24 Questions
What is the coefficient of the second degree term in the polynomial function f(x) = 2x^2 - 20x + 10?
2
What is the vertex form of a quadratic function?
f(x) = a(x - h)^2 + k
What is the axis of symmetry of a parabola?
x = -b / 2a
What is the y-coordinate of the vertex of the parabola f(x) = 2x^2 - 20x + 10?
-40
If ' is positive, what is the shape of the graph of the function f(x) = x^2 + 'x + *?
U-shaped, opening upwards
What is the definition of a parabola?
The graph of a quadratic function
How do you determine the characteristics of a parabola?
By graphing the function
What is the purpose of rewriting a quadratic function in vertex form?
To find the vertex of the parabola
What is a polynomial function of the second degree defined as?
A function defined on ℝ with a form of ax^2 + bx + c
What is another name for a polynomial function of the second degree?
Trinôme
What is the form of a polynomial function of the second degree?
f(x) = ax^2 + bx + c
What is the condition for the coefficients of a polynomial function of the second degree?
a ≠ 0
What is the canonical form of a polynomial function of the second degree?
f(x) = a(x - α)^2 + β
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
Which of the following is a polynomial function of the second degree?
f(x) = 3x - 7x + 3
Which of the following is NOT a polynomial function of the second degree?
f(x) = 5x - 3
What is the value of a in the equation of the parabola?
2
What are the coordinates of the vertex of the parabola?
(3, -17)
What is the equation of the axis of symmetry of the parabola?
x = 3
What is the value of the function f(x) when x = 0?
0
What is the value of the function f(x) when x = 1?
3
What is the maximum value of the function f(x)?
4
What is the x-coordinate of the vertex of the parabola?
3
What is the y-coordinate of the vertex of the parabola?
-17
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.
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.
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