Quadratics and Their Properties

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Questions and Answers

What is the domain of the parabola y = x^2 - 5?

All real numbers

What is the range of the parabola y = x^2 - 5?

y > -5

What is the range of the parabola y = -x^2 - 5?

y < -5

Describe the shifts of the parabola y=x^2 if its vertex is at (-2,3).

Horizontal shift left 2, vertical shift up 3 Signup and view all the answers

Describe the shifts of the parabola y = 7 (x - 2)^2 + 3 relative to the parent parabola y = x^2.

Horizontal shift right 2, vertical shift up 3 Signup and view all the answers

How do you find the axis of symmetry for the standard form equation y = ax^2 + bx + c?

x = -b/2a Signup and view all the answers

How do you find the axis of symmetry for the vertex form y = a(x-h)^2 + k?

x = h Signup and view all the answers

How do you find the axis of symmetry for the intercept form y = a(x-p)(x-q)?

x = (p + q) / 2 Signup and view all the answers

Provide an example of an even function.

y = 7x^4 + 3x^2 Signup and view all the answers

Provide an example of an odd function.

y = 4x^7 + 2x^3 Signup and view all the answers

Provide an example of a function that is neither even nor odd.

y = 4x^4 + 3x^3 Signup and view all the answers

How can you find the y-value of the vertex for any parabola?

Find the x-value of the axis of symmetry and plug it into the original equation. Signup and view all the answers

How can you find the y-intercept for any parabola?

Plug in x = 0 into the original equation. Signup and view all the answers

How can you use a table to graph any parabola?

Determine the axis of symmetry, use it as the center value with two smaller numbers and two larger numbers. Signup and view all the answers

What a-values cause vertical stretches?

Bigger than 1, or smaller than -1 (e.g., 3 or -1.7) Signup and view all the answers

What a-values cause vertical shrinks?

In between -1 and 1 (e.g., 1/3 or -0.25) Signup and view all the answers

When a parabola opens upward, the vertex is a:

minimum Signup and view all the answers

When a parabola opens downward, the vertex is a:

maximum Signup and view all the answers

What causes a parabola to open downwards?

A negative a-value Signup and view all the answers

What causes a parabola to open upwards?

A positive a-value Signup and view all the answers

How can you tell that a function is even by looking at its graph?

The graph is symmetrical across the y-axis Signup and view all the answers

How can you tell that a function is odd by looking at its graph?

The graph has rotational symmetry about the origin. Signup and view all the answers

How can you tell that a function is neither even nor odd by looking at its graph?

The graph is not symmetrical across the y-axis and it does not have rotational symmetry about the origin. Signup and view all the answers

Study Notes

Parabolas and Their Properties

The domain of the parabola ( y = x^2 - 5 ) includes all real numbers.
The range for ( y = x^2 - 5 ) is ( y > -5 ).
The range for ( y = -x^2 - 5 ) is ( y < -5 ).

Vertex and Shifts

A parabola with a vertex at (-2, 3) experiences a horizontal shift left by 2 units and a vertical shift up by 3 units from the parent function ( y = x^2 ).
For the vertex form equation ( y = 7(x - 2)^2 + 3 ), the shifts are a horizontal shift right by 2 units and a vertical shift up by 3 units.

Finding Axis of Symmetry

In standard form ( y = ax^2 + bx + c ), the axis of symmetry can be calculated with ( x = -\frac{b}{2a} ).
From vertex form ( y = a(x-h)^2 + k ), the axis of symmetry is ( x = h ).
In intercept form ( y = a(x-p)(x-q) ), the axis of symmetry is found using ( x = \frac{(p + q)}{2} ) (average of the intercepts).

Function Types: Even, Odd, Neither

An example of an even function is ( y = 7x^4 + 3x^2 ).
An example of an odd function is ( y = 4x^7 + 2x^3 ).
A function that is neither even nor odd is represented by ( y = 4x^4 + 3x^3 ).

Vertex and Intercepts

The y-value of the vertex for any parabola can be found by substituting the x-value of the axis of symmetry into the original equation.
The y-intercept of any parabola can be determined by inputting ( x = 0 ) into the equation.

Graphing Parabolas Using a Table

To graph a parabola using a table, identify the axis of symmetry as the center value, choosing two numbers less than and two greater than this value for x.

Effects of "a" Values on Parabolas

Values of ( a ) greater than 1 or less than -1 cause vertical stretches in the parabola, such as ( a = 3 ) or ( a = -1.7 ).
Values of ( a ) between -1 and 1 cause vertical shrinks, like ( a = \frac{1}{3} ) or ( a = -0.25 ).

Opening Direction

A parabola that opens upward will have its vertex as a minimum point.
Conversely, a downward-opening parabola has its vertex as a maximum point.
A negative ( a )-value results in a downward-opening parabola.
A positive ( a )-value leads to an upward-opening parabola.

Function Symmetry

A graph indicates an even function if it proves symmetrical across the y-axis.
An odd function graph exhibits rotational symmetry about the origin, meaning it maintains its appearance when rotated half a circle around the origin.
A graph that lacks symmetry across the y-axis and does not show rotational symmetry about the origin is categorized as neither even nor odd.

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Description

Explore the key concepts of parabolas, including their properties, vertex shifts, and axis of symmetry in this quiz. Understand the differences between even, odd, and neither functions through various examples. Perfect for students studying quadratic functions in mathematics.