Quadratic Functions 1

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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)).


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