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Graphing from Factored Form

What is factored form?

Factors are numbers multiplied together, to give you the answer. So, factored form are two terms multiplied together to give you (y).

ex. y=(x-3)(x+1)

ex. y=2(x-3)(x+1)

ex. y=-(x+3)(x+5)

Steps to take in order to graph:

1. Find your zeros

  • set the x-intercepts to 0

  • ex. your original equation is y=0.5(x+3)(x-9), you would take (x+3) and (x-9) and make it equal to 0.

  • you do this in order to isolate the x (so you can find what your zeros are)

  • so, x-9=0    and      x+3=0

   x=9                   x=-3

  (9,0)                   (-3,0)

(add 9 to             (subtract 3

both sides,            to both

to cancel               sides)

the 9 out

on the left

side, so x

is alone)

 

2. Use your zeros, to find the axis of symmetry/the x-value of the vertex

  • add you zeros and divide them by 2/ x=(r+s)÷2

  • x=(-3+9)÷2

       x=(6)÷2

       x=3

 

3. Plug your axis of symmetry into the original equation, to find the optimal value/the y-value of the vertex

  • y=0.5(x+3)(x-9)

       y=0.5(3+3)(3-9)

       y=0.5(6)(-6)

       y=0.5(-36)

       y=-18

 

4. Graph your vertex and your zeros

  • the vertex is (3,-18)

  • the zeros are (-3,0) and (9,0)

  • Note: the graph will open up because the (a-which is 0.5) is positive

 

 

Equation: y=a(x-r)(x-s)

Graphing from Standard Form

Equation: y=ax +bx+c

2

Steps to take in order to graph:

1. Factor the equation as much as possible and set it to 0

  • ex. the original equation may be y=x +5x+6

   the factored equation (that's set to 0) would be 0=(x+2)(x+3)

 

2. Solve/ find the zeros by setting the x-intercepts to 0

  • (x+2)=0      and      (x+3)=0

        x=-2                       x=-3

       (-2,0)                      (-3,0)

 

3. Use your zeros, to find the axis of symmetry/the x-value of the vertex

  • add you zeros and divide them by 2/plug the values of the original equation in the expression -b divided by 2a

  • x=(-2+-3)÷2

       x=(-5)÷2

       x=-2.5

 

3. Plug your axis of symmetry into the original equation, to find the optimal value/the y-value of the vertex

  • y=x +5x+6

       y=-2.5 +5(-2.5)+6

       y=6.25-12.5+6

       y=-6.25+6

       y=-0.25

 

4. Graph your vertex and your zeros

  • the vertex is (-2.5,-0.25)

  • the zeros are (-2,0) and (-3,0)

  • Note: the graph will open up because the (a-which is 1) is positive

 

 

2

2

2

Graphing from Vertex Form (Finding the Y-intercept and Zeros)

Note: Remember that there are 2 solutions because there are typically 2 zeros (sometimes there can be one or none). Also, there are two possible answers because when sqaure-rooting, you could square-root 2 negatives or 2 postivies

 

Steps to take in order to graph:

 

To find your y-intercept

1. Set your (x) to zero.

2.Then, use bedmas in order to find out what they y-intercept is.

Equation: y=a(x-h) +k

2

To find your x-intercepts/ zeros

3.Set your (y) to zero.

4. Isolate the (x)

-move the (k)

-divide by (a)

-sqaure-root

-move (h)-add/subtract

-solve two solutions (by setting x to zero)

Lastly, graph your x-intercepts/zeros. Connection: Then use your zeros to find the axis of symmetry (add, then divide by 2). Now you have the x-value of your vertex and you already had the y-value (which is the y-intercept). Graph your vertex.

Quadratic Formula

You use the quadratic fromula in order to find the x-intercepts/zeros. You use this formula in the standard form, when you can't factor (ex. common factoring, simple trionomials etc.)

Formula: -b +/-  b -4ac

2

Approximate solutions are written as points. This answer is uaully rounded because it occurs after you have square-rooted and divided.

Exact soultions are the answers before you square-root it and divide (it usually looks like a number +/- by sqaure-root of a number, divided by a number)

Note: if there is a negative within the square-root, it is a mathematical erorr and the there would be no x-intercepts

 

Steps to take in order to graph:

1. Set your equation to zero

2. Write out the formula

3. Plug in the values

4. solve to find your x-intercepts

 

Connection: Once you find your x-intercepts/zeros, you can find your axis of symmetry (by adding the zeros, then divinding them by 2). After this, you can use your axis of symmetry to find the y-intercept/ the y value. Now you would have both zeros and your vertex.

Discriminants

You use the discriminants in order to find how many x-intercepts/zeros you have.

The discriminant is the number inside of the square-root of the quadratic formula.

 

If the disriminant is less than zero (a negative discriminant), you will have no solutions and therefore no x-intercepts.

If the discriminant is more than zero (a positive), you will have two solutions and therefore two x-intercepts.

If the discriminant is zero, you will have one solution and therefore one x-intercept.

 

In order to find your discriminant, you plug in the equation into the quadratic formula and solve. Note: you only have to plug numbers into the terms within the square-root.

Connection: Use the quadratic formula in order to find the discriminant.

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