How do you do quadratic equations and functions. please list the steps.?

Question

Example equation:

y= 4x2 + 2x + 2

Ohh the 4x2 is really 4x to the second power.

Tell me the steps to find the vertex, y-intercept, and the axis of symmetry.

Answer 1

y = 4x^2 + 2x + 2

To find the vertex, you must complete the square.
Our goal is to put the function in the form
y = a(x - h)^2 + k

y = 4(x^2 + (1/2)x) + 2
y = 4(x^2 + (1/2)x + (1/16)) + 2 - 4/16
y = 4(x + (1/4))^2 + 2 - 1/4

y = 4[x + (1/4)]^2 + (7/4)

This tells us that the vertex is at (-1/4, 7/4). Remember to take the negation of what's inside the bracket, to get the x-coordinate of the vertex.

To obtain the y-intercept, just make x = 0.

y = 4[0 + (1/4)]^2 + (7/4)
y = 4[1/4]^2 + 7/4
y = 4[1/16] + 7/4
y = 1/4 + 7/4
y = 8/4 = 2

y-intercept at (0, 2)

To find the x-intercepts, make y = 0.

y = 4[x + (1/4)]^2 + (7/4)
0 = 4[x + (1/4)]^2 + (7/4)

-7/4 = 4[x + (1/4)]^2

Which will have no solution (the right hand side is a positive value because it is squared and multiplied by 4, another positive value). Therefore, there are no x-intercepts.

The axis of symmetry is found by taking the x-coordinate of the vertex. x = -1/4

Answer 2

For power, use the symbol ^
so
y= 4x2 + 2x + 2 turns to
y= 4x^2 + 2x + 2

The easiest part to find first is the y-intercept
which is the point where the function hits the y-axis
The y-axis is the graph of x=0
so you set x=0 in your equation and get
y = 4(0)^2 +2(0) +2
and that is y=2 so thats the y=intercept

Now to find the axis of symmetry and vertex, there's a formula
x = -b/(2a), in the equation y = ax^2 +bx + c
this will give us the axis of symmetry
looking at our equation
y= 4x^2 + 2x + 2
b= 2
a=4
so x= -(2)/(2*4) = -2/8 = -1/4
and that's the axis of symmetry

axis at which it is symmetrical also happens to have the vertex so we plug in that value of x in the original to find the y value of the vertex.

so
y= 4(-1/4)^2 + 2(-1/4) + 2
y= 4(-1/16) + (-2/4) + 2
simplify more and solve gives
y= 1/4 -2/4 + 2 = 2- 1/4 = 8/4 - 1/4 = 7/4
so the vertex lies on
(-1/4, 7/4)

Answer 3

Just to add on:

A faster way of finding the y-intercept: it is the last (constant) term of your quadratic equation (provided it is in the form ax^2+bx+c) which in this case is 2

Also, the equation of the axis of symmetry is x= -b/2a, so for this one it's x= -2/(2*2) = -1/4

oh, btw, you can also find the vertex by first finding the x-value for the axis of symmetry and then inserting it into the equation