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 formula, After 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).

3.  Determinethe symmetrical axis (Xp) and the extreme point (Xy) isobtained from the top point of the graph of the quadratic function that is,

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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