1. to create your own quadratic you have to just make up any numbers. so off the top of my head:
y = -5x^2 + 10x - 3
2. the equation to find the x value of the vertex uses is the equation -b/2a. You may notice that this is similar to the quadratic formula except for you take away the plus or minus square root part. if you think about it, it makes sense that this would be the vertex because the plus or minus parts are an equal distance away from the center of the parabola. so using the form ax^2 + bx + c we can plug in the a and b:
x = -10/2 (-5)
x = -10/-10
x = 1
so we have x and can plug into equation to get y value:
y = -5 (1)^2 + 10(1) -3
y= -10 + 10 - 3
y = -3
vertex is at point (1, -3)
3. the axis of symmetry is the line that runs straight up and down at the center of a parabola. the center of the parabola is the vertex so the line runs straight up and down through vertex. for a line to run straight up and down it is a constant x value. since the x of the vertex is 1, the line is:
x = 1
4. using quadratic formula :
x = [-b (+-) sqrt (b^2 - 4ac)]/2a
we get:
x = [-10 (+-) sqrt (10^2 - 4 (-5)(-3)]/2 (-5)
x = [-10 (+-) sqrt (100 - 60)]/-10
x= [-10 (+-) sqrt (40)]/-10
x= [-10 (+-) 2* sqrt (10)]/-10
x= [5 (+-) sqrt (10)]/5
that is the answer. You can use a calculator to solve easily.
5. the discriminant is the solution to what is under the squareroot sign. so the square root part was:
sqrt (b^2 - 4ac)
so discriminant is :
b^2 - 4ac
so for this function it is:
10^2 - 4 (-5)(-3)
solving this the discriminant is 40