Introduction of Quadratic function Concepts

Introduction of Quadratic function

A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two. Since the highest degree term in a quadratic function is of the second degree, therefore it is also called the polynomial of degree 2. A quadratic function has a minimum of one term which is of the second degree. It is an algebraic function.

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a Parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

Properties of the quadratic function

Three properties that are universal to all quadratic functions:

• The graph of a quadratic function is always a parabola that either opens upward or downward (end behavior);
• The domain of a quadratic function is all real numbers; and
• The vertex is the lowest point when the parabola opens upwards; while the vertex is the highest point when the parabola opens downward.
• The standard form of the quadratic function is f(x) = ax²+bx+c where a ≠ 0.
• The graph of the quadratic function is in the form of a parabola.
• The quadratic formula is used to solve a quadratic equation ax²+ bx + c = 0 and is given by x = −b±√b2−4ac/2a
• The discriminant of a quadratic equation ax²+ bx + c = 0 is given by b²-4ac. This is used to determine the nature of the solutions of a quadratic function.

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Quadratic function - Formula

A quadratic function can always be factorized, but the factorization process may be difficult if the zeroes of the expression are non-integer real numbers, or non-real numbers. In such cases, we can use the quadratic formula to determine the zeroes of the expression. The general form of a quadratic function is given as: f(x) = ax² + bx + c, where a, b, and c are real numbers with a ≠ 0. The roots of the quadratic function f(x) can be calculated using the formula of the quadratic function which is:

x=−−b±√b2−4ac/2a

Examples of quadratic function

The quadratic function equation is f(x) = ax² + bx + c, where a ≠ 0. Let us see a few examples of quadratic functions:

• f(x) = 2x²+ 4x - 5; Here a = 2, b = 4, c = -5
• f(x) = 3x²- 9; Here a = 3, b = 0, c = -9
• f(x) = x²- x; Here a = 1, b = -1, c = 0

Now, consider f(x) = 4x-11; Here a = 0, therefore f(x) is not a quadratic function.

Example 1

Sketch the graph of y = x2/2. Starting with the graph of y = x², we shrink by a factor of one-half. This means that for each point on the graph of y = x², we draw a new point that is one-half of the way from the x-axis to that point.

Example 2

Sketch the graph of y = (x - 4) ^2 - 5. We start with the graph of y = x², shift 4 units right, then 5 units down.

Example 3: Alternate method of finding the vertex

In some cases, completing the square is not the easiest way to find the vertex of a parabola. If the graph of a quadratic function has two x-intercepts, then the line of symmetry is the vertical line through the midpoint of the x-intercepts.

The x-intercepts of the graph above are at -5 and 3. The line of symmetry goes through -1, which is the average of -5 and 3. (-5 + 3)/2 = -2/2 = -1. Once we know that the line of symmetry is x = -1, then we know the first coordinate of the vertex is -1. The second coordinate of the vertex can be found by evaluating the function at x = -1.

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Most Popular Questions Searched By Students:

Q.1. What is quadratic function?

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbered with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

Q.2. What are the 3 quadratic equations?

There are three commonly-used forms of quadratics:

1. Standard Form: y = ax²+bx+cy
2. Factored Form: y = a(x-r₁) (x-r₂)
3. Vertex Form: y = a (x-h)²+k

Q.3. How do you find the quadratic function?

There are several methods you can use to solve a quadratic equation:

• Factoring
• Completing the Square
• Quadratic Formula
• Graphing

Q.4. What are the 3 quadratic functions?

• Standard form: f(x) = ax²+ bx + c, where a ≠ 0.
• Vertex form:f(x) = a (x - h)² + k, where a ≠ 0 and (h, k) is the vertex of the parabola representing the quadratic function.
• Intercept form: f(x) = a (x - p) (x - q), where a ≠ 0 and (p, 0) and (q, 0) are the x-intercepts of the parabola representing the quadratic function.

Q.5. What is quadratic and not quadratic?

A quadratic is an expression of the form ax² + bx + c, where a, b and c are given numbers and a ≠ 0. The standard form of a quadratic equation is an equation of the form

ax² + bx + c = 0, where a, b and c are given numbers and a ≠ 0.

A non-monic quadratic equation is an equation of the form ax² + bx + c = 0, where and are given numbers, and a ≠ 1 or 0. This is the general case. Thus 2x² + 5x + 3 = 0 is an example of a non-monic quadratic equation.

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