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Quadratic Functions A quadratic function has the form
f (x) = ax2 + bx + c
Where a, b and c are real numbers and a is not equal to 0.
The domain of this function is the set of all real numbers. The y intercept of the graph of f is given by f(0) = c. The x intercepts are found by solving the equation
ax2 + bx + c = 0
There are several methods to solve this equation. One of these methods is the use of the quadratic formulas. The two solutions are given by
x1 = ( - b + sqrt (D) ) / 2a
x2 = ( - b - sqrt (D) ) / 2a
where D is the discriminant given
D = b2 - 4ac
To find the range of the quadratic
function, we first rewrite it in the form
f (x) = a(x - h)2 + k
- If we expand the square above in f(x) above, we obtain
f (x) = ax2 - 2ahx + ah2 + k
- It is the same function, therefore we need to have
-2ah = b (first equation)
and
ah2 + k = c (second equation)
- Solve the first equation for h to obtain
h = -b / 2a
- Substitute h by -b/2a in the second equation and solve it for k to obtain
k = c - b2 / 4a
- Hence any quadratic function can be written in the form f (x) = a(x - h)2 + k, with h and k given in terms of a, b and c as shown above. The form above leads to very interesting results. A x changes, The term (x - h)2 is either positive or zero. Hence
(x - h)2 >= 0 ( >= means greater that or equal)
- 1 - If a > 0, multiply both sides of the inequality above by a
a(x - h)2 >= 0
- Add k to both sides of the inequality
a(x - h)2 + k >= k
- The left side is the formula of the
function, f (x) = a(x - h)2 + k , hence
f (x) >= k
- The above result tells us that f (x) has a minimum value equal to k. It also tells us that the range of f (x) is given by
[ k , + infinity)
- 2 - If a < 0, multiply both sides of the inequality (x - h)2 >= 0 a and change the inequality symbol.
a(x - h)2 <= 0
- The above result tells us that f (x) has a maximum value equal to k. It also tells us that the range of f (x) is given by
(- infinity , k]
It is also important to note that k = f(h). The graph of a quadratic function is called a parabola and the point with coordinates (h , k) is called the vertex of the parabola which can be a maximum or a minimum point as we have
shown above.
Example 1: f is a quadratic function given by
f (x) = 2x2 + 2 x - 4
- Find the x and y intercepts of the graph of f.
- Find the vertex of the graph of f.
- Find the domain and range of f.
- Sketch the graph of f.
Solution to Example 1
- a - The y intercept is given by
(0 , f(0)) = (0 , -4)
- The x coordinates of the x intercepts are the solutions to
2x2 + 2 x - 4 = 0
- The discriminant D of the above quadratic equation is given by
D = (2)2 - 4(2)(-4)
= 36
- The solutions are
x1 = (-2 + 6) / 4
= 1
x2 = (-2 - 6) / 4
= - 2
- The x intercepts are at the points (1 , 0) and (-2 , 0).
- b - The x coordinate h and the y coordinate k of the
vertex are given by
h = -b / 2a
= -2/4
= -1/2
k = f (h)
= f(-1/2)
= 2(-1/2)2 + 2 (-1/2) - 4
= -9/2
The vertex is at the point (-1/2 , -9/2)
- c - The domain of f (x) is the set of all real numbers.
- Since coefficient a is positive, f has a minimum value equal to k. The range is given by the set of real values in the interval [-9/2 , +infinity).
- d - Find extra points if necessary;
(2 , f(2)) = (2 , 8) and (-3 , f(-3)) = ( -3 , 8 ) as an example.
Plot the vertex (lowest
point), the x and y intercepts and the extra points as shown
below. From the vertex, which is a minimum point, the left and right
sides of the graph of f should rise upward.
Matched Problem :
f is a quadratic function given by
f (x) = x2 - 2 x - 3
- Find the x and y intercepts of the graph of f.
- Find the vertex of the graph of f.
- Find the domain and range of f.
- Sketch the graph of f.
More references and links to graphing, graphs of functions and quadratic functions.
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