“QUADRATIC FUNCTIONS”Definition:

A quadratic function f is a function of the form f(x) = ax2 + bx + c where a, b and c are real numbers and a not equal to zero. The graph of the quadratic function is called a parabola. It is a "U" shaped curve that may open up or down depending on the sign of coefficient a. Examples of quadratic functions f(x) = -2x2 + x - 1 f(x) = x2 + 3x + 2

Deriving Vertex form from the Standard Form of a Quadratic Function and vice versa

The standard form of a parabola's equation is generally expressed: y = ax 2 + bx + c

The role of 'a' If a> 1, the parabola opens upwardsif a< 1, it opens downwards.

The axis of symmetryThe axis of symmetry is the line x = -b/2a

Vertex Form of Equation  

The vertex form of a parabola's equation is generally expressed as : y= a(x-h)2+k (h,k) is the vertex

•If a is positive then the parabola opens upwards like a regular "U". •If a is negative, then the graph opens downwards like an upside down

•If |a| > 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways . •If |a| < 1, the graph of the graph becomes narrower(The effect is the opposite of |a| > 1).

 From Vertex To Standard Form  

Equation in vertex form : y = (x – 1)² To convert equation to standard form simply expand and simplify the binomial square (Refresher on FOIL to multiply binomials)

The Graph of A Quadratic functionRegardless of the format, the graph of a quadratic function is a parabola (as shown above).•If a>0 (or is a positive number), the parabola opens upward. •If a <0 (or is a negative number), the parabola opens downward. •The coefficient a controls the speed of increase (or decrease) of the quadratic function from the vertex, bigger positive a makes the function increase faster and the graph appear more closed•The coefficients b and a together control the axis of symmetry of the parabola (also the x-coordinate of the vertex) which is at .

The vertex of a parabola is the point where the parabola crosses its axis.   If the coefficient of the x2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “U”-shape.  If the coefficient of the x2 term is negative, the vertex will be the highest

point on the graph, the point at the top of the “U”-shape.

The standard equation of a parabola is y = ax2 + bx + c.

But the equation for a parabola can also be written in "vertex form":

y = a(x – h)2 + kIn this equation, the vertex of the

parabola is the point (h, k).

The Vertex

Example: Find the vertex of the parabola.y = 3x2 + 12x – 12 Here, a = 3 and b = 12. So, the x-coordinate of the vertex is:

Substituting in the original equation to get the y-coordinate, we get:y = 3(–2)2 + 12(–2) – 12= –24 So, the vertex of the parabola is at (–2, –24).

THE AXIS OF SYMMETRY

Every parabola has an axis of symmetry which is the line that runs down its 'center'. This line divides the graph into two perfect halves.

In the picture of on the left, the axis of symmetry is the line x =1.

Axis of Symmetry Formula

There are two different formulas that you can use to find the axis of symmetry. One formula works when the parabola's equation is in vertex form and the other works when the parabola's equation is in standard form. •If your equation is in vertex form, then the axis of is •x= h in the general vertex form equation y = (x-h)2 + k •If your equation is in standard form, then the formula for the axis of symmetry is: •x = -b/2a from the general standard form equation y = ax2+bx + c

•THE Y-INTERCEPT

A parabola is a visual representation of a quadratic function. Each parabola contains a y-intercept, the point at which the function crosses the y-axis.How to Find the y-interceptThis article introduces the tools for finding the y-intercept.The graph of a quadratic functionThe equation of a quadratic function

Graphing y = x 2

Data Table for y = x 2 And graph the points, connecting them with a smooth curve:

Graph of y = x 2 The shape of this graph is a parabola.

Note that the parabola does not have a constant slope. In fact, as x increases by 1 , starting with x = 0 , y increases by 1, 3, 5, 7,… . As x decreases by 1 , starting with x = 0 , y again increases by 1, 3, 5, 7,… .

Graphing y = (x - h)2 + k

In the graph of y = x 2 , the point (0, 0) is called the vertex. The vertex is the minimum point in a parabola that opens upward. In a parabola that opens downward, the vertex is the maximum point. We can graph a parabola with a different vertex. Observe the graph of y = x 2 + 3 :

Graph of y = x 2 + 3

The graph is shifted up 3 units from the graph of y = x 2 , and the vertex is (0, 3) .

Observe the graph of y = x 2 - 3 :

Graph of y = x 2 - 3

The graph is shifted down 3 units from the graph of y = x 2 , and the vertex is (0, - 3) . We can also shift the vertex left and right. Observe the graph of y = (x + 3)2 :

Graph of y = (x + 3)2

The graph is shifted left 3 units from the graph of y = x 2 , and the vertex is (- 3, 0) .

Observe the graph of y = (x - 3)2 :

Graph of y = (x - 3)2

The graph is shifted to the right 3 units from the graph of y = x 2 , and the vertex is (3, 0) . In general, the vertex of the graph of y = (x - h)2 + k is (h, k) . For example, the vertex of y = (x - 2)2 + 1 is (2, 1) :

Graph of y = (x - 2)2 + 1

The axis of symmetry is the line which divides the parabola into two symmetrical halves. It is given by the equation x = h . For example, the line of symmetry in the graph of y = (x - 2)2 + 1 is x = 2 :

Graphing y = - a(x - h)2 + k

Sometimes, the coefficient in front of (x - h) is negative. If this is the case, the parabola opens downward. In the graph of y = - a(x - h) + k , the vertex and axis of symmetry are still (h, k) and x = k , but as x increases or decreases by units of 1 starting from the vertex, y decreases by 1a, 3a, 5a, 7a,… . For example, here are the data table and graph for y = - (x - 2)2 + 3 :

ZEROS OF A Q.F.

In your textbook, a quadratic function is full of x's and y's. This article focuses on the practical applications of quadrati The graph of a quadratic function is a parabola. A parabola can cross the x-axis once, twice, or never. These points of intersection are called x-intercepts or zeros.  c functions. In the real world, the x's and y's are replaced with real measures of time, distance, and money. To avoid confusion, this article focuses on zeros and not x-intercepts.

Four Methods of Finding the Zeros Quadratic Formula Factoring Completing the Square Graphing

The Quadratic Formula Explained

Often, the simplest way to solve "ax2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. But sometimes the quadratic is too messy, or it doesn't factor at all, or you just don't feel like factoring. While factoring may not always be successful, the Quadratic Formula can always find the solution.

Here are some examples of how the Quadratic Formula works: Solve x2 + 3x – 4 = 0 This quadratic happens to factor: x2 + 3x – 4 = (x + 4)(x – 1) = 0 ...so I already know that the solutions are x = –4 and x = 1. How would my

solution look in the Quadratic Formula? Using a = 1, b = 3, and c = –4, my solution looks like this:

Then, as expected, the solution is x = –4, x = 1. Suppose you have ax2 + bx + c = y, and you are told to plug zero in for y.

The corresponding x-values are the x-intercepts of the graph. So solving ax2 + bx + c = 0 for x means, among other things, that you are trying to find x-intercepts. Since there were two solutions for x2 + 3x – 4 = 0, there must then be two x-intercepts on the graph. Graphing, we get the curve below:

Factoring Quadratics:

A "quadratic" is a polynomial that looks like "ax2 + bx + c", where "a", "b", and "c" are just numbers. For the easy case of factoring, you will find two numbers that will not only multiply to equal the constant term "c", but also add up to equal "b", the coefficient on the x-term. For instance:•Factor x2 + 5x + 6.

need to find factors of 6 that add up to 5. Since 6 can be written as the product of 2 and 3, and since 2 + 3 = 5, then I'll use 2 and 3. I know from multiplying polynomials that this quadratic is formed from multiplying two factors of the form "(x + m)(x + n)", for some numbers m and n. So I'll draw my parentheses, with an "x" in the front of each:(x       )(x      )Then I'll write in the two numbers that I found above:(x + 2)(x + 3)THEN:x=-2 and x=-3

Completing the Square:     Solving Quadratic functions

Some quadratics are fairly simple to solve because they are of the form "something-with-x squared equals some number", and then you take the square root of both sides. An example would be:(x – 4)2 = 5 x – 4 = ± sqrt(5) x = 4 ± sqrt(5) x = 4 – sqrt(5)  and  x = 4 + sqrt(5)

Determining the Nature of the Roots

Determining the Nature of the RootsIf you have a quadratic equation ax2 + bx + c = 0 , then it is true thatx =[-b (b2 - 4ac)]/2a. That (b2 - 4ac) is known as the discriminant ().If > 0, then the roots are real and distinct. Further, if > 0 and is a perfect square, then the roots are rational.If = 0, there is one solution to the quadratic equation, or two equal roots.If < 0, then the roots not real, incorporating an imaginary

Example: If -5 is a root of the quadratic equation 2x2 + px - 15 = 0 and the quadratic equation p(x2 + x) + k = 0 has equal roots, find the value of k.Solution: Since -5 is a root of the equation 2x2 + px - 15 = 0. Therefore,2(-5)2 - 5p - 15 - 0 = 0=> 50 - 5p - 15 = 0=> 5p = 35=> p = 7.Putting p = 7 in p(x2 + x) + k = 0, we get7x2 + 7x + k = 0This equation will have equal roots, if discriminant = 0=> 49 - 4 x 7 x k = 0=> k = 49/28=> k = 7/4

Formula: Sum & Product of Roots

sum of roots: −b/a product of roots: c/a As you can see from the derivation below, when you are trying to solve aquadratic equations in the form of ax2+bx +c. The sum and product of the roots can be rewritten using the two formulas above.

Example 1 The example below illustrates how this formula applies to the quadratic equation x2 + 5x +6. As you can see the sum of the roots is indeed -b/a and the product of the roots is c/a.

Solving Quadratic Inequalities:

The parabola graph was drawn using a solid line since the inequality was "greater than or equal to".The point (0,0) was tested into the inequality and found to be true.

The point (0,-2) was tested into the inequality and found to be false.

The graph was shaded in the region where the true test point was located. 

ANSWER:  The shaded area (including the solid line of the parabola) contains all of the points that make this inequality true.