Algebra II Quadratic Functions

Questions and Answers

What is a quadratic function?

A function that can be written in standard form $ax^2+bx+c$ where $a$ is not zero (correct)

A linear function

A logarithmic function

A function involving trigonometric ratios

What is the graph of a quadratic equation called?

parabola

What is the lowest or highest point on a parabola known as?

vertex

What divides a parabola into mirror images and passes through the vertex?

axis of symmetry Signup and view all the answers

What is an expression that is either a number, a variable, or the product of a number and one or more variables with whole number exponents called?

monomial Signup and view all the answers

What do you call the sum of two monomials?

binomial Signup and view all the answers

What is the vertex form of a quadratic equation?

y=a(x-h)^2 + k Signup and view all the answers

What is the intercept form of a quadratic equation?

y=a(x-p)(x-q) Signup and view all the answers

What is the y-coordinate of the vertex of a quadratic function if $a$ is greater than 0?

minimum value Signup and view all the answers

What is the y-coordinate of the vertex of a quadratic function if $a$ is less than 0?

maximum value Signup and view all the answers

What is the sum of three monomials called?

trinomial Signup and view all the answers

What kind of equation can be written in the form $ax^2 +bx+c$?

quadratic equation Signup and view all the answers

What is an x-value for which $f(x)=0$ called?

zero of a function Signup and view all the answers

If $b^2 = a$, what does $b$ represent?

√a Signup and view all the answers

What is an expression of the form $√a$ called?

radical Signup and view all the answers

What is the number or expression beneath a radical sign called?

radicand Signup and view all the answers

What is the process of eliminating a radical expression in the denominator of a fraction called?

rationalizing the denominator Signup and view all the answers

What is a complex number of the form $a+bi$ where $b$ does not equal zero called?

imaginary number Signup and view all the answers

What does the imaginary unit $i$ equal?

√-1 Signup and view all the answers

What is a number of the form $a+bi$ where $a$ and $b$ are real numbers called?

complex number Signup and view all the answers

What are two complex numbers of the form $a+bi$ and $a-bi$ known as?

complex conjugates Signup and view all the answers

What is the process of adding a term to a quadratic expression of the form $x^2 + bx$ to make it a perfect square trinomial called?

completing the square Signup and view all the answers

What is the expression used to find the solutions of a quadratic equation?

quadratic formula Signup and view all the answers

What is the expression under the radical sign in the quadratic formula called?

Discriminant Signup and view all the answers

Study Notes

Quadratic Functions

Quadratic functions are expressed in the standard form ( ax^2 + bx + c ), where ( a \neq 0 ).
They describe relationships where the highest power of the variable is 2.

Graphing Quadratic Functions

The graph of a quadratic function is a parabola, which opens upward if ( a > 0 ) or downward if ( a < 0 ).
The vertex of a parabola represents its highest or lowest point.

Important Features of Parabolas

The axis of symmetry divides the parabola into two mirror-image halves and passes through the vertex.
If ( a > 0 ), the vertex's y-coordinate indicates the minimum value; if ( a < 0 ), it indicates the maximum value.

Polynomial Terminology

A monomial is a single term consisting of a number, variable, or product of numbers and variables with whole number exponents.
A binomial is the sum of two monomials, while a trinomial is the sum of three monomials.

Forms of Quadratic Equations

The vertex form of a quadratic equation is expressed as ( y = a(x-h)^2 + k ), where ( (h, k) ) is the vertex.
The intercept form is given by ( y = a(x-p)(x-q) ), where ( p ) and ( q ) are the x-intercepts.

Solving Quadratic Equations

The quadratic formula ( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) provides the solutions for ( ax^2 + bx + c = 0 ).
The discriminant ( b^2 - 4ac ) determines the nature of the roots: two real roots, one real root, or no real roots.

Radical Expressions

A square root ( b ) of ( a ) is denoted as ( b^2 = a ).
A radical expression is of the form ( \sqrt{a} ), and the radicand refers to the value inside the radical sign.

Complex Numbers

Complex numbers are expressed as ( a + bi ), where ( a ) and ( b ) are real numbers, and ( i ) is the imaginary unit.
The imaginary unit is defined as ( i = \sqrt{-1} ) and satisfies ( i^2 = -1 ).
Complex conjugates consist of a pair of numbers ( a + bi ) and ( a - bi ).

Algebraic Manipulations

Rationalizing the denominator involves eliminating radicals from the denominator by multiplying the fraction’s numerator and denominator by an appropriate radical.
Completing the square transforms a quadratic expression into a perfect square trinomial by adding a specific term.

These notes cover the fundamental concepts and properties of quadratic functions, essential for understanding their application in algebra.

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Test your knowledge of quadratic functions with these flashcards. Learn key concepts such as parabolas, vertices, and axes of symmetry. Perfect for students studying Algebra II!