The graph of y = ax 2 + bx + c is a parabola that opens up and has a vertex at (-2, 5). what is the solution set of the related equation 0 = ax 2 + bx + c?

a nonlinear function that can be written on the standard form

ax2+bx+c,wherea≠0ax2+bx+c,wherea≠0

is called a quadratic function.

all quadratic functions has a u-shaped graph called a parabola. the parent quadratic function is

y=x2y=x2

the lowest or the highest point on a parabola is called the vertex. the vertex has the x-coordinate

x=−b2ax=−b2a

the y-coordinate of the vertex is the maximum or minimum value of the function.

a > 0                                    parabola opens up                                      minimum value

a < 0                                      parabola opens down                          maximum value

a rule of thumb reminds us that when we have a positive symbol before  x2we get a happy expression on the graph and a negative symbol renders a sad expression.

the vertical line that passes through the vertex and divides the parabola in two is called the axis of symmetry. the axis of symmetry has the equation

x=−b2ax=−b2a

the y-intercept of the equation is c.

when you want to graph a quadratic function you begin by making a table of values for some values of your function and then plot those values in a coordinate plane and draw a smooth curve through the points.

example

graph

x=x2+2x+1x=x2+2x+1

make a table of value for some values of x. use both positive and negative values!

xy = x2  + 2x + 1-34-21-10011429316

graph the points and draw a smooth line through the points and extend it in both directions

notice that we have a minimum point which was indicated by a positive a value (a = 1). the vertex has the coordinates (-1, 0)  which is what you will get if you use the formula for the x-coordinate of the vertex

x=−b2a=−22⋅1=−1x=−b2a=−22⋅1=−1

and that the line has an y-intercept of (0, 1) which could have been determined from the c-value which is 1.

if you have an absolute value of a that is greater than 1 the parabola will be narrower than the parental quadratic function. and the opposite that if you have a absolute value of a that is less than 1 then the parabola will be wider than the parental quadratic function.

here  you can get a visual of your quadratic equations

hope this

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Here are the answers to the ones that i could see.