Quadratic Function

Let two non empty set A and B. A function f is defined as the pairing of every element x in A to exactly one value f(x) in B. The set of numbers x for which a function f is defined, is called the domain of f.  The value of f(x) is called the image of x. The set of all image is called the range of f.

The expression such as f(x) = ax + b, where a and b are real elements, or f(x) = ax^2 + bx + c, where a is non zero, a, b, and c are real elements, are examples of function of x. Having chosen a real value of x, we can get a unique real value of f(x) or y from it.

A Function f(x) = ax + b, where a and b are real elements is called a linear function. It’s a injection function.

A function f(x) = ax^2 + bx + c, where a is non zero, a, b, and c are real elements is called a quadratic function. It’s surjection function.

When you sketch a quadratic function, You set up coordinat system using x- and y- axis, which is called Cartesian coordinat. Choose a scale on each axis. But unfortunately,  you cannot show the whole graph. However small the scale or however large the paper, the graph will eventually  spill over the edge. This because the value of x can be any real number.  The skill is to choose the value of x  between which to draw the graph so that you include all the important features.

All the quadratic functions have the same general shape, which is called a parabola. These parabola have a vertical axis of symetry.  The point where a parabola meets its axis of symetry is called the vertex.

You can predict  the equation of a quadratic function from its graph. Let f(x) = ax^2 + bx + c is a quadratic function.

(1)       The value of a

If the vertex is at the lowest point of the graph, then the value of a is positive. If the vertex is at the highest point, then the value of a is negative.

(2)       The value of b

Changing the value of b moves the axis of symetry of the graph in the x- direction.

If the axis of symetry is to the right of the y-axis, then the value of a and b have opposite signs.

If the axis of symetry is to the left of the y-axis, then the value of a and b have same sign.

(3)       The value of c

Changing the value of c moves the graph up and down in the y- direction.

If the graph through (0, c) then the value of c > 0.

If  the graph through (0, 0) then the value of c = 0

If the graph through (0, -c) then the value of c < 0

Here is an exercise for you. And its answer_key.

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