Quadratic equations and functions

Vertex form of quadratic function

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Quadratic function

A quadratic function is a function that can be written in the form $ y = ax ^ 2 + bx + c $.
E. g. $ y = 5x^2−4x − 12$ Here a = 5, b = 4 and c = −12

The first term (a) is called quadratic. It is never equal to zero. The second term is linear (b), and the third term is constant (c).

The terms b, c can be 1 or 0. $ y = 4x^2 −16 $ or $ y = x^2 + 10x $ are still quadratic functions.

The graph of a quadratic function is always a parabola. In comparison, a graph of every linear function (y = ax + b) is a straight line. The quadratic term a determines how steeply the function will rise (the higher the absolute value of a is, the steeper the parabola rises). If a is positive, the function is convex (U-shaped - opens upward), if negative, the function is concave (U opens downward).

xy-5-4-3-2-1 13412354-1-2-3-42y=3xy=xy=2221x2y=(−1)x2

Linear term b affects the position of the function and c determines at which point the function intersects the y-axis (when we set x=0 we will receive y-coordinate of intersection $ y = a×0^2 + b×0 + c$). The figure below shows functions with different values a, b and c .

xy-5-4-3-2-1 13412354-1-2-3-42y=x +4x+1y=xy=2523x2y=(−3)x +6x−42−1

To determine at which point the quadratic function intersects the x-axis let y=0 ($ax^2+bx+c=0$). Now it is a quadratic equation which can have two roots (blue, green), one (pink), or none (red).

Each parabola has one vertex. The vertex of a parabola is the point where the parabola crosses its axis of symmetry. If the quadratic term a is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “U”-shape. If the quadratic term a is negative, the vertex will be the highest point on the graph. Determining the coordinates of a vertex can be obtained by converting the function to the vertex/standard form : $ y = a(x − h) + k $ where h, k are the coordinates of the parabola vertex. We do this by completing the square using the formulae:

$$(a + b)^2 = a ^ 2 + 2ab + b^2 $$

or

$$ (a−b)^2 = a^2−2ab + b^2 $)$$

E.g. for $x^2 + 4x + 1 $ (blue) we will consider $x^2+4x $ for $a^2+2ab $ and we must calculate b from 2ab, then add $b^2$ (and subtract it at the same time), so rewriting the function to the vertex form will look like this:

$$ x^2 + 4 + 1 = x^2+4x+4−4+ 1 =$$
$$= (x + 2)^2 − 3 $

When $ y = (x + 2)^2−3 $ is compared to $ y = a (x − h) + k $, then the coordinates of the vertex [h, k] can be determined as [−2; −3] .

Another option is to use the formula. Vertex coordinates V [h; k] is calculated as:

$$
V [{−b} / {2a}; c − b ^ 2 / {4a}]
$$

E.g.

$$
y = x^2+4+1
$$ $$
[h; k] =[{−b} / {2a}; c − b ^ 2 / {4a}] =
$$ $$
= [{−4} / 2 ;; ; 1−4 ^ 2/4] = [−2; −3]
$$

The k (y-coordinate of the vertex) can be also calculated as:

$$
h = {−b} / {2a}; k = f (h)
$$

E.g.

$$
y = x^2 + 4 + 1
$$ $$
[h; k] = [{−b}/{2a}; f (h)] =
$$ $$
=[-2; (−2)^2 + 4 × (−2) +1] = [−2; −3]
$$

The line of symmetry of quadratic function is the vertical line x = h

Properties of the quadratic function

The domain (set of all x-values) is all real numbers, the range of values (set of all y-values) depends on the parameters of the function, but always goes from the vertex to infinity towards the positive or negative part of the y-axis.

If a < 0, the value range is (− ∞; y-coordinate of the vertex]; if a> 0, it is [y-coordinate of the vertex; ∞).

The quadratic function is even (symmetric about the y-axis) if the linear term (b) is zero.



Quadratic function

Copy link to this topic. expand learning text

Quadratic function

A quadratic function is a function that can be written in the form $ y = ax ^ 2 + bx + c $.
E. g. $ y = 5x^2−4x − 12$ Here a = 5, b = 4 and c = −12

The first term (a) is called quadratic. It is never equal to zero. The second term is linear (b), and the third term is constant (c).

The terms b, c can be 1 or 0. $ y = 4x^2 −16 $ or $ y = x^2 + 10x $ are still quadratic functions.

The graph of a quadratic function is always a parabola. In comparison, a graph of every linear function (y = ax + b) is a straight line. The quadratic term a determines how steeply the function will rise (the higher the absolute value of a is, the steeper the parabola rises). If a is positive, the function is convex (U-shaped - opens upward), if negative, the function is concave (U opens downward).

xy-5-4-3-2-1 13412354-1-2-3-42y=3xy=xy=2221x2y=(−1)x2

Linear term b affects the position of the function and c determines at which point the function intersects the y-axis (when we set x=0 we will receive y-coordinate of intersection $ y = a×0^2 + b×0 + c$). The figure below shows functions with different values a, b and c .

xy-5-4-3-2-1 13412354-1-2-3-42y=x +4x+1y=xy=2523x2y=(−3)x +6x−42−1

To determine at which point the quadratic function intersects the x-axis let y=0 ($ax^2+bx+c=0$). Now it is a quadratic equation which can have two roots (blue, green), one (pink), or none (red).

Each parabola has one vertex. The vertex of a parabola is the point where the parabola crosses its axis of symmetry. If the quadratic term a is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “U”-shape. If the quadratic term a is negative, the vertex will be the highest point on the graph. Determining the coordinates of a vertex can be obtained by converting the function to the vertex/standard form : $ y = a(x − h) + k $ where h, k are the coordinates of the parabola vertex. We do this by completing the square using the formulae:

$$(a + b)^2 = a ^ 2 + 2ab + b^2 $$

or

$$ (a−b)^2 = a^2−2ab + b^2 $)$$

E.g. for $x^2 + 4x + 1 $ (blue) we will consider $x^2+4x $ for $a^2+2ab $ and we must calculate b from 2ab, then add $b^2$ (and subtract it at the same time), so rewriting the function to the vertex form will look like this:

$$ x^2 + 4 + 1 = x^2+4x+4−4+ 1 =$$
$$= (x + 2)^2 − 3 $

When $ y = (x + 2)^2−3 $ is compared to $ y = a (x − h) + k $, then the coordinates of the vertex [h, k] can be determined as [−2; −3] .

Another option is to use the formula. Vertex coordinates V [h; k] is calculated as:

$$
V [{−b} / {2a}; c − b ^ 2 / {4a}]
$$

E.g.

$$
y = x^2+4+1
$$ $$
[h; k] =[{−b} / {2a}; c − b ^ 2 / {4a}] =
$$ $$
= [{−4} / 2 ;; ; 1−4 ^ 2/4] = [−2; −3]
$$

The k (y-coordinate of the vertex) can be also calculated as:

$$
h = {−b} / {2a}; k = f (h)
$$

E.g.

$$
y = x^2 + 4 + 1
$$ $$
[h; k] = [{−b}/{2a}; f (h)] =
$$ $$
= [-2; (−2)^2 + 4 × (−2) +1] = [−2; −3]
$$

The line of symmetry of quadratic function is the vertical line x = h

Properties of the quadratic function

The domain (set of all x-values) is all real numbers, the range of values (set of all y-values) depends on the parameters of the function, but always goes from the vertex to infinity towards the positive or negative part of the y-axis.

If a < 0, the value range is (− ∞; y-coordinate of the vertex]; if a> 0, it is [y-coordinate of the vertex; ∞).

The quadratic function is even (symmetric about the y-axis) if the linear term (b) is zero.



   
   

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