Graphing Quadratic Functions

The quadratic function is the polynomial function with the highest variable power is 2.
The general form of quadratic function equations is:

f(x) = ax2 + bx + x or
y = ax2 + bx + c; a ≠ 0

a function is always related to the function graph. Likewise with the quadratic function where the graph of the quadratic function is called a satellite dish. To illustrate the graph of the quadratic function, it must be determined at the intersection with the axis of the coordinate and the extreme point (peak point / maximum point. / Minimum point). Where the complete step is
Steps Describe the Graph of Quadratic Functions y = ax2 + bx + c
1.  Determine the direction of the parabolic chart up or down by looking at the value of a
  • If the value of a> 0 then the satellite dish opens up and has a minimum extreme value.
  • If the value of a <0 then the satellite dish opens down and has a maximum extreme value.


2.  Determine the intersection with the coordinate axis.
The intersection of the x axis if y = 0. So
The values ​​of x1 and x2 will be known to use factoring, if it is difficult to get the previous results, check the value first.

  • If D < 0, then the function does not have the roots of quadratic equations so the graph sketch of the quadratic function does not cut the X-axis.
  • If D> 0, then the function has the roots of the quadratic function equation but we have difficulty solving the solution because the number that is factored is a decimal number. Where the values ​​of these roots can be obtained by the abc formulaAfter we get the values ​​x1 and x2, the intersection points of the quadratic function are (x1, 0) and (x2, 0)
  • The intersection with the Y axis if x = 0 because x = 0 then y = c and the intersection point is (0, c).


4.  Determine the position of the graph of the square function of the X axis.
  1. If D > 0 then the graph of the quadratic function intersects the X-axis at two points.
  2. If D = 0, then the graph of the quadratic function alludes to the X-axis at one point
  3. If D <0 then the graph of the quadratic function does not cut the X-axis.

Example 1.

Find the deskriminan value, the intersection point, the extreme point of the quadratic equation of f(x) = x2 - 6x + 5?

Answer:

The intersection of the X-axis is obtained if y = 0, then the form of the quadratic equation becomes
x2 - 6x + 5 = 0

To ensure that the quadratic equation above has a square root, we must look for the discriminant value.

D = b2 – 4ac = (-6) – 4(1)(5) = 36 – 20 = 16

Because the discriminant value is 16 (positive / more than 0) the quadratic equation must have two (square roots) different real and two intersections of the x axis. The intersection of the X axis is obtained from the roots of the quadratic equation,

x2 – 6x + 5 = 0
(x – 1)(x – 5)
x = 1 or x = 5

So, the intersection of the x-axis is (1, 0) and (5, 0).

The intersection of the Y-axis
The intersection with the Y-axis is obtained if the value of x = 0.
y = x2 – 6x + 5
y = (0)2 – 6 (0) + 5 = 5
So, the intersection of the Y axis is (0, 5).

The Extreme Point
The extreme point of the quadratic function f(x) = y = ax2 + bx + c  is


The symmetrical axis is x = 3 and the extreme value is - 4.

After obtaining the intersection of the X axis, the intersection of the Y axis, the extreme point, we can draw a graph of the quadratic function. Where the quadratic f(x) = x2 - 6x + 5 has a intersection of the X axis: and (5, 0), the intersection of the Y axis: (0, 5) and extreme points (3, -4). Draw the points on the Cartesian coordinates as shown below,


Then connect the points with a smooth curve, so that the quadratic function curve f(x) = x2 - 6x + 5 will be obtained as follows,
 


Example 2. 
If the function f(x) = qx2 – (q+2)x – 6 reaches the highest value for x = -1, specify the value of q?
Answer:
x = -1 is a symmetrical axis, then


Example 3. 
Find the extreme point and intersection of the X axis for the square function f(x) = x2 – 20x + 75.

Answer:
If the intersection of X with the condition y = 0, the extreme point for the quardat function y = x2 – 20x + 75 is


The intersection of the X-axis
x2 – 20x + 75 = 0
(x – 5)(x – 15)
X= 5, x = 15

So, the intersection of the x-axis is (5, 0) and (15, 0).

Example 4. 
It is known that f(x) = -x2 + 5x + c, if the ordinate is 6 then the value of c is

Answer:
then the ordinate value of the cusp,

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