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Questions and Answers
What is a quadratic function?
What is a quadratic function?
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?
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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?
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What do you call the sum of two monomials?
What do you call the sum of two monomials?
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What is the vertex form of a quadratic equation?
What is the vertex form of a quadratic equation?
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What is the intercept form of a quadratic equation?
What is the intercept form of a quadratic equation?
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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?
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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?
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What is the sum of three monomials called?
What is the sum of three monomials called?
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What kind of equation can be written in the form $ax^2 +bx+c$?
What kind of equation can be written in the form $ax^2 +bx+c$?
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What is an x-value for which $f(x)=0$ called?
What is an x-value for which $f(x)=0$ called?
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If $b^2 = a$, what does $b$ represent?
If $b^2 = a$, what does $b$ represent?
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What is an expression of the form $√a$ called?
What is an expression of the form $√a$ called?
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What is the number or expression beneath a radical sign called?
What is the number or expression beneath a radical sign called?
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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?
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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?
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What does the imaginary unit $i$ equal?
What does the imaginary unit $i$ equal?
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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?
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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?
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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?
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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?
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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?
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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!
