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Questions and Answers
What is a quadratic function?
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?
What is the graph of a quadratic equation called?
parabola
What is the lowest or highest point on a parabola known as?
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?
What divides a parabola into mirror images and passes through the vertex?
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?
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?
What do you call the sum of two monomials?
What do you call the sum of two monomials?
What is the vertex form of a quadratic equation?
What is the vertex form of a quadratic equation?
What is the intercept form of a quadratic equation?
What is the intercept form of a quadratic equation?
What is the y-coordinate of the vertex of a quadratic function if $a$ is greater than 0?
What is the y-coordinate of the vertex of a quadratic function if $a$ is greater than 0?
What is the y-coordinate of the vertex of a quadratic function if $a$ is less than 0?
What is the y-coordinate of the vertex of a quadratic function if $a$ is less than 0?
What is the sum of three monomials called?
What is the sum of three monomials called?
What kind of equation can be written in the form $ax^2 +bx+c$?
What kind of equation can be written in the form $ax^2 +bx+c$?
What is an x-value for which $f(x)=0$ called?
What is an x-value for which $f(x)=0$ called?
If $b^2 = a$, what does $b$ represent?
If $b^2 = a$, what does $b$ represent?
What is an expression of the form $√a$ called?
What is an expression of the form $√a$ called?
What is the number or expression beneath a radical sign called?
What is the number or expression beneath a radical sign called?
What is the process of eliminating a radical expression in the denominator of a fraction called?
What is the process of eliminating a radical expression in the denominator of a fraction called?
What is a complex number of the form $a+bi$ where $b$ does not equal zero called?
What is a complex number of the form $a+bi$ where $b$ does not equal zero called?
What does the imaginary unit $i$ equal?
What does the imaginary unit $i$ equal?
What is a number of the form $a+bi$ where $a$ and $b$ are real numbers called?
What is a number of the form $a+bi$ where $a$ and $b$ are real numbers called?
What are two complex numbers of the form $a+bi$ and $a-bi$ known as?
What are two complex numbers of the form $a+bi$ and $a-bi$ known as?
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?
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?
What is the expression used to find the solutions of a quadratic equation?
What is the expression used to find the solutions of a quadratic equation?
What is the expression under the radical sign in the quadratic formula called?
What is the expression under the radical sign in the quadratic formula called?
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!
