Graphing a Quadratic Function?

Question

F(x)=3x^2+4x-1
How do you solve by completing the square to find the vertex and axis of symetry

Answer 1

Given: F(x)=3x^2+4x-1 We want form A [x-b]^2 + c

where ( b,c) will be your vertex. Sign of A determines whether the parabola opens up or down.

F(x)=3 [x^2+4x/3] -1 Now we add a term inside the bracket to complete the square: [half the coefficient of x]^2

which is [2/3]^2

F(x) "modified" =3 [x^2+4x/3 + (2/3)^2] -1 BUT sticking that term INSIDE the bracket added 3 times (2/3)^2 = 4/3 to the original function so we have to subtract it (in order to not change the function):

back to ORIGINAL F(x)=3 [x^2+4x/3 + (2/3)^2] -1 - 4/3

Finally we factor:

F(x)=3 [x - (-2/3)]^2 + (-1 - 4/3)

F(x)=3 [x - (-2/3)]^2 + (- 7/3)

so vertex is (-2/3, -7/3) opens up, line of symmetry is x = -2/3.

Answer 2

don't worry, i learnt this in add. maths :

1. Determine wheter it has a minimum/maximum point. Do this by looking at a. If a>0, then it has a minimum point, if a<0, it has a maximum point. In your question, a =3 which means a>0 and it has a minimum point. If a equation has a minimum point, your graph will be like a smiling curve, if maximum point, then your graph will be like a frowning curve.

2. Next, determine the value of b^2 - 4ac. If b^2 - 4ac>0, then it has two roots. If b^2 - 4ac=0, then it has one root. If b^2 - 4ac<0 it has no roots. for your question , a=3 ,b=4, c= -1.

b^2 - 4ac
=(4)^2 - 4(3)(-1)
=16-4(-3)
=16+12
=28.

28>0 means your b^2 - 4ac >0, therefore your equation has two roots.

3. Next, determine the roots. Set your scientific calculator in EQN mode under Degree 2 (if its a CASIO Scientific Calculator fx-570MS). Then key in 3 for a, 4 for b, and c for -1. You'll get X1=0.215 and X2=-1.549.

4. Then by completing the square determine the minimum point (vertex) coordinate.

f(x)=3x^2+4x-1

Factor out 3. Now if a=1, you don't need to factor out.You'll get something like below. Now, a=1, b=4/3, c=-1/3 because I factored 3 out.

=3[ x^2 + 4/3x - 1/3 ]

Now add + (4/3 / 2)^2 - (4/3 / 2)^2 between the 4/3x and -1/3. I do this because to complete the square you would need to add +(b/2)^2-(b/2)^2 between b and c. Now, b=4/3 and c= -1/3 (only for this here) because I factored out 3.

=3[ x^2 + 4/3x + (4/3 / 2)^2 - (4/3 / 2)^2 -1/3]

Then, take x^2+4/3x+(4/3 /2)^2 and make it like this (see below) : ( x + 4/3)^2. Do this by using this format (x+b)^2. Then, solve the rest in the [ ] in which you will get -19/9.

=3[ (x+4/3)^2-19/9]

Now, multiply 3 , you'll get something like below :

=3( x + 4/3)^2 - 19/3

A coordinate is (x,y). Now you get x by doing this. Take x+4/3 from up there and do this
x+4/3=0
x=-4/3 (this is the axis of symetry) .
To get y, take the last digit from 3( x + 4/3)^2 - 19/3 which is -19/3. There you have it , x=-4/3, y=-19/3. So the minimum coordinate is (-4/3,-19/3). Therefore the axis of symetry is
x= -4/3.

You asked how to find the vertex and axis of symetry but i'll help you draw the graph.

5. Now, you must find the y-intercept. To do this,

Substitute x = -4/3 in the given equation (3x^2+4x-1), we get:

y=3(4/3)^2+4(4/3)-1
y=29/3.
don't be confused, why y is used instead of f(x) , its the same, since you are finding for y-intercept, use y.

Remember that y-intercept coordinate is (0,y), therefore the y-intercept coordinate for your question is (0,29/3).

Now, mark on an empty graph, the roots, x=0.215 ,-1.549 on the graph. Remember its y=0 bcause its on the x-axis. Then mark the mini e- point and the y-intercept. Then , draw the graph. Make sure your graph intersects with all points. The minimum point will be the graph's bottom most part To see the graph, go to http://gotoscience.com/Calculator.html, type 3x^2+4x-1 and the you can see the graph.

Any math problems, click my avatar and e-mail me. i'll help you free of charge. i love maths and i currently help my sister with add. maths. If the explanation above is not clear, i'll explain on paper, scan it and e-mail u. (ask me)

Answer 3

f(x) = 3x^2 + 4x -1
f(x) + 1 = 3x^2+4x
f(x) + 1 +16/36 = 3(x^2+(4/3)x+16/36)
f(x) + 13/9 = 3(x+4/6)^2
f(x) = 3(x+2/3)^2 - (13/9)

Vertex(-2/3, -13/9)
a.o.s. x = -2/3