Questions and Answers
What are the roots, zeros, x-intercepts, or solutions of a quadratic function?
Where the parabola crosses the x-axis.
What is a perfect square trinomial?
A trinomial expression where the 'a' and 'c' values are perfect squares and 'b' is twice the square root of 'a' and 'c'.
What does the 'c' value in the standard form of a quadratic represent?
The y-intercept.
What do 'r' and 's' represent in the factored form of a quadratic?
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What is the vertex of a parabola?
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What is the quadratic formula used for?
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What is a parabola?
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What do (h, k) represent in the vertex form of a parabola?
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What are vertical shifts of a parabola?
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What are horizontal shifts of a parabola?
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What is the difference of squares?
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What does the vertex of a parabola indicate?
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What are reflections of a parabola?
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What does dilation of a parabola refer to?
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What is the axis of symmetry in relation to a parabola?
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What is the discriminant in quadratic equations?
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What is the y-intercept of a parabola?
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What is factoring in the context of quadratics?
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Study Notes
Roots/Zeros/X-Intercepts/Solutions
- Points where a parabola intersects the x-axis, indicating the solutions of the quadratic equation.
- A quadratic function can have zero, one, or two x-intercepts.
Perfect Square Trinomial
- A trinomial where both "a" and "c" values are perfect squares.
- The "b" value is calculated as twice the product of the square roots of "a" and "c".
Standard Form of a Quadratic
- Represented as ( ax^2 + bx + c ).
- The "c" value corresponds to the y-intercept of the quadratic function.
Factored Form of a Quadratic
- Written as ( a(x - r)(x - s) ), with "r" and "s" representing the zeros of the quadratic function.
Vertex of a Parabola
- The vertex is the peak (maximum) or trough (minimum) of the parabola.
- It marks the point of inflection where the curve changes direction.
Quadratic Formula
- A pivotal formula given by ( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} ) used to determine the zeros of a quadratic equation.
Parabola
- The graphical representation of a quadratic function, characterized by its U-shaped curve.
Vertex Form of a Parabola
- Expressed as ( y = a(x - h)^2 + k ), where (h, k) is the vertex.
- Vertex form reveals transformations applied to the basic parabola.
Vertical Shifts of a Parabola
- Movement up or down the y-axis.
- Represented as ( f(x) = x^2 + a ) (upward) or ( f(x) = x^2 - a ) (downward).
Horizontal Shifts of a Parabola
- Movement along the x-axis.
- Represented as ( f(x) = (x + 4)^2 ) (left shift) or ( f(x) = (x - 4)^2 ) (right shift).
Difference of Squares
- A factoring method applicable when both "a" and "c" are perfect squares and their relationship involves subtraction.
- It is expressed as ( a^2 - b^2 = (a + b)(a - b) ).
Minimum or Maximum
- Referred to as the vertex; points can be either the lowest (minimum) or the highest (maximum) in a quadratic function.
Reflections of a Parabola
- Flipping the parabola over the x-axis changes its orientation.
- Represented by ( f(x) = -x^2 ).
Dilation of a Parabola
- Refers to stretching or compressing the parabola vertically.
- Increases or decreases the width of the parabola based on the coefficient ( a ) in ( f(x) = ax^2 ) or ( f(x) = \frac{1}{a}x^2 ).
Axis of Symmetry
- The vertical line (( x = h )) that divides the parabola into two equal parts through the vertex.
Discriminant
- A formula ( b^2 - 4ac ) that indicates the number of real zeros (solutions) in a quadratic equation before solving it.
Y-intercept of a Parabola
- The point where the parabola intersects the y-axis, corresponding to the value of ( c ) in the standard form.
Factoring
- The method of re-expressing a quadratic equation from its standard form to its factored form.
- Applicable for different values of "a", including when ( a = 1 ) and special cases.
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Description
Test your knowledge on modeling and analyzing quadratic functions with our flashcards. This quiz covers essential terms such as roots, perfect square trinomials, and standard form of quadratic equations.
