Writing Quadratic Equations Given Different Information

By: Melissa Light, Devon Moran, Christy Ringdahl and Jess Ward

When given roots of a graph, you will have to use the FOIL method to find the intercept form.

FOIL Method: First Outer

Inner Last

1) (3x)(x) = 3x2

2) (3x)(2) = 6x3) (1)(x) = 1x = x4) (1)(2) = 2Final Answer: 3x2 +7x+2

(6,11), (2,3), (-2,19)

• To solve this, you need to use the standard form of a quadratic equation which is y=ax^2+bx+c.

• Plug in the x and y values for each point, and get three equations.

• With this system of three equations, solve using one of the methods you have previously learned, for example elimination of a variable or substitution. 

The equations for each point would be:1. 11=36a+6b+c2. 3=4a+2b+c3. 19=4a-2b+c

In this problem, eliminate ‘c’ in all three equations and solve to get the a, b, and c values for your final quadratic.

Example: Find the y intercept of the graph of the following quadratic functions. F1. (x) = x2 + 2x - 32. h(x) = -x2 + 4x + 4

Solution: 1.f(0) = -3. The graph of f has a y intercept at (0,-3).  2. h(0) = 4. The graph of h has a y intercept at (0,4).

y=ax^2+bx+c

a=leading coefficient, determines width of parabola and whether parabola opens up or down

b=linear coefficient, determines whether the slope is negative or positive

c=constant, determines how much the graph will be shifted up or down

When given three points on a graph you can use those to write the standard form of a quadratic equation.

When given the points (-1,1), (-3,1), and (-4, 4) on a graph you can use them to write the standard form by doing these steps.

Substitute the values of the points into the standard form equation, ax^2+bx+c.

1=1a-1b+c

1=9a-3b+c

4=16a-4b+c

Then you can put these numbers into a matrix to find what the values of the coefficients are.

When putting these in a matrix the matrix must be 3x4. The top row should be [1 -1 1 1], the middle row should be [9 -3 1 1] and the bottom row should be [16 -4 1 4]. In place of the C there is a 1. And the number to the left of the equal sign must be moved to the end of the equation before being put in the matrix. When those numbers are put into the matrix then use the rref( button to solve for the coefficients. After you use that button the matrix should read

[1 0 0 1]

[0 1 0 4]

[0 0 1 4]

In the first row the first 1 represents the A coefficient and the second 1 is the value of the A coefficient. In the second row the 1 represents the b coefficient and the 4 represents the value of the b coefficient. And finally in the third row the 1 represents the C coefficient and the 4 represents the value of it. So the final equation is y=1x^2+4x+4.

When given a quadratic function in intercept form you can write it in standard form by using the FOIL method.

y=-(x+4)(x-9) intercept formy=-(x^2-9x+4x-36) multiply the terms by

each other by using the FOIL method (first, inner, outer, last)

y=-(x^2-5x-36) combine like termsy=-x^2+5x+36 use the distributive

property

When given a quadratic function in vertex form you can write it in standard form.

y=3(x-1)^2+8 vertex formy=3(x-1)(x-1)+8 rewrite (x-1)^2y=3(x^2-x-x+1)+8 multiply using FOIL

methody=3(x^2-2x+1)+8 combine like termsy=3x^2-6x+3+8 use distributive propertyy=3x^2-6x+11combine like terms

y=a(x-p)(x-q)

p,q =p and q are the x-interceptsThe axis of symmetry is halfway between

(p,0) and (q,0).

When you are given the x-intercepts and another given point on the graph of an equation you can use the points to write the intercept form of the equation.

The x-intercepts are (-2,0) and (3,0). The other point is (-1,2)

y=a(x+2)(x-3) substitute values for p & q2=a(-1+2)(-1-3) substite values for other

point2=-4a simplify coefficient of “a”-1/2=a divide both sides by -4y=-1/2(x+2)(x-3)