Quadratic Functions and Characteristics

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Questions and Answers

What is a quadratic function?

A function in which the greatest power of the variable is 2.

What is the vertex of a parabola?

The maximum or minimum point of a parabola.

What is the axis of symmetry?

A line that divides a plane figure or a graph into two congruent reflected halves.

What does maximum refer to in a parabola?

The greatest value on a parabola occurs when the number in front of the x squared is negative. Signup and view all the answers

What does minimum refer to in a parabola?

The lowest value on a parabola occurs when the number in front of the x squared is positive. Signup and view all the answers

What is the domain in a quadratic function?

All possible x values. Signup and view all the answers

What is the range in a quadratic function?

All possible y values. Signup and view all the answers

What is a solution in the context of a graph?

The place where the graph crosses the x-axis. Signup and view all the answers

What is the standard form of a quadratic function?

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What does the equation y = x^2 + k do?

Translates the graph up or down k spaces. Signup and view all the answers

What does the equation y = (x + h)^2 do?

Translates the graph left or right h spaces. Signup and view all the answers

What does the equation y = ax^2 when a is a fraction do?

Makes the graph wider. Signup and view all the answers

What does the equation y = ax^2 when a is a whole number do?

Makes the graph narrower. Signup and view all the answers

What happens to the graph when a in y = ax^2 is negative?

Flips the graph upside down. Signup and view all the answers

What is completing the square?

A process used to make a quadratic expression into a perfect square trinomial. Signup and view all the answers

What is the Quadratic Formula?

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What are first differences in a table?

If the first difference in a table is a constant rate, then the table represents a linear function. Signup and view all the answers

What are second differences in a table?

If the second difference in a table is a constant, then the table represents a quadratic function. Signup and view all the answers

What is a step function?

A function whose graph is a series of line segments. Signup and view all the answers

What is an absolute value function?

A function written in the form y = |x|, and the graph is always in the shape of a v. Signup and view all the answers

What is an exponential function?

y = ab^x. Signup and view all the answers

What is a linear function?

A function in which the graph of the solutions forms a line. Signup and view all the answers

Study Notes

Quadratic Functions and Characteristics

Quadratic function: A mathematical function characterized by the variable raised to the greatest power of 2.
Vertex: The turning point of a parabola, representing either its maximum or minimum value.
Axis of Symmetry: A vertical line that divides a parabola into two mirror-image halves.

Maximum and Minimum Values

Maximum: The highest point on a downward-opening parabola, occurring when the coefficient of x squared is negative.
Minimum: The lowest point on an upward-opening parabola, occurring when the coefficient of x squared is positive.

Domain and Range

Domain: All possible values for the variable x in a function.
Range: All possible values for the variable y generated by a function.

Solutions and Forms of Quadratic Functions

Solution: The x-coordinate(s) where the graph intersects the x-axis.
Standard form of quadratic function: Typically represented as y = ax^2 + bx + c, where a, b, and c are constants.

Transformations of Quadratic Functions

y = x^2 + k: Translates the graph vertically by k units.
y = (x+h)^2: Translates the graph horizontally by h units.
y = ax^2 when a is a fraction: Makes the graph wider due to the smaller value of a.
y = ax^2 when a is a whole number: Makes the graph narrower as a increases.
y = ax^2 when a is negative: Inverts the graph, flipping it upside down.

Completing the square: A method used to transform a quadratic expression into a perfect square trinomial of the form (x+b)^2 = x^2 + 2bx + b^2.
Quadratic Formula: A tool used for finding roots of quadratic equations; however, no specific definition provided.

Differences in Functions

First Differences: If the differences of consecutive outputs in a table are consistent, the relationship is linear.
Second Differences: If the differences of consecutive first differences are consistent, the relationship is quadratic.

Additional Function Types

Step function: Exhibits discrete jumps and is represented graphically by a series of connected line segments.
Absolute Value Function: Expressed as y = |x|, produces a V-shaped graph.
Exponential function: Defined as y = ab^x, where a is a constant and b is the base.
Linear function: A function that generates a straight line graph.

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Description

Explore the key concepts of quadratic functions through this quiz. Understand the properties of parabolas, including their vertex, axis of symmetry, and the calculation of maximum and minimum values. Test your knowledge on the domain, range, and standard form of quadratic functions.