Quadratic Functions

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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


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