Quadratic Functions 1
taking square roots
a method that can be used to solve a quadratic when the variable is isolated in an expression that is squared
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.
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.
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.
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.
quadratic function
A function that can be written in the form
axis of symmetry
A line that divides a plane figure into two congruent reflected halves
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).
Root of an equation
A solution to an equation of the form f(x) = 0. Also known as a zero of an function.
Range
All y-values for a function.
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.
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.
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.
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)
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.
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.)
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
Vertex
The highest or lowest point on the parabola
parabola
The shape of the graph of a quadratic function.
Domain
The x-values for a function
maximum
The y-value of the highest 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!)
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).
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.
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
factoring a quadratic equation
a popular method of solving a quadratic equation which involves breaking it down into two factors
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.
quadratic function
an equation, graph or data that can be modeled by a degree two polynomial
Discriminant
b²-4ac
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.
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
initial velocity
the initial speed when an object is launched; it becomes the "b" value, or the coefficient of the linear term
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
vertex of a parabola
the point where a parabola makes a turn
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
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)).
