Quadratic Functions: Representations and Properties

10 Questions

What does the value of 'a' determine in a quadratic function?

Shape of the parabola

In which form is a quadratic function given by the equation y = a(x - h)² + k?

Vertex form

What does the vertex of a parabola represent in a quadratic function?

Minimum or maximum point

What is the axis of symmetry of a parabola?

-b/2a

How does the value of 'h' affect the graph of a quadratic function?

Horizontal shift

What is the significance of the sign of 'a' in a quadratic function?

Determines if the parabola opens upwards or downwards

What is the factored form of a quadratic function?

y = a(x - r)(x - s)

How can a table of values be created for a quadratic function?

By plugging in values of x into the equation and finding the corresponding y values

What do the values of x and y represent in a table of values for a quadratic function?

x represents the independent variable and y represents the dependent variable

What does the vertex of a parabola represent?

The maximum or minimum point of the parabola

Study Notes

Quadratic Functions and Their Representations

Quadratic Functions

Quadratic functions are mathematical functions that involve the square of a variable. They are important because they are used to model a wide range of real-world phenomena, such as projectile motion and the behavior of electric circuits. Quadratic functions are defined by the formula:

y = a(x - h)² + k

where a, h, and k are constants, and x is the independent variable. The value of a determines the shape of the parabola, with positive values resulting in an upward-opening parabola and negative values resulting in a downward-opening one. The value of h represents the horizontal shift of the parabola, and k represents the vertical shift.

Graphs

The graph of a quadratic function is a parabola, which is a "u"-shaped curve. The graph of the equation y = ax² + bx + c is a parabola with vertex at (-b/2a, c - b²/4a) and opening up if a > 0 and down if a < 0. The x-axis is the axis of symmetry, and the parabola is symmetric with respect to the x-axis.

Equations

Quadratic functions can be represented in various forms, including vertex form, standard form, and factored form. The vertex form is y = a(x - h)² + k, where a, h, and k are constants and (h, k) is the vertex of the parabola. The standard form is y = a(x - h)² + k, where a, h, and k are constants and (h, k) is the vertex of the parabola. The factored form is y = a(x - r)(x - s), where a, r, and s are constants and (r, s) is the vertex of the parabola.

Tables

Quadratic functions can be represented in a table of values. To create a table of values for a quadratic function, choose values for x and substitute them into the equation to find the corresponding values of y. For example, for the function y = x² - 2x + 1, a table of values could be:

x | y
---|---
1 | 0
2 | 1
3 | 4
4 | 9
5 | 16

Values

The values of a quadratic function can be found by plugging in values of x into the equation and simplifying. For example, for the function y = 3x² - 4x + 2, the value of y when x = 2 is:

y = 3(2)² - 4(2) + 2
y = 3(4) - 8 + 2
y = 12 - 8 + 2
y = 4

In conclusion, quadratic functions are an important part of mathematics, and their representations, including graphs, equations, tables, and values, are essential for understanding and working with these functions.