximum of a graph
The highest point (vertex) on a graph when a parabola opens down: the point where the graph changes from increasing to decreasing
Minimum of a graph
The lowest point (vertex) on a graph when a parabola opens up: the point where the graph changes from decreasing to increasing
Vertex
The maximum or minimum point of a parabola
Reflecting a function
The process of multiplying a function by negative one
Domain
The set values of x used for the input of the function
Range
The set values of y calculated from the domain or the output of the function
Parabola
The shape of a the graph of a quadratic function
Standard Form
a(x-h)²+k
General Form
ax²+bx+c
principal square root
for any positive real number b the principal square root of the negative number -b is √-b=i√b
How do you find A.O.S. in standard form?
h
If b≠- then...
the complex number i called an irrational number
definition of axis of symmetry
the line that cuts the graph into two symmetrical parts
A quadratic equation will have no real solutions when...
the parabola does not cross the x axis
y intercept
the point where the graph crosses the y axis
A quadratic equation will have one solution when...
the vertex of the parabola is on the x axis
Describe the solutions when the discriminant equals a negative perfect square
two imaginary solutions
Describe the solutions when the discriminant equals a positive non-perfect square
two real irrational solutions
Describe the solutions when the discriminant equals a positive perfect square
two real rational solutions
Solve (x + 10)(x + 2) = 0
x = -10, x = -2
Solve (x + 5)(x - 1) = 0
x = -5, x = 1
Formula for the axis of symmetry
x = -b/2a
Solve (x - 10)(x - 2) = 0
x = 2, x = 10
Solve (x - 5)(x + 1) = 0
x = 5, x = -1
To solve a quadratic equation by factoring first write the equation in the form ax to the power of 2 plus BX plus C equals zero, and then factor the left side. Next set each factor equal to zero, and solve for the unknown
x²+8x=-12 x²+8x+12=0 (x-2)(x+6)=0 x+2=0 or x+6=0 x=-2 x=-6
Factor x^2 - 20x + 100
(x - 10)(x - 10)
Factor x^2 - 10x + 25
(x - 5)(x - 5)
i²=
-1
What does A.O.S. equal in general form?
-b/2a
What does 1 solution look like?
-crosses the x axis at 1 point
What does 2 solutions look like?
-crosses the x axis at 2 points
What does no solution look like?
-does not cross the x axis
How do you solve via factoring?
1. Factor trinomial 2. Set them equal to zero 3. Solve for x
How do you solve via square roots?
1. Isolate the squared term 2. Take the square root of both sides 3. Solve for the variable
How do you solve via completing the square?
1. Make sure a= 1 2. Find (b/2)² 3. Add/subtract c 4. Add/subtract (b/2)² to both sides 5. Factor the left side and simplify the right 6. Solve by square roots
Factor try no meals of the form a(P)²+b(P) +C, where a≠0 and P is any expression, by replacing the expression for P with a single variable. Then substitute the expression for P back into the factored expression. Simplify the final factors, if possible.
2(x+3) to the power of 2 -11(x+3) +15 by letting r = x+3. 2(x+3) to the power of two -11(x+3)+15 =2r²-11r+15 = 2r²-5r-6r+15 =(2r²-5r)+(-6r+15) =r(2r-5)-3(2r-5) =(2r-5)(r-3) =[2(x+3)-5][(x+3)-3] =(2x+1)(x) =x(2x+1)
To make x^2 + 10x + _____ a perfect square trinomial, add
25
To make x^2 + 50x + _____ a perfect square trinomial, add
625
To make x^2 -6x + _____ a perfect square trinomial, add
9
If two factors of a quadratic equation have a product of zero, then by the zero product property one of the factors must be equal to zero
The zero product property states that if the product of two real numbers is zero, then one or both or both Of the numbers must be zero. This means that if DE equals zero, then at least one of the D & E is 0
Standard Form
This quadratic function is written in ...
Vertex Form
This quadratic function is written in....
Decreasing interval
When reading a graph from left to right, as the x values increase, the y values decrease
Increasing intervals
When reading a graph from left to right, as the x values increase, the y values increase
