A quadratic function is a function whose equation is a second degree polynomial. Thus, if f is a quadratic function, then

f ( x ) = a x² + b x + c

for constants a, b, and c, with a not equal to zero.

Exercise 2.2.1

Why will a not equal zero for quadratic functions?

Solution

The graph of a quadratic function is a parabola.

The parabola opens upwards if a > 0 and opens downward if a < 0.

The graph is symmetric about the vertical line x = - b / ( 2a ).

The vertex of the graph has x coordinate - b / ( 2a ).

Exercise 2.2.2

Find the range of the quadratic function

f ( x ) = 2 x² + 8 x + 1

Solution

The zeros of a function are the input values which result in an output value of zero.

Exercise 2.2.3

What formula did you learn in algebra which allows you to find the zeros of a quadratic function?

Solution

Exercise 2.2.4

Find the zeros of the quadratic function in exercise 2.2.2.

Solution
Theres another way to find the zeros of a quadratic function when the quadratic expression can be factored.

For example, let f ( x ) = x² - x - 6 = ( x - 3 ) ( x + 2 ).

Then, clearly, the zeros are 3 and - 2, since those are the input values which will result in an output value of zero.

Exercise 2.2.5

Find the zeros of f ( x ) = - x² + 4 x + 5

Solution

Exercise 2.2.6

Sketch the graph of the function in exercise 2.2.5. Specify the coordinates of the vertex and the x and y intercepts.

Solution

There is an alternate way to write the equation of a quadratic function using the technique of completing the square which you learned in algebra.

For example, let f ( x ) = - x² + 4 x + 5

First, subtract the 5 from both sides, then divide both sides by the coefficient of x² to get

[ f ( x ) - 5 ] / ( - 1 ) = x² - 4 x

Next, add the square of half the coefficient of x to each side of the equation to get

[ f ( x ) - 5 ] / ( - 1 ) + 4 = x² - 4 x + 4

The expression on the right is now a perfect square--the square of ( x - 2 ).

[ f ( x ) - 5 ] / ( - 1 ) + 4 = ( x - 2 )²

Next, subtract the 4 from each side and multiply both sides by - 1 to get

f ( x ) - 5 = - ( x - 2 )² + 4

Finally, add 5 to both sides to get

f ( x ) = - ( x - 2 )² + 9

The advantage of this form of the function is that we can now see that the graph of f is just the graph of y = - x² shifted 2 units to the right and 9 units upwards. We can also see that the vertex of the graph is at the point (2,9).

The process of completing the square can be used to re-write a quadratic function in the form

f ( x ) = a ( x - h )² + k

where ( h,k ) is the vertex and the graph is just the graph of y = a x² shifted h units horizontally and k units vertically.

Exercise 2.2.7

Complete the square on the function and sketch the graph:

f ( x ) = 2 x² - 4 x + 3

Solution

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