Quadratics: Finding Canonical Form

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Questions and Answers

What is the coefficient of the second degree term in the polynomial function f(x) = 2x^2 - 20x + 10?

-2

2 (correct)

What is the vertex form of a quadratic function?

f(x) = a(x - h)^2 + k (correct)

f(x) = ax + b

f(x) = ax^2 + bx + c

f(x) = x^2 + bx + c

What is the axis of symmetry of a parabola?

x = b / 2a

x = a / b

x = -a / b

x = -b / 2a (correct)

What is the y-coordinate of the vertex of the parabola f(x) = 2x^2 - 20x + 10?

-40 Signup and view all the answers

If ' is positive, what is the shape of the graph of the function f(x) = x^2 + 'x + *?

U-shaped, opening upwards Signup and view all the answers

What is the definition of a parabola?

The graph of a quadratic function Signup and view all the answers

How do you determine the characteristics of a parabola?

By graphing the function Signup and view all the answers

What is the purpose of rewriting a quadratic function in vertex form?

To find the vertex of the parabola Signup and view all the answers

What is a polynomial function of the second degree defined as?

A function defined on ℝ with a form of ax^2 + bx + c Signup and view all the answers

What is another name for a polynomial function of the second degree?

Trinôme Signup and view all the answers

What is the form of a polynomial function of the second degree?

f(x) = ax^2 + bx + c Signup and view all the answers

What is the condition for the coefficients of a polynomial function of the second degree?

a ≠ 0 Signup and view all the answers

What is the canonical form of a polynomial function of the second degree?

f(x) = a(x - α)^2 + β Signup and view all the answers

What is the property of a polynomial function of the second degree?

It can be written in the form f(x) = a(x - α)^2 + β Signup and view all the answers

Which of the following is a polynomial function of the second degree?

f(x) = 3x - 7x + 3 Signup and view all the answers

Which of the following is NOT a polynomial function of the second degree?

f(x) = 5x - 3 Signup and view all the answers

What is the value of a in the equation of the parabola?

2 Signup and view all the answers

What are the coordinates of the vertex of the parabola?

(3, -17) Signup and view all the answers

What is the equation of the axis of symmetry of the parabola?

x = 3 Signup and view all the answers

What is the value of the function f(x) when x = 0?

0 Signup and view all the answers

What is the value of the function f(x) when x = 1?

3 Signup and view all the answers

What is the maximum value of the function f(x)?

4 Signup and view all the answers

What is the x-coordinate of the vertex of the parabola?

3 Signup and view all the answers

What is the y-coordinate of the vertex of the parabola?

-17 Signup and view all the answers

Study Notes

Polynomial Function of Second Degree

A polynomial function of second degree is defined as a function f(x) on ℝ, where f(x) = ax^2 + bx + c, with a, b, and c being real numbers and a ≠ 0.
It is also known as a "trinôme".

Canonical Form

A polynomial function of second degree can be written in the canonical form: f(x) = a(x - h)^2 + k, where h and k are real numbers.
The canonical form is obtained by completing the square.

Properties

If a > 0, the function is first decreasing, then increasing.
If a < 0, the function is first increasing, then decreasing.
The graph of the function is a parabola.

Parabola

The graph of a polynomial function of second degree is a parabola.
The parabola has a vertex (h, k) and an axis of symmetry x = h.
The vertex corresponds to the minimum or maximum of the function.

Determining the Characteristics of a Parabola

To determine the vertex of a parabola, use the formula: h = -b / 2a and k = f(h).
To determine the axis of symmetry, use the formula: x = h.

Graphical Representation

To graph a polynomial function of second degree, start by writing the function in canonical form.
Determine the vertex and axis of symmetry.
Calculate the coordinates of some points on the curve and plot them.
Reflect the points across the axis of symmetry to obtain the complete graph.

Variations

The function can be increasing or decreasing depending on the sign of a.
The vertex can be a minimum or maximum depending on the sign of a.

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Description

Determine the canonical form of a quadratic polynomial function. Learn how to express the function in the form f(x) = a(x-h)^2 + k, where a, h, and k are real numbers.