A quadratic function is a type of polynomial function of degree 2, which can be written in the general form:
f(x) = ax2 + bx + c
where:
• x is the variable,
• a, b, and c are constants with a ≠ 0 (if a = 0, the function would be linear, not quadratic),
• The highest exponent of x is 2 (hence the term "quadratic").
Quadratic FunctionQuadratic functions are important in various mathematical fields and real-life applications, particularly because their graphs are parabolas. They are commonly used in contexts where parabolic shapes and properties are needed.
Some examples of Quadratic Functions:
- f(x) = 3x2 + 7x + 2
- g(x) = x2 – 2
- h(x) = 9x2 + 5x
General Form of a quadratic function: f(x) = ax2 + bx + c where a ≠ 0. To find the roots of the given quadratic function we apply the quadratic function formula.
x = [-b ± √ (b2 - 4ac)] / 2a
Where,
(b2 - 4ac) is called the Discriminant of the quadratic function, and
a, b, and c are coefficients in the quadratic function.
Learn More: Quadratic Formula
Key terms in Quadratic Function
Some of the key features of Quadratic Functions are Vertex, Axis of Symmetry, Domain, and Range, Maximum or Minimum Value.
Let's learn about these features in detail as follows:
Vertex
The vertex is the point where the axis of symmetry intersects the parabola. It represents the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards.
- Coordinates of the vertex = \big( \,\frac{-b}{2a}, f \big( \frac{-b}{2a} \big) \big )
- Direction of the parable:
- If a > 0, the parabola opens upwards.
- If a < 0, the parabola opens downwards.
Read More: Vertex Form of Quadratic Equation
Axis of Symmetry
The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two symmetric halves.
- Equation of the axis of symmetry: x = \frac{-b}{2a}
Learn More: Axis of Symmetry of a Parabola
Domain and Range
- Domain: The set of all real values for x, so the domain of a quadratic function is (−∞, ∞).
- Range: The set of possible values for f(x), or the y-values, which depend on the vertex and the direction of the parabola:
- If a > 0, the range is\left[f\left(\frac{-b}{2a}\right), \infty\right).
- If a < 0, the range is(-\infty, f\left(\frac{-b}{2a}\right)].
Maximum or Minimum Value
The maximum or minimum value of a quadratic function occurs at the vertex. The sign of the leading coefficient aaa determines whether the value is a maximum or minimum:
- Minimum value: If a > 0, the function has a minimum value at x = \frac{-b}{2a}, with the minimum value beingf\left(\frac{-b}{2a}\right) = \frac{-D}{a}, where D is the discriminant.
- Maximum value: If a < 0, the function has a maximum value at x = \frac{-b}{2a} with the maximum value being f\left(\frac{-b}{2a}\right) = \frac{-D}{a}.
The standard or general form of a quadratic function is given as follows:
f(x) = ax2 + bx + c
Where,
a, b, and c are real numbers and a ≠ 0.
Learn more: Standard form of Quadratic Equation
In the vertex form of quadratic function, the quadratic function is of the form:
f(x) = a(x - h)2 + k
Where a ≠ 0 and (h, k) is the vertex of the parabola that represents quadratic function.
In the intercept form of quadratic function, the quadratic function is of the 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.
Note: For the standard form of quadratic function i.e., f(x) = ax2 + bx + c
Vertex of quadratic function = (h, k) = ((- b / 2a), f (- b / 2a))
Types of Quadratic Functions
There are three types of quadratic functions:
- Univariate quadratic functions
- Bivariate quadratic functions
- Multivariate quadratic functions
Univariate Quadratic Functions
The quadratic function that involves only one variable is called the univariate quadratic function and all the discussion throughout this article involves only this type of quadratic function. Thus, the general form of univariate quadratic function is given as:
f(x) = px2 + qx + r
Where,
x is the variables, and p, q, and r are the coefficients of variables.
Bivariate Quadratic Function
The quadratic function that involves two variables is called the bivariate quadratic function and the general form of bivariate quadratic function is given as:
f(x) = ax2 + by2 + cx + dy + exy + f
Where,
x, and y are variables, and a, b, c, d, e, and f are the coefficients of variables.
Multivariate Quadratic Function
The quadratic function that involves three or more variables is called the multivariate quadratic function. The general form of a multivariate quadratic function with three variables is given as:
f(x) = ax2 + by2 + cz2 + dx + ey + fxy + gyz + hxz + i
Where,
x, y, and z are variables, and a, b, c, d, e, f, g, h, and i are the coefficients of variables.
Graphing Quadratic Function
The graph of the quadratic function is a U-shaped parabola whose direction is either upwards or downwards.
Steps to plot a graph of a quadratic function:
- Find the vertex of the quadratic function.
- Construct the table for different values of x and substitute it to find the value of quadratic function f(x) i.e., y.
- Plot the points in the graph and join them to get a graph for the given quadratic function.
Example: Plot the graph for the quadratic function f(x) = x2 - x - 6.
Solution:
For function, f(x) = x2 - x - 6
Here, a = 1, b = -1, c = -6
Step 1: The vertex of above quadratic function = (-b / a, f(-b/a))
f(-b/a) = f [-(-1)/1] = f(1) = -6
The vertex of above quadratic function = (1, -6)
Step 2: Following is the quadratic function table
Step 3: Plot the graph from above table.
Shifting of Graph
By changing the vertex we can shift the graph in the cartesian plane anywhere, we can make two kinds of shifts by changing the vertex parameters i.e.,
Horizontal Shift
The quadratic function graph of f(x) = (x - h)2 shifts the graph of f(x) = x2 by h units horizontally.
- If h>0, then shift the parabola h units towards the left.
- If h<0, then shift the parabola h units towards the right.
Vertical Shift
The quadratic function graph of f(x) = x2 + k shifts the graph of f(x) = x2 by k units vertically.
- If k > 0, then shift the parabola k units upwards.
- If k < 0, then shift the parabola |k| units downwards.
How to Solve Quadratic Equations
For a quadratic function, f(x) we can form it into any quadratic equation by equating it to any quadratic, linear of constant function i.e., f(x) = g(x) is a quadratic equation if g(x) is a function with at most degree 2. To solve such an equation, we have various methods such as:
- Factorization Method,
- Completing Square Method,
- Quadratic Formula Method.
Learn the details of the solution by reading Quadratic equations
Real and Complex Solutions
In quadratic equation ax2 + bx + c where, a ≠ 0, the discriminant of the equation is given by:
Discriminant (D) = b2 - 4ac
The nature of the roots depends on the discriminant of the quadratic equation.
- If D > 0 then, the roots are real and distinct.
- If D = 0 then, the roots are real and equal.
- If D < 0 then, there are no real roots of the given equation or only complex or imaginary roots exist.
Solved Examples of Quadratic Function
Problem 1: Find the vertex of the quadratic function f(x) = 5(x - 3)2 + 6
Solution:
f(x) = 5(x - 3)2 + 6
The above quadratic function represents the vertex form of the quadratic equation which can be written as:
f(x) = a(x - h)2 + k
where, (h, k) is the vertex of quadratic function.
Here, h = 3 and k = 6
The vertex of the quadratic equation f(x) = 5(x - 3)2 + 6 is (3, 6).
Problem 2: Find the roots of the quadratic function f(x) = x2 + 5x + 6 using the quadratic function formula.
Solution:
f(x) = x2 + 5x + 6
The quadratic equation formula: x = [-b ± √ (b2 - 4ac)] / 2a
Here, a =1, b =5 and c= 6
x = [-5 ± √ (52 - 4 × 1 × 6)] / 2 × 1
x = [-5 ± √ (25 - 24)] / 2
x = [-5 ± √1)] / 2
x = [-5 ± 1)] / 2
x = [-5 + 1)] / 2 or x = [-5 - 1)] / 2
x = -4 / 2 or x = -6 / 2
x = -2 or x = -3
Problem 3: Convert the quadratic function f(x) = (x - 4) (x + 5) in standard form.
Solution:
f(x) = (x - 4) (x + 5)
Multiplying both brackets
f(x) = x(x + 5) - 4(x + 5)
f(x) = x2 + 5x - 4x - 20
f(x) = x2 + x - 20
The quadratic function f(x) = (x - 4) (x + 5) in standard form is f(x) = x2 + x - 20
Problem 4: Solve: f(x) = x2 + 4x - 45 using quadratic formula.
Solution:
f(x) = x2 + 4x - 45
The quadratic equation formula: x = [-b ± √ (b2 - 4ac)] / 2a
Here, a =1, b = 4 and c = -45
x = [-4 ± √ (42 - 4 × 1 × -45)] / 2 × 1
x = [-4 ± √ (16 + 180)] / 2
x = [-4 ± √196)] / 2
x = [-4 ± 14)] / 2
x = [-4 + 14)] / 2 or x = [-4 - 14)] / 2
x = -10 / 2 or x = -18 / 2
x = -5 or x = -9
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