General Form of quadratic function:
In general the quadratic function is of the form f(x) = ax^2 + bx + c,
Where a, b, and c are numbers and a not equal to zero.
The graph of a quadratic function is a curve and it is called as a parabola. Parabolas may open upward or downward and different length in "width" or "steepness", but they all have the same basic "U" shape. For example, y = 2x^2is a quadratic function since we have the x-squared term. y = x^2+1/x would not be a quadratic function because the 1/x term is x-1 which does not valid the form.
Sketching Parabola
To sketch the graph of the quadratic equation, following steps are used :
(i)Verify if a > 0 or a < 0 to decide if it is opened upwards or downwards respectively.
(ii)If a>0, the quadratic function has a minimum value, and if a<0 a="" br="" function="" has="" maximum="" quadratic="" the="" value.="">
(iii)The x-coordinate of the minimum or maximum point is given by
X=-b/2a
We need to substitute x-value into our quadratic function Then we will have the (x, y) coordinates of the minimum or maximum point. This is called the vertex of the parabola.
(iv)To find the he coordinates of the y-intercept, substitute x = 0. This is always simple to find!
(v)We substitute y = 0 to find the values of y-intercepts in the quadratic equation.
Example for Quadratic functions
Sketch the graph of the function y = 2x^2 ? 8x + 6
Solution:
We first identify that a = 2, b = -8 and c = 6.
Step (i) Since a = 2, a > 0, it is a parabola with a minimum point and opens upwards
Step (ii) The x co-ordinate of the minimum point is
X= -b /2a = -8/ 2(2)= 2
y value of the minimum point is
y = 2(2)2 - 8(2) + 6 = -2
Hence the minimum point is (2, -2)
Step (iii) the y-intercept is found by substituting x = 0
y = 2(0)2 - 8(0) + 6 = 6
Hence (0, 6) is the y-intercept.
Step (iv) The x-intercepts are found by setting y = 0
2(x^2 - 4x + 3) = 0
2(x - 1)(x - 3) = 0
Hence x = 1, or x = 3.
In general the quadratic function is of the form f(x) = ax^2 + bx + c,
Where a, b, and c are numbers and a not equal to zero.
The graph of a quadratic function is a curve and it is called as a parabola. Parabolas may open upward or downward and different length in "width" or "steepness", but they all have the same basic "U" shape. For example, y = 2x^2is a quadratic function since we have the x-squared term. y = x^2+1/x would not be a quadratic function because the 1/x term is x-1 which does not valid the form.
Sketching Parabola
To sketch the graph of the quadratic equation, following steps are used :
(i)Verify if a > 0 or a < 0 to decide if it is opened upwards or downwards respectively.
(ii)If a>0, the quadratic function has a minimum value, and if a<0 a="" br="" function="" has="" maximum="" quadratic="" the="" value.="">
(iii)The x-coordinate of the minimum or maximum point is given by
X=-b/2a
We need to substitute x-value into our quadratic function Then we will have the (x, y) coordinates of the minimum or maximum point. This is called the vertex of the parabola.
(iv)To find the he coordinates of the y-intercept, substitute x = 0. This is always simple to find!
(v)We substitute y = 0 to find the values of y-intercepts in the quadratic equation.
Example for Quadratic functions
Sketch the graph of the function y = 2x^2 ? 8x + 6
Solution:
We first identify that a = 2, b = -8 and c = 6.
Step (i) Since a = 2, a > 0, it is a parabola with a minimum point and opens upwards
Step (ii) The x co-ordinate of the minimum point is
X= -b /2a = -8/ 2(2)= 2
y value of the minimum point is
y = 2(2)2 - 8(2) + 6 = -2
Hence the minimum point is (2, -2)
Step (iii) the y-intercept is found by substituting x = 0
y = 2(0)2 - 8(0) + 6 = 6
Hence (0, 6) is the y-intercept.
Step (iv) The x-intercepts are found by setting y = 0
2(x^2 - 4x + 3) = 0
2(x - 1)(x - 3) = 0
Hence x = 1, or x = 3.
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