Podcast
Questions and Answers
The maximum value of a parabola that opens downwards is the y coordinate of the ____ point on the graph.
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.
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?
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?
How do you find the y coordinate of the vertex of the equation y = ax^2 + bx + c using the value of x?
Find the minimum value of the parabola y = x^2 + 6.
Find the minimum value of the parabola y = x^2 + 6.
Find the minimum value of the parabola y = 3x^2 - 6x.
Find the minimum value of the parabola y = 3x^2 - 6x.
The vertex of a parabola that opens upwards is the ____ point.
The vertex of a parabola that opens upwards is the ____ point.
The vertex of a parabola that opens downwards is the ____ point.
The vertex of a parabola that opens downwards is the ____ point.
What is the y coordinate of the point where the parabola intersects the y axis?
What is the y coordinate of the point where the parabola intersects the y axis?
What is the line that passes through the vertex and divides the parabola into two halves that are mirror images?
What is the line that passes through the vertex and divides the parabola into two halves that are mirror images?
Find the y-intercept of the parabola y = x^2 + 19/5.
Find the y-intercept of the parabola y = x^2 + 19/5.
Find the equation of the axis of symmetry for the parabola y = x^2 - 4x.
Find the equation of the axis of symmetry for the parabola y = x^2 - 4x.
What is the graph of a quadratic polynomial called?
What is the graph of a quadratic polynomial called?
What is the vertex form of a parabola that opens up or down?
What is the vertex form of a parabola that opens up or down?
When the equation is written in vertex form, the vertex is (h, ____).
When the equation is written in vertex form, the vertex is (h, ____).
(In vertex form) If a > 0, the parabola opens ____.
(In vertex form) If a > 0, the parabola opens ____.
(In vertex form) If a < 0, the parabola opens ____.
(In vertex form) If a < 0, the parabola opens ____.
Match the graph to its function:
Match the graph to its function:
Solve by using square roots: 181z^2 - 134 = -157z^2 - 132. What are the solutions?
Solve by using square roots: 181z^2 - 134 = -157z^2 - 132. What are the solutions?
Solve by factoring: 4y^2 - 36y + 81 = 0. What is the solution?
Solve by factoring: 4y^2 - 36y + 81 = 0. What is the solution?
Solve by completing the square: m^2 - 30m - 1 = 0. What are the solutions?
Solve by completing the square: m^2 - 30m - 1 = 0. What are the solutions?
Solve using the quadratic formula: 2j^2 = -3j + 3. What are the solutions?
Solve using the quadratic formula: 2j^2 = -3j + 3. What are the solutions?
Find the discriminant for the equation 4u = -7u^2 - 6. What type of solutions does this equation have?
Find the discriminant for the equation 4u = -7u^2 - 6. What type of solutions does this equation have?
What are the steps for solving by completing the square?
What are the steps for solving by completing the square?
What are the steps for solving by factoring?
What are the steps for solving by factoring?
What is this? (the quadratic formula)
What is this? (the quadratic formula)
The term underneath the square in the quadratic formula is known as the ____.
The term underneath the square in the quadratic formula is known as the ____.
The discriminant tells the number of ____ of solutions to a quadratic.
The discriminant tells the number of ____ of solutions to a quadratic.
What is the discriminant formula?
What is the discriminant formula?
If the solution to the discriminant is > 0, then there are... ____ roots.
If the solution to the discriminant is > 0, then there are... ____ roots.
If the solution to the discriminant is = 0, then there is... ____ root.
If the solution to the discriminant is = 0, then there is... ____ root.
If the solution to the discriminant is < 0, then there are... ____ roots.
If the solution to the discriminant is < 0, then there are... ____ roots.
Since there are two real roots, the discriminant is... ____.
Since there are two real roots, the discriminant is... ____.
Since there is only one real root, the discriminant is... ____.
Since there is only one real root, the discriminant is... ____.
Since there are no real roots, the discriminant is... ____.
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
- Completing the Square:
- Steps include rewriting in standard form, factoring, and utilizing the zero product property.
- 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:
-
0 indicates two real roots,
- = 0 indicates one real root,
- < 0 indicates no real roots (complex solutions).
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