Questions and Answers
The maximum value of a parabola that opens downwards is the y coordinate of the ____ point on the graph.
highest
The minimum value of a parabola that opens upwards is the y coordinate of the ____ point on the graph.
lowest
What is the x coordinate of the vertex of the equation y = ax^2 + bx + c?
x = -b/2a
How do you find the y coordinate of the vertex of the equation y = ax^2 + bx + c using the value of x?
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Find the minimum value of the parabola y = x^2 + 6.
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Find the minimum value of the parabola y = 3x^2 - 6x.
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The vertex of a parabola that opens upwards is the ____ point.
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The vertex of a parabola that opens downwards is the ____ point.
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What is the y coordinate of the point where the parabola intersects the y axis?
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What is the line that passes through the vertex and divides the parabola into two halves that are mirror images?
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Find the y-intercept of the parabola y = x^2 + 19/5.
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Find the equation of the axis of symmetry for the parabola y = x^2 - 4x.
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What is the graph of a quadratic polynomial called?
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What is the vertex form of a parabola that opens up or down?
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When the equation is written in vertex form, the vertex is (h, ____).
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(In vertex form) If a > 0, the parabola opens ____.
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(In vertex form) If a < 0, the parabola opens ____.
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Match the graph to its function:
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Solve by using square roots: 181z^2 - 134 = -157z^2 - 132. What are the solutions?
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Solve by factoring: 4y^2 - 36y + 81 = 0. What is the solution?
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Solve by completing the square: m^2 - 30m - 1 = 0. What are the solutions?
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Solve using the quadratic formula: 2j^2 = -3j + 3. What are the solutions?
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Find the discriminant for the equation 4u = -7u^2 - 6. What type of solutions does this equation have?
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What are the steps for solving by completing the square?
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What are the steps for solving by factoring?
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What is this? (the quadratic formula)
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The term underneath the square in the quadratic formula is known as the ____.
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The discriminant tells the number of ____ of solutions to a quadratic.
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What is the discriminant formula?
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If the solution to the discriminant is > 0, then there are... ____ roots.
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If the solution to the discriminant is = 0, then there is... ____ root.
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If the solution to the discriminant is < 0, then there are... ____ roots.
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Since there are two real roots, the discriminant is... ____.
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Since there is only one real root, the discriminant is... ____.
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Since there are no real roots, the discriminant is... ____.
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Study Notes
Parabola Characteristics
- The highest point on a downward-opening parabola is the vertex's y-coordinate, known as the maximum value.
- The lowest point on an upward-opening parabola is the vertex's y-coordinate, known as the minimum value.
- The vertex of a parabola can be identified using the formula x = -b/2a, which represents the x-coordinate in the quadratic equation y = ax^2 + bx + c.
Finding Coordinates and Values
- To find the y-coordinate of the vertex, substitute x = -b/2a into the original quadratic equation.
- The minimum value of the parabola y = x^2 + 6 is 6, occurring at the vertex.
- The minimum value of y = 3x^2 - 6x is -3, identified similarly.
Key Parabola Terminology
- The vertex of an upward-opening parabola is termed the "lowest point."
- The vertex of a downward-opening parabola is termed the "highest point."
- The y-intercept is the point where the parabola crosses the y-axis, expressed as the y-coordinate at x = 0.
- The axis of symmetry is a vertical line that runs through the vertex and divides the parabola into two symmetrical halves.
Vertex and Standard Forms
- The standard form of a parabola can be written as y = a(x-h)^2 + k, where (h, k) is the vertex.
- In vertex form, if a > 0, the parabola opens upwards; if a < 0, it opens downwards.
Solving Quadratic Equations
- The quadratic formula, used for solving equations, is derived from standard form, and is essential when other methods are inapplicable.
- The discriminant (b^2 - 4ac) determines the nature of the roots of quadratic equations:
- Greater than 0: two distinct real roots.
- Equal to 0: one real root.
- Less than 0: two complex (non-real) roots.
Methods for Solving Quadratics
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Completing the Square:
- Steps include rewriting in standard form, factoring, and utilizing the zero product property.
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Factoring:
- Involves writing the equation in standard form, factoring into binomials, and applying the zero product property.
Solutions and Discriminants
- The discriminant is crucial in understanding the quantity of solutions to a quadratic equation:
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0 indicates two real roots,
- = 0 indicates one real root,
- < 0 indicates no real roots (complex solutions).
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Test your knowledge of key concepts in Pre-Calculation with these flashcards covering the highest and lowest points of parabolas, as well as the vertex formula. Perfect for students preparing for assessments in this fundamental area of mathematics.