Questions and Answers
What is a quadratic function?
a function that has the form y = ax^2 + bx + c where a ≠ 0
What is the shape of the graph of a quadratic function?
U-shaped
What is a vertex in the context of a quadratic function?
the lowest or highest point on the graph
What is the axis of symmetry for graphs of quadratic functions?
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The parabola opens up if ______ and opens down if ______.
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What is the formula for the vertex form of a quadratic function?
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What is the formula for the intercept form of a quadratic function?
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What are the characteristics of the graph of a quadratic function?
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What are binomials?
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What are trinomials?
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What is factoring in the context of polynomials?
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What is the standard form of a quadratic equation?
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What does the zero product property state?
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Study Notes
Quadratic Functions
- A quadratic function is expressed as y = ax² + bx + c, where a ≠ 0.
- Graphing a quadratic function results in a U-shaped figure known as a parabola.
Parabola Characteristics
- The vertex is the highest or lowest point on the parabola.
- An axis of symmetry divides the parabola into two mirror-image halves; for y = x² and y = ax², symmetry exists around the y-axis.
- A parabola opens upwards if a > 0 and downwards if a < 0.
Vertex and Axis of Symmetry
- The x-coordinate of the vertex can be found using x = -b/2a.
- The axis of symmetry is the vertical line x = -b/2a.
Forms of Quadratic Functions
- Vertex form: y = a(x-h)² + k, where (h, k) is the vertex.
- Intercept form: y = a(x-p)(x-q), where p and q represent x-intercepts.
Graph Characteristics
- In vertex form, the vertex is (h, k) and the axis of symmetry is x = h.
- The x-intercepts p and q are located on the x-axis, with the axis of symmetry situated midway between these two points.
Binomials and Trinomials
- Binomials are algebraic expressions consisting of two terms, e.g., x + 3, x + 5.
- Trinomials consist of three terms, e.g., x² + 8x + 15.
Factoring Methods
- Factoring involves expressing a trinomial as a product of binomials: x² + bx + c = (x + m)(x + n), where the sum m + n = b and the product mn = c.
Special Factoring Cases
- Difference of Two Squares formula: a² - b² = (a + b)(a - b).
- Perfect Square Trinomial formulas:
- a² + 2ab + b² = (a + b)²
- a² - 2ab + b² = (a - b)²
Standard Form of Quadratic Equations
- A quadratic equation can be represented as ax² + bx + c = 0, known as the standard form, where a ≠ 0.
Zero Product Property
- If the left side of the equation ax² + bx + c = 0 is factorable, the Zero Product Property applies: If AB = 0, then A = 0 or B = 0.
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Description
Explore the characteristics and forms of quadratic functions. This quiz covers the vertex, axis of symmetry, and graphing techniques to identify parabolic shapes. Understand the differences between vertex and intercept forms and their implications in graphing.
