• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Investigating the Quadratic Function

Extracts from this document...


Investigating the Quadratic Function

This investigation is focused on how to solve quadratic functions by putting them into a perfect square.  The basic form of a quadratic function is y = ax² + bx + c. When drawn on a graph these functions create a parabola that opens down or opens up. The ‘a’ in the function refers to the leading coefficient; the value of this number decides wheather the parabola will be negative or positive. Positive parabolas open up whereas negative parabolas open down.

Here are a few graphs that illustrate positive parabolas that shift vertically due to a different value of ‘c’. The scale of all the graphs is 1 on both the y-axis and x-axis.

        A        B        C

y =x²                 y = x² + 3        y = x² -2


        The “vertex” is the co-ordinate where the parabola turns; this is also referred to as the “turning point”. In these three graphs we can see a clear transition of the parabola on the vertical y-axis. If the value being added to the ax² is positive the parabola shifts up, above the origin of the x-axis. Thusly if the value being added to the ax² is negative the parabola shifts down under the origin of the x-axis.

...read more.



A noticeable pattern is seen as the vertex of these parabolas shift from left to right on the x-axis. Now that the value being added is directly linked to the x value the position of the vertex changes on the x-axis, as this is the value being changed. If a value is added to ‘x’ the parabola moves to the left and the value on the x-axis is negative. To prove this I am going to use the function in graph F:

 The graph shows the x value to be –3 whereas the value being added to x is +3.

I am simply going to work out the equation by moving the numbers to the other side to leave x alone- y = (x + 3)²

        y = x² + 9         

         √-9 = √ x²                

         -3 = x

 So we see that positive values added to x will always end up negative therefore the opposite, all negative values, will show positive values on the x-axis for the parabola.

If the form ‘y = (x + b)²’ is modified to ‘y = a(x + b)² + c’ this is now “turning point form”. We already established that adding values to x² will move the vertex of a parabola up and down the y-axis.

...read more.


So to get the basic shape of a parabola first find the vertex by moving ‘h’ to the right and ‘g’ up, then substitute ‘0’ for ‘x’ and work out what ‘y’ will be to get the y-intercept. Once the y-intercept is found a curve can be drawn from the vertex to the y-intercept and since the line is a reflection through the line of the vertex the other side of the parabola can be draw as well.

        Some of the rules that apply to all the forms discussed in this investigation also apply to other forms. To translate a linier function on the y-axis you add or subtract a value directly to x just like mentioned before. The slope will stay the same but the interception of the y-axis will be different as the value implied on x changes.

y = 2x

y = 2x + 6


Observe how the blue line shifts its y- intercept as 6 is added to the equation

If x is raised to any even number the parabola will observe the exact same rules as x².

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. Marked by a teacher

    The Gradient Function

    5 star(s)

    80 3 81 405 4 256 1280 Points Gradient 1,1 5 2,32 80 3,243 405 4,(1024) 1280 The general pattern here between the two values of x and the gradient, the value of the gradient function here is 5x4, as seen when comparing the two smaller tables above.

  2. Marked by a teacher

    Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    3 star(s)

    However, the process of OLS is a sampling method, the coefficient calculates will not be precise value. Thus, additional test is required to calculate the estimated value of the coefficient relative to its error. This test known as t-test which is very useful, the equation is : t=�/s.e.(�).

  1. The open box problem

    X 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 V 242 244.944 247.192 248.768 249.696 250 249.704 248.832 247.408 245.456 243 This table now suggests that the maximum volume is 250 and that x is 2.5. To prove this again I will construct two graphs.

  2. Triminoes Investigation

    I'm doing this because to find what "c" is worth when substituting a = 2, b = 3 into equation 6. 19a + 5b + c = 10 19 x 1/6 + 5 x 1 + c = 10 19/6 + 5 + c = 10 3 + 1/6 +

  1. Investigation into combined transoformations of 6 trigonometric functions

    The d value changes the y value of the co-ordinates; the value seems to rotate the graph around the point where the graph passes through the x-axis. From this I conclude the graph is translated first then is rotated. The fourth combined transformation I'm going to be looking at will

  2. Investigation of the Phi Function

    1, 5, 7, 11, 13, 17, 19 and 23, which are not divisible by 2 or 3, so all of them are co-prime with 24. b) i ?(18) = {1, 5, 7, 11, 13, 17} = 6 ii ?(41) = 40, because it is a prime number, so it has

  1. Math Portfolio Type II - Applications of Sinusoidal Functions

    1 305 6.87h December 1 335 7.50h Toronto Location: 44?N 79?W Date Day Time of Sunset January 1 1 16.80h February 1 32 17.43h March 1 60 18.07h April 1 91 18.72h May 1 121 19.30h June 1 152 19.85h July 1 182 20.05h August 1 213 19.68h September 1

  2. Fractals. In order to create a fractal, you will need to be acquainted ...

    and the turtle will draw out the fractal by drawing its trail. To see a Netlogo-based L-system generator, go to: http://ccl.northwestern.edu/netlogo/models/run.cgi?L-SystemFractals.722.481 . Another field in fractals research is the theory of fractal dimensions. We know that the first dimension consists of a line, the second consists of a plane, and the third consists of a cube.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work