Quadratic Functions
No Solutions (quadratic)
A quadratic function that does not have an x-intercept. The discriminant is a negative value.
No Solutions (quadratic)
A quadratic function that does not have an x-intercept. There are NO REAL solutions, but there are 2 COMPLEX solutions. The discriminant is a negative value.
Two Solutions (quadratic)
A quadratic function that has two x-intercepts. The discriminant is a positive value.
Concavity
Concave up - min - happy parabola Concave down - max - sad parabola
Find vertex given factored form
Find x-intercepts (set each factor = 0 and solve). The x-value of the vertex is always half-way between the roots, as parabolas are symmetric. When you have the x-value, substitute it back into function to find the y-value.
zeros of a function
For any function, the number x such that f(x)=0. ( x intercept)
Zero of a function
For the function f, any number x such that f(x)=0. Also known as the roots of an equation.
Vertex
Highest or lowest point on a parabola.
Binomial Distribution
Multiplying two binomial factors to get a trinomial
How to Factor for Quadratics
Steps: 1.) Factor out greatest common factor (GCF) if possible 2.) Find 2 numbers that when you multiply them you get a*c, and when you add them you get b 3.) Rewrite original expression with middle term separated into the two numbers you found for step 2 4.) Find GCF between first 2 terms, and last 2 terms 5.) Rewrite in factored form OR: If you already have 4 terms you need to factor, factor by grouping (GCF between the first 2 terms, last 2 terms, etc.)
Minimum
The lowest point on a parabola
Radical
The number of times the radicand is multiplied by itself.
X-Intercept
The point at which the graph of a relation intercepts the x-axis. The ordered pair for this point has a value of y = 0.
Parabola
The u-shaped graph of a quadratic function.
What is the max or min value of a quadratic function?
The y coordinate of the vertex.
Factor
To break up into numbers or expressions that multiply together to get get the original number or expression.
Quadratic Formula
Used to find the solutions of a quadratic (whether real or imaginary).
X-Intercept
Where a quadratic equation crosses the x-axis
Y-Intercept
Where a quadratic equation crosses the y-axis
Roots
Where the graph crosses the x-axis. Where the y-value is 0. Also called x-intercepts or zeros.
Vertex Form
a = dilation (h, k) = coordinate of vertex
Vertex Form
a = dilation (h, k) = vertex
polynomial
an algebraic expression with 2 or more terms
In Vertex form of a quadratic function, what is the vertex?
(h,k)
Factors of a Quadratic
(x-p)(x-q) = 0
Transformations: Reflect over x-axis
-f(x) For quadratics: e.g. f(x)=-3(x-6)^2-1, the quadratic function shifted right 6, down 1, vertically stretched by a factor of 3, and is reflected over the x-axis. e.g. f(x)= -x^2, the function just reflected over the x-axis.
How many points of intersection can a linear-quadratic system have?
2 points, 1 point, or no points of intersection
quadratic function
A function that can be written in the form
Quadratic Function
A function whose input is squared. This means there will always be two inputs that yield the same output.
Quadratic Function
A function whose input is squared. This means there will be two inputs that yield the same output.
Axis Of Symmetry
A line that divides a figure in half so that each half is the mirror image of the other
axis of symmetry
A line that divides a plane figure into two congruent reflected halves
Axis of Symmetry
A line that divides a plane figure or a graph into two congruent reflected halves.
coefficient
A number multiplied by a variable in an algebraic expression.
Rational
A number or expression that can be written as a fraction. E.g. 0.3333..., 8, 2.5, because 0.333...=1/3, 8 = 8/1, 2.5= 5/2.
Prime
A number or expression that can only be divided by 1 and itself.
Irrational
A number or expression that cannot be written as a fraction. They also cannot be expressed as terminating or repeating decimals. E.g. sqrt(2), pi, sqrt(3), e, sqrt(10).
Rational Number
A number that can be written as a quotient of two integers.
Square Root
A number that when multiplied by itself equals a given number.
symmetry
A plane figure that can be folded along a line so the two parts match
One Solution (quadratics)
A quadratic function that has one x-intercept. The discriminant has a value of zero. This indicates that the zero is repeating or is said to have a multiplicity of 2.
Function
A relationship where each input (independent variable) has exactly one output (dependent variable).
Root of an equation
A solution to an equation of the form f(x) = 0. Also known as a zero of an function.
Parabola
A symmetrical curve formed by a quadratic function.
Reflection
A transformation that "flips" a figure over a mirror or reflection line.
Dilation
A transformation that changes the size of an object, but not the shape.
Shift
A transformation that moves a graph horizontally or vertically without changing its size or shape.
Axis of Symmetry
A vertical line (x=a) that divides a parabola into two symmetrical halves.
Real Number
All rational and irrational numbers.
Range
All y-values for a function.
Quadratic Equation
An equation where the highest power is 2
Function Notation
An equation written as f(x) to represent the output.
Roots
Another name for the solutions of a quadratic equation.
Zeros
Another name for the solutions of a quadratic equation.
How do you solve by Isolating the Variable?
Bring constant to the other side. Divide by a coefficient, if applicable. Take the positive and negative square root. x = ±#
How do you solve by Completing the Square?
Bring variable terms to one side and constants to the other side. Find the new c term (half of b, then square it) and add it to both sides. Factor one side, combine like terms on the other. Take the positive/negative square root. "T" it up and solve for x.
Transformation
Changes made to a graph.
Rewrite a Quadratic from Vertex Form to Standard Form
Expand, multiply, combine like terms. Example: y=3(x-1)^2+5 1.) y=3(x-1)(x-1)+5 (Expand binomial) 2.) y=3(x^2-2x+1)+5 (FOIL) 3.) y=(3x^2-6x+3)+5 (Distribute "a") 3.) y=3x^2-6x+8 (Combine like terms)
Extrema
Extrema are the minimum(s) and maximum(s) of a function on a certain interval.
How can you solve a linear-quadratic system algebraically?
Get both equations equal to y. Set both equations equal to each other. Solve by bringing all terms to one side and factoring.
How do you find Axis of Symmetry in a Quadratic Function?
It is the vertical line through the x coordinate of the vertex. x=h
If the graph of a parabola opens down, does the function have a maximum value or a minimum value?
Maximum
Vertex
Maximum or minimum point. Turning point.
Find vertex given standard form
Method 1: Complete the square to rewrite in vertex form. Vertex is (h, k). Method 2: x-value of vertex is -b/2a. Then substitute that value into original function to find the y-value.
Rewrite a Quadratic from Standard Form to Vertex Form
Method 1: Find the vertex, or if you know the vertex, rewrite as y=a(x-h)^2+k Method 2: Complete the square. Example: y=3x^2-6x+8 1.) y-8=3x^2-6x (Move "c" to other side) 2.) y-8=3(x^2-2x) (Factor out "a") 3.) (y-8)/3=x^2-2x (Divide by "a") 4.) (y-8)/3+1=x^2-2x+1(Take 1/2 of b, square it, then add to both sides. 4.) (y-8)/3+1=(x-1)^2 (Complete the square, which is factoring on the right). 5.) y=3(x-1)^2+5 (Solve for y)
Quadratic Formula
Method used to find the x-intercepts (solve) a quadratic equation that cannot be factored.
Quadratic Formula
Method used to solve a quadratic equation that cannot be factored.
If the graph of a parabola opens up, does the function have a maximum value or a minimum value?
Minimum
Irrational Number
Non-repeating, non-terminating decimal numbers. Examples: pi and the square root of 2
Complex Conjugate
Numbers of the form (a+bi) and (a-bi). Their product is a² - b².
Factors
Numbers or expressions that are being multiplied to form a product. [Ex: 2 x 6 = 12 or (x-3)(x+2) = x²-x-6 ]
Complex Number/Expression
Numbers or expressions that have i, the imaginary number, in them. Typically written as a+bi, where the real term is written first and the imaginary term is written second.
Complex Numbers
Numbers which have a real and imaginary component (a +bi).
How do you solve by Factoring?
Set the quadratic equal to 0. Factor into 2 bubbles (use GCF, AC, FBG, or Double Bubble). "T" it up and set each bubble equal to 0. Solve for x.
Simplify a radical (for square roots)
Steps: 1.) Make a factor tree 2.) Circle pairs of the same number or variable 3.) Take the square root of each pair, and bring outside of radical 4.) Any number or variable without a pair stays under radical
Discriminant
The expression under the radical sign in the quadratic formula.
Discriminant
The expression under the radical sign in the quadratic formula. It indicates the number and the type (rational, irrational, complex) of solutions a quadratic function will have.
Vertex
The highest or lowest point on the parabola
Maximum Value
The highest point of a parabola ( when a<0)
Maximum
The highest point on a parabola
Input
The independent variable used to evaluate a function at a specific point.
Vertex
The input that yields the minimum or maximum output value of a quadratic function. Turning point on a parabola and where the axis of symmetry is located.
Vertex
The input that yields the minimum or maximum output value of a quadratic function. Turning point or axis of symmetry for a parabola.
Greatest Common Factor
The largest number or variable expression that divides evenly into all terms of an equation
Minimum Value
The lowest point of a parabola (when a>0)
Vertex
The minimum or maximum point on a parabola
Vertex of a parabola
The point on the parabola that lies on the axis of symmetry that is either the highest or lowest point of the function.
Intercept Form
The product of two expressions. Yields the roots: (p, 0) & (q, 0)
Intercept Form
The product of two expressions. a = dilation (p, 0) & (q, 0) = x-intercepts
Output
The result obtained by entering an input into a function. The dependent variable.
Product
The result of multiplying 2 or more factors together.
Domain
The set of input values of a function.
Range
The set of output values of a function.
range
The set of output values of a function.
Parabola
The shape of the graph of a quadratic function.
parabola
The shape of the graph of a quadratic function.
Parent Function
The simplest, most general function in a family of functions.
Zeros of a Function, Roots, Solutions, x-intercepts
The value of the input when the output is zero. The location where the graph of a function touches the x-axis. (4 terms for the same thing)
x-intercepts (zeros, roots, solutions)
The value of the input when the output is zero. The location where the graph of a function touches the x-axis. (4 terms for the same thing)
y-intercept
The value of the output when the input is zero. The location where the graph of a function touches the y-axis.
Axis Of Symmetry
The vertical line that cuts through the vertex of a parabola, x = h
x intercept
The x coordinates of the points where a graph intersects the x axis.
Domain
The x-values for a function
y intercept
The y coordinate of the point where a graph intersects the y axis.
maximum
The y-value of the highest point on the graph of the function.
Maximum/minimum value of a function
The y-value of the highest/lowest point on the graph of the function.
minimum
The y-value of the lowest point on the graph of the function.
Solve quadratic
This means to find roots of a quadratic (x-intercepts or imaginary ones) Method 1: Factor, if possible. Set each expression = 0 and solve. If not possible to factor, you MUST use method 2. Method 2: Quadratic formula. Method 3: Complete the square to turn into vertex form, then solve for x (don't forget to +/- when you take the square root!)
Solutions
Where y=o in a quadratic equation.
Axis of symmetry (mathematically)
X = - b/2a
Y-intercept
Y-intercept for any function is on the y-axis, where x is always 0. So substitute 0 into x and solve for y.
What part of the quadratic equation tells you if the parabola is concave up or concave down?
a - the lead coefficient. If a is positive, it will be concave up (have a min) If a is negative, it will be concave down (have a max)
Standard Form
a = dilation c = y-intercept (0, c)
discriminant of quadratic equation
a formula found under the radical in the quadratic formula that is used to determine the nature of its roots
quadratic formula
a formula that can be used to solve a quadratic equation; requires careful attention to detail
projectile motion
a formula used to model the path of an object that is dropped, thrown or launched
Exponential Function
a function in which an independent variable appears as an exponent
Linear Function
a function in which the graph of the solutions forms a line
Quadratic Function
a function that can be written in the form f(x)=ax^2+bx+c, where a, b & c are real numbers and a is not equal to zero
taking square roots
a method that can be used to solve a quadratic when the variable is isolated in an expression that is squared
Parabola
a plane curve formed by the intersection of a right circular cone and a plane parallel to an element of the curve
factoring a quadratic equation
a popular method of solving a quadratic equation which involves breaking it down into two factors
function
a relation where every input has only one output
Perfect Square
a square of a whole number
Zero Of a Function
a value of x that makes the function's value zero
Axis of Symmetry
a vertical line that divides the parabola into two congruent halves. The axis of symmetry always passes through the vertex of the parabola
Transformations: Vertical Compression (also called Horizontal Stretch)
af(x), when 0<|a|<1 For quadratics: e.g. f(x)=1/4(x-6)^2-1, the quadratic function shifted right 6, down 1, and vertically compressed by a factor of 1/4, making it look wider. e.g. f(x)= 1/4x^2, the function just vertically compressed.
Transformations: Vertical stretch (also called Horizontal Compression)
af(x), when |a|>1 For quadratics: e.g. f(x)=3(x-6)^2-1, the quadratic function shifted right 6, down 1, and vertically stretched by a factor of 3, making it look narrower. e.g. f(x)= 3x^2, the function just vertically stretched.
monomial
an algebraic expression with only one term
trinomial
an algebraic expression with three terms
binomial
an algebraic expression with two terms
quadratic function
an equation, graph or data that can be modeled by a degree two polynomial
Quadratic Equation
ax² + bx + c = 0
Discriminant
b²-4ac
What part of the quadratic equation tells you the y-intercept?
c - the constant If you substitute 0 in for x, the constant will be equal to y, representing the y-intercept.
Transformation: Shift Up
f(x)+k For quadratics: e.g. f(x)=(x-6)^2+8, the quadratic function shifted right 6, and up 8. e.g. f(x)=x^2+8, the function just shifted up 8.
Transformations: Shift Down
f(x)-k For quadratics: e.g. f(x)=(x-6)^2-1, the quadratic function shifted right 6, and down 1. e.g. f(x)=x^2-1, the function just shifted down 1.
Standard form of a quadratic equation
f(x)=ax^2+bx+c
Transformations: Shift Left
f(x+h) For quadratics: e.g. f(x)=(x+5)^2+8, the quadratic function shifted left 5, and up 8. e.g. f(x)=(x+5)^2, the function just shifted left 5.
Transformations: Shift Right
f(x-h) For quadratics: e.g. f(x)=(x-6)^2+8, the quadratic function shifted right 6, and up 8. e.g. f(x)=(x-6)^2, the function just shifted up 8.
quadratic
having to do with the second power
Imaginary Unit
i, or the principle square root of -1.
Quadratic Formula
method used to solve a quadratic equation that cannot be factored (ax2 + bx + c = 0, a/b/c integers)
zeros of quadratic equation
solution to a quadratic equation when it is set equal to zero. synonyms are roots, solutions
roots of a quadratic function
solution to a quadratic equation when it is set equal to zero. synonyms are zeros, solutions
quadratic term
the ax^2 term in a function
linear term
the bx term in a function
parabola
the graph of a quadratic function
How can you tell if an equation is quadratic?
the highest degree is 2.
maximum
the highest point of a parabola
initial velocity
the initial speed when an object is launched; it becomes the "b" value, or the coefficient of the linear term
minimum
the lowest point of a parabola
Parent Function
the most basic function of a family of functions, or the original function before a transformation is applied
constant term
the numerical value that is not part of a variable expression
factor
the opposite of the distributive property
Vertex
the point of intersection of lines or the point opposite the base of a figure
vertex of a parabola
the point where a parabola makes a turn
If I ask you to find the time a rocket lands back to the ground, what am I asking you to find mathematically?
the root (besides x=0)
domain
the set of input values of a function
Roots
the solutions to a quadratic equation
vertex
turning point in a quadratic equation
vertex is a maximum point
when the value of the coefficient of the x^2 term is negative, the parabola opens downward and the vertex is a maximum
vertex is a minimum point
when the value of the coefficient of the x^2 term is positive, the parabola opens upward and the vertex is a minimum
one real root
when the value of the discriminant of a quadratic equation equals zero; its graph will touch the x-axis once and turn around
two real roots
when the value of the discriminant of a quadratic equation is greater than zero; its graph will cross the x-axis twice
no real roots
when the value of the discriminant of a quadratic equation is less than zero; its graph will not touch or cross the x-axis
x-intercept
where a line crosses the x axis, also called a solution, root or zero
y-intercept
where a line crosses the y axis
Decreasing Function
where the slope is negative, can be either: x ≥ x-value of vertex x ≤ x-value of vertex
Increasing Function
where the slope is positive, can be either: x ≥ x-value of vertex x ≤ x-value of vertex
x-intercept
x coordinate of a point where a graph crosses the x axis/ y coordinate of this point is zero
These words all mean the same.
x intercepts, zeros, roots, solutions
If I ask you to find the time a rocket reaches it's maximum height, what am I asking you to find mathematically?
x-value of the vertex (use the axis of symmetry formula)
Domain
x-values
If I ask you to find the maximum height of a rocket, what am I asking you to find mathematically?
y-value of the vertex (substitute the axis of symmetry into the function and solve for y)
Range
y-values
Vertex form of a quadratic equation
y=a(x-h)^2+k
Find vertex given vertex form
y=a(x-h)^2+k Vertex is (h, k) Think of how the quadratic shifted (e.g. left 2, up 4) by looking at its vertex form, then you can picture the vertex (e.g. left 2, up 4 means vertex is at (-2, 4)).
Square Root
√(xy) = √x · √y, a number that when multiplied by itself equals a given number