Chapter 4 - Functions
Page 9 of 15
|Functions - Quadratic Functions - Contents|
A quadratic function has the form y equals a times x squared, plus b times x, plus c. This is known as the quadratic equation. In the first term the x variable is taken to the power of two using the power function, then multiplied and then added to the second term, according to operator precedence. The letters a, b and c are used here as numeric constants. The quadratic equation has two solutions for x called roots which can be arrived upon by solving the equation or plugging the numeric constants into the general quadratic formula. The general quadratic formula is: negative b, plus or minus the square root of the quantity b squared minus 4 times a times c, all divided by 2 times a. Solving the specific equation by using this formula provide the two roots. These roots of the equation is where the y variable equals zero. If there is a negative number in the square root when using the general quadratic equation the solution requires complex numbers. Mapping a quadratic equation on a Cartesian Plane results in a certain type of curved lines. These special curved lines are geometric conic sections called parabolas. The domain of a quadratic function is the set of all real numbers and the range of the quadratic function depends on the specific equation.
|Functions - Quadratic Functions - Examples|
Quadratic function form:
y = ax² + bx + c
y equals a times x squared, plus b times x, plus c
Factoring a quadratic function into its roots:
If y = x² + 5x + 6
the same equation is y = (x + 3) x (x + 2),
then the roots of the equation are -2 and -3,
since they cause y to be zero.
Using the quadratic equation to find the roots:
If y = x² + 2x - 3,
the roots are found through the formula -2 plus or minus the square root of the quantity 2 squared minus 4 times -3 all divided by 2.
This is equal to -2 plus or minus the square root of 16 all over by two, which is -2 plus or minus 4 all over 2,
(-2 + 4)/2 = 1,
(-2 - 4)/2 = -3
and then the roots of the equation are 1 and -3.
Notice that y = (x - 1) x (x + 3)
is the same as the original equation y = x² + 2x - 3.
|Functions -Sections - Chapters|
|1 - Function Definition||2 - Function Notation||3 - Function Domain|
|4 - Function Range||5 - Composite Functions||6 - Inverse Functions|
|7 - Linear Functions||8 - Power Functions||9 - Quadratic Functions|
|10 - Logarithmic Functions||11 - Exponential Functions||12 - Factorial Functions|
|13 - Limit Functions||14 - Summation Functions||15 - Percentage Functions|
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